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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Loop spaces fibrations\n",
+    "\n",
+    "## The canonical loop space fibration\n",
+    "\n",
+    "Let $X$ be a  simplicial set. The software `Kenzo` implements the *canonical* twisted cartesian product of $X$ by its loop space $GX$, denoted $X \\times_\\tau GX$, giving the Kan (**Daniel M. Kan**. *A combinatorial definition of homotopy groups*, Ann. of Math., 1958, vol. 67, pp 282-312.) model of the contractible path space of $X$. The *canonical fibration*\n",
+    "$$ GX\\quad \\hookrightarrow\\quad X \\times_\\tau GX  \\hskip 1.2em \\longrightarrow \\hskip -1.2em \\longrightarrow  \n",
+    "   \\quad X,$$\n",
+    "is defined by the trivial application $\\tau$, where $\\tau(x)$ is the *letter* $x$ in $GX$. This twisted cartesian product has the same simplices as the non-twisted one. So both associated chain complexes are equal as graded modules. The essential difference  resides in the differential morphism. More precisely, the face operator $\\partial_n$ behaves differently in the twisted cartesian product. Let $(x,g) \\in X \\times_\\tau GX$, the rule for the face operators are:\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "\\partial_i(x,g) & =  & (\\partial_i x, \\partial_i g), \\qquad i<n, \\\\\n",
+    "\\partial_n(x,g) & =  & (\\partial_n x, \\tau(x).\\partial_n g), \\qquad i=n,\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "where $\\tau(x)$ is the *letter* $x$ in $GX$ and '**.**' is the group product in $GX$. Let us recall that, from the mathematical definition of $GX$, if $x$ is $n$-degenerate, for instance $x=\\eta_n x'$, the $\\tau(x)$ is the null element of $GX_n$.\n",
+    "The behaviour of $\\partial_n$ induces a change in the differential morphism of the chain complex associated to the twisted cartesian product.\n",
+    "\n",
+    "## The associated tensor twisted product\n",
+    "\n",
+    "We recall that around the cartesian product $X \\times GX$,  the Eilenberg-Zilber theorem allows to build the reduction (see the function `ez`):\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "{{\\cal C}_*(X \\times GX)} \\quad& \\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(X \\times GX)} \\cr\n",
+    " {\\scriptstyle f} \\downarrow \\uparrow {\\scriptstyle g} \\qquad& & \\cr\n",
+    " \\qquad{{\\cal C}_*(X) \\otimes {\\cal C}_*(GX)} & &\\cr\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "On the other hand, the differential morphism $d_\\tau$ of the chain complex ${\\cal C}_*(X \\times_\\tau GX)$ may be considered as a perturbed differential morphism $d$ of the chain complex ${\\cal C}_*(X \\times GX)$, the perturbation $\\delta$ being: $\\delta= d_\\tau - d$. Using the basic perturbation lemma, we may construct a new reduction:\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "{{\\cal C}_*(X \\times_\\tau GX)} \\quad& \\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(X \\times_\\tau GX)} \\cr\n",
+    " {\\scriptstyle f} \\downarrow \\uparrow {\\scriptstyle g} \\qquad& & \\cr\n",
+    " \\qquad{{\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)} & & \\cr\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "In fact, the interesting object produced by this machinery is the bottom chain complex  ${\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)$ which is nothing but the *twisted tensor product* (**Edgar H. Brown Jr.** *Twisted tensor products, I*,\n",
+    "Ann. of Math., 1959, vol. 69, pp. 223-246.).\n",
+    "\n",
+    "## Main functions for the twisted products\n",
+    "\n",
+    "`twisted-crts-prdc` *space* *[Function]*\n",
+    "> From the simplicial set $X$, here the argument *space*,  return  the twisted cartesian product $X \\times_\\tau GX$ (a simplicial set). This function calls internally the functions `loop-space` and `crts-prdc` described in preceding chapters. The slot `grmd` of the resulting instance is set with the ordinary cartesian product $X \\times GX$.\n",
+    "\n",
+    "`dtau-d` *space* *[Function]*\n",
+    "> From the simplicial set $X$, here the argument *space*, build the morphism perturbation (degree -1)\n",
+    "$$d_\\tau-d: X \\times GX \\longrightarrow X \\times GX.$$\n",
+    "The work is essentialy done by the  lisp function `dtau-d-intr` which computes the difference between  the differential morphisms taking in account the discrepancy between the last face operator ($\\partial_n$) in both chain complexes, $X \\times_\\tau GX$ and $X \\times GX$.\n",
+    "\n",
+    "`szczarba` *space* *[Function]*\n",
+    "> Starting from the simplicial set $X$, (the argument *space*), return two values:  \n",
+    "1- The reduction\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "{{\\cal C}_*(X \\times_\\tau GX)}\\quad & \\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(X \\times_\\tau GX)} \\cr\n",
+    " {\\scriptstyle f} \\downarrow \\uparrow {\\scriptstyle g}\\qquad  \\cr\n",
+    " \\quad{{\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)} \\cr\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "obtained by perturbing by $d_\\tau-d$ the top chain complex of the reduction \n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "{{\\cal C}_*(X \\times GX)}\\quad & \\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(X \\times GX)} \\cr\n",
+    " {\\scriptstyle f} \\downarrow \\uparrow {\\scriptstyle g}\\qquad  \\cr\n",
+    " \\quad{{\\cal C}_*(X) \\otimes {\\cal C}_*(GX)} \\cr\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "2- The morphism corresponding to the perturbation of the bottom chain complex induced by the perturbation $d_\\tau-d$ upon the top chain complex of the above reduction.\n",
+    "\n",
+    "This algorithm implements in particular  in a very efficient way the formulas of Szczarba (**R.H. Szczarba.** *The homology of twisted cartesian products.* Transactions of the American Math. Society, 1961, vol. 100, pp. 197-216).\n",
+    "\n",
+    "`twisted-tnsr-prdc` *space* *[Function]*\n",
+    "> Return the twisted tensor product (a chain complex)  $X \\otimes_t GX$. This is done simply by extraction of the bottom chain complex from the reduction  `(szczarba` *space*`)`.\n",
+    "\n",
+    "### Examples\n",
+    "\n",
+    "For testing the functions above, we shall use a \"soft\" version of $\\bar{\\Delta}$ analogous to `soft-delta` or `soft-delta-infinity`. The input of the simplices will be done in clear with the help of the macro `code`. As this version does not exists in `Kenzo`, we have modified some functions of `soft-delta`, as indicated."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "SOFT-DELTAB-CMPR"
+      ]
+     },
+     "execution_count": 1,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(DEFUN SOFT-DELTAB-CMPR (gmsm1 gmsm2)\n",
+    "    (if (= 1 (logcount (cdr gmsm1))) \n",
+    "        :EQUAL\n",
+    "        (f-cmpr (cdr gmsm1)(cdr gmsm2)) ))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "SOFT-DELTAB-BNDR"
+      ]
+     },
+     "execution_count": 2,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(DEFUN SOFT-DELTAB-BNDR (dmns gmsm)\n",
+    "   (declare\n",
+    "      (fixnum dmns)\n",
+    "      (type soft-dlop gmsm))\n",
+    "   (the cmbn\n",
+    "      (if (< dmns 2)\n",
+    "         (zero-cmbn (1- dmns))\n",
+    "         (make-cmbn\n",
+    "         :degr (1- dmns)\n",
+    "         :list (mapcar #'(lambda (term)\n",
+    "                         (with-term (cffc gmsm) term\n",
+    "                            (term cffc (d gmsm))))\n",
+    "               (cmbn-list (delta-bndr dmns (cdr gmsm))))))))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"undefined ~(~A~): ~S\" {1005353B63}>\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "SOFT-DELTAB"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(DEFUN SOFT-DELTAB ()\n",
+    "   (the simplicial-set\n",
+    "      (build-smst\n",
+    "         :cmpr #'soft-deltab-cmpr\n",
+    "         :basis :locally-effective\n",
+    "         :bspn (d 1)\n",
+    "         :face #'soft-delta-face\n",
+    ";;;      :intr-dgnl #'soft-delta-dgnl :dgnl-strt :gnrt\n",
+    "         :intr-bndr #'soft-deltab-bndr :bndr-strt :gnrt\n",
+    "         :orgn `(soft-deltab))))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "CODE"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(defmacro code (arg) `(d (dlop-ext-int ,arg)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K1 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def db (soft-deltab))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 6,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K6 Simplicial-Group]"
+      ]
+     },
+     "execution_count": 6,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def gdb (loop-space db))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Let us compare the faces of a cartesian product in $\\bar{\\Delta} \\times \\Omega(\\bar{\\Delta})$ and the faces of the same object in $\\bar{\\Delta} \\times_\\tau \\Omega(\\bar{\\Delta})$:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K18 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def crts-pr (crts-prdc db gdb))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 8,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K23 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 8,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def tw-crts-pr (twisted-crts-prdc db))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 9,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<CrPr - (DELT . 31) - <<Loop[(DELT . 63)]>>>"
+      ]
+     },
+     "execution_count": 9,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def smpx (crpr 0 (code '(0 1 2 3 4)) \n",
+    "                 0 (loop3 0 (code '(0 1 2 3 4 5)) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 10,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - (DELT . 30) - <<Loop[(DELT . 62)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 29) - <<Loop[(DELT . 61)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 27) - <<Loop[(DELT . 59)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 23) - <<Loop[(DELT . 55)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 15) - <<Loop[(DELT . 31)\\-1][(DELT . 47)]>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 10,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (j 5)\n",
+    "         (print (face crts-pr j 4 smpx)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 11,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - (DELT . 30) - <<Loop[(DELT . 62)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 29) - <<Loop[(DELT . 61)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 27) - <<Loop[(DELT . 59)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 23) - <<Loop[(DELT . 55)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 15) - <<Loop[(DELT . 47)]>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 11,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (j 5)\n",
+    "         (print (face tw-crts-pr j 4 smpx)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 12,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<CrPr - (DELT . 62) - <<Loop[(DELT . 2016)]>>>"
+      ]
+     },
+     "execution_count": 12,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def smpx2 (crpr 0 (code '(1 2 3 4 5)) \n",
+    "                  0 (loop3 0 (code '(5 6 7 8 9 10)) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 13,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - (DELT . 60) - <<Loop[(DELT . 1984)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 58) - <<Loop[(DELT . 1952)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 54) - <<Loop[(DELT . 1888)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 46) - <<Loop[(DELT . 1760)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 30) - <<Loop[(DELT . 992)\\-1][(DELT . 1504)]>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 13,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (j 5)\n",
+    "         (print (face crts-pr j 4 smpx2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 14,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - (DELT . 60) - <<Loop[(DELT . 1984)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 58) - <<Loop[(DELT . 1952)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 54) - <<Loop[(DELT . 1888)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 46) - <<Loop[(DELT . 1760)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 30) - <<Loop[(DELT . 62)][(DELT . 992)\\-1][(DELT . 1504)]>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 14,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (j 5)\n",
+    "         (print (face tw-crts-pr j 4 smpx2)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The user will have noted that we may obtain the same twisted cartesian product by the fibration theory. Let us define a general function `loop-fbr` with  a reduced simplicial set, *space*, as argument. This function uses the function `fibration-total` that we have seen in the  chapter about fibrations. Here, we see that the lisp function for the simplicial morphism, (keyword `:sintr`), implements straightforwardly the canonical twisting operator which applies a simplex of $X$ to the corresponding 1-letter word of $GX$. Note that the execution time by the fibration function is a little longer than for the direct twisted cartesian product and it requires more memory."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 15,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"The variable ~S is defined but never used.\" {1005C2D303}>\n",
+      "\n",
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"undefined ~(~A~): ~S\" {1005C5C533}>\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "LOOP-FBR"
+      ]
+     },
+     "execution_count": 15,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(defun loop-fbr (space)\n",
+    "             (fibration-total\n",
+    "                (build-smmr\n",
+    "                  :sorc space \n",
+    "                  :trgt (loop-space space) \n",
+    "                  :degr -1\n",
+    "                  :sintr #'(lambda (dmns gmsm)\n",
+    "                               (absm 0 (loop3 0 gmsm 1)))\n",
+    "                  :orgn `(total-fibration ,space))))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 16,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K29 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 16,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def fb-tt-db (loop-fbr db))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 17,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - (DELT . 60) - <<Loop[(DELT . 1984)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 58) - <<Loop[(DELT . 1952)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 54) - <<Loop[(DELT . 1888)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 46) - <<Loop[(DELT . 1760)]>>>> \n",
+      "<AbSm - <CrPr - (DELT . 30) - <<Loop[(DELT . 62)][(DELT . 992)\\-1][(DELT . 1504)]>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 17,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (j 5)\n",
+    "         (print(face fb-tt-db j 4 smpx2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 18,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 18,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(time (dotimes (j 1000) (face tw-crts-pr 4 4 smpx2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 19,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 19,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(time (dotimes (j 1000) (face fb-tt-db 4 4 smpx2)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The four following statements show the relationship between the differentials:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 20,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K34 Morphism (degree -1): K18 -> K18]"
+      ]
+     },
+     "execution_count": 20,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def perturb (dtau-d db))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 21,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <CrPr - (DELT . 30) - <<Loop[(DELT . 62)][(DELT . 992)\\-1][(DELT . 1504)]>>>>\n",
+       "<-1 * <CrPr - (DELT . 30) - <<Loop[(DELT . 992)\\-1][(DELT . 1504)]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 21,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? perturb 4 smpx2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 22,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <CrPr - (DELT . 30) - <<Loop[(DELT . 992)\\-1][(DELT . 1504)]>>>>\n",
+       "<-1 * <CrPr - (DELT . 46) - <<Loop[(DELT . 1760)]>>>>\n",
+       "<1 * <CrPr - (DELT . 54) - <<Loop[(DELT . 1888)]>>>>\n",
+       "<-1 * <CrPr - (DELT . 58) - <<Loop[(DELT . 1952)]>>>>\n",
+       "<1 * <CrPr - (DELT . 60) - <<Loop[(DELT . 1984)]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 22,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? crts-pr 4 smpx2)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 23,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <CrPr - (DELT . 30) - <<Loop[(DELT . 62)][(DELT . 992)\\-1][(DELT . 1504)]>>>>\n",
+       "<-1 * <CrPr - (DELT . 46) - <<Loop[(DELT . 1760)]>>>>\n",
+       "<1 * <CrPr - (DELT . 54) - <<Loop[(DELT . 1888)]>>>>\n",
+       "<-1 * <CrPr - (DELT . 58) - <<Loop[(DELT . 1952)]>>>>\n",
+       "<1 * <CrPr - (DELT . 60) - <<Loop[(DELT . 1984)]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 23,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? tw-crts-pr 4 smpx2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The same type of tests may be done with another reduced simplicial set, namely $P^3\\mathbb{R}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 24,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K35 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 24,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def pri3 (R-proj-space 3))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 25,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K57 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 25,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def tw-pri3 (twisted-crts-prdc pri3))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 26,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       ":LOCALLY-EFFECTIVE"
+      ]
+     },
+     "execution_count": 26,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(basis tw-pri3)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 27,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K52 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 27,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def crt-pri3 (crts-prdc pri3 (loop-space pri3)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 28,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<CrPr - 4 - <<Loop[5]>>>"
+      ]
+     },
+     "execution_count": 28,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def s (crpr 0 4 0 (loop3 0 5 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 29,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - 3 - <<Loop[4]>>>> \n",
+      "<AbSm 0 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm 1 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm 2 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm - <CrPr - 3 - <<Loop[4\\-1]>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 29,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (i 5) (print (face crt-pri3 i 4 s)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 30,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - 3 - <<Loop[4]>>>> \n",
+      "<AbSm 0 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm 1 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm 2 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm - <CrPr - 3 2-1-0 <<Loop>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 30,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (i 5) (print (face tw-pri3 i 4 s)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 31,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K63 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 31,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def fb-tt-pri3 (loop-fbr pri3))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 32,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "<AbSm - <CrPr - 3 - <<Loop[4]>>>> \n",
+      "<AbSm 0 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm 1 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm 2 <CrPr - 2 - <<Loop[3]>>>> \n",
+      "<AbSm - <CrPr - 3 2-1-0 <<Loop>>>> "
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 32,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(dotimes (j 5) (print (face fb-tt-pri3 j 4 s)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Let us check now the Szczarba reduction, with the simplicial set $\\bar{\\Delta}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 33,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K90 Reduction K23 => K88]"
+      ]
+     },
+     "execution_count": 33,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def szc-reduc (szczarba db))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 34,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "---done---"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 34,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(pre-check-rdct szc-reduc)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 35,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<1 * <CrPr - (DELT . 7) - <<Loop[(DELT . 480)]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 35,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(setf *tc* (cmbn 2 1 (crpr 0 (code '(0 1 2)) \n",
+    "                           0 (loop3 0 (code '(5 6 7 8)) 1))))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 36,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<1 * <TnPr (DELT . 1) <<Loop[(DELT . 480)]>>>>\n",
+       "<10 * <TnPr (DELT . 3072) <<Loop[(DELT . 7)]>>>>\n",
+       "<100 * <TnPr (DELT . 7) <<Loop[(DELT . 3072)]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 36,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(setf *bc* (cmbn 2 1 (tnpr 0 (code '(0)) 2 (loop3 0 (code '(5 6 7 8)) 1))\n",
+    "                  10 (tnpr 1 (code '(10 11)) 1 (loop3 0 (code '(0 1 2)) 1))\n",
+    "                 100 (tnpr 2 (code '(0 1 2)) 0 (loop3 0 (code '(10 11)) 1))))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 37,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "---done---"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 37,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(pre-check-rdct szc-reduc)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 38,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "*TC* => \n",
+      "----------------------------------------------------------------------{CMBN 2}\n",
+      "<1 * <CrPr - (DELT . 7) - <<Loop[(DELT . 480)]>>>>\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "*BC* => \n",
+      "----------------------------------------------------------------------{CMBN 2}\n",
+      "<1 * <TnPr (DELT . 1) <<Loop[(DELT . 480)]>>>>\n",
+      "<10 * <TnPr (DELT . 3072) <<Loop[(DELT . 7)]>>>>\n",
+      "<100 * <TnPr (DELT . 7) <<Loop[(DELT . 3072)]>>>>\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *TDD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 0}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *BDD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 0}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *DF-FD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 1}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *DG-GD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 1}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *ID-FG* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 2}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *ID-GF-DH-HD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 2}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *HH* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 4}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *FH* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 3}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *HG* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 3}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "---done---"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 38,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(check-rdct-no-wait)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Exercice with the twisting cochain.\n",
+    "\n",
+    "Let $B$ be a reduced simplicial set and its loop space $GB$ (a simplicial group). In the following examples, we shall use the efficient `Kenzo` version of the simplicial set $\\bar{\\Delta}$, i.e. the simplices are input and printed under their binary form coding.\n",
+    "\n",
+    "### The twisting cochain.\n",
+    "\n",
+    "The twisting cochain $t$ is the morphism\n",
+    "$$t: aug \\circ d_{\\otimes t} \\circ coaug$$\n",
+    "where,\n",
+    "- $coaug$ is the morphism \n",
+    "$$ coaug: B \\rightarrow B \\otimes GB,$$\n",
+    "$$ coaug(b) = b \\otimes 1_{GB}. $$\n",
+    "- $d_{\\otimes t}$ is the differential morphism of the twisted tensor product $B \\otimes_t GB$.\n",
+    "- $aug$ is the morphism\n",
+    "$$aug: B \\otimes GB  \\longrightarrow GB ,  $$\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "aug(b_i \\otimes g_j)&  = g_j, & b_i \\in B_0, \\\\\n",
+    "aug(b_i \\otimes g_j)&  = 0, & b_i \\in B_k,\\, k\\not=0.\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "We recall that, as graded modules, $B\\otimes GB$ and $B \\otimes_t GB$ are identical.\n",
+    "\n",
+    "First, we create $\\bar{\\Delta}$ and $G\\bar{\\Delta}$. The function `twisted-tnsr-prdc` extracts the twisted tensor product $\\bar{\\Delta} \\otimes_t G\\bar{\\Delta}$ from the reduction obtained by a call to the function `szczarba`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 39,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K112 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 39,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def d (deltab))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 40,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K117 Simplicial-Group]"
+      ]
+     },
+     "execution_count": 40,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def gd (loop-space d))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 41,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K160 Chain-Complex]"
+      ]
+     },
+     "execution_count": 41,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def d-twtp-gd (twisted-tnsr-prdc d))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Let us define the  coaugmentation and augmentation morphisms. As usual, it is sufficient to define the algorithm on a generator."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 42,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K163 Morphism (degree 0): K112 -> K160]"
+      ]
+     },
+     "execution_count": 42,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def coaug (build-mrph \n",
+    "               :sorc d \n",
+    "               :trgt d-twtp-gd \n",
+    "               :degr 0\n",
+    "               :intr #'(lambda (degr gnrt)\n",
+    "                         (term-cmbn degr 1 (tnpr degr gnrt 0 +null-loop+)))\n",
+    "               :strt :gnrt \n",
+    "               :orgn '(coaug. d --> d-twtp-gd)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 43,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 15 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 43,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? coaug 3 15)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 44,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<1 * <TnPr 31 <<Loop>>>>\n",
+       "<1 * <TnPr 62 <<Loop>>>>\n",
+       "<1 * <TnPr 124 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 44,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? coaug (cmbn 4 1 31 1 62 1 124))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 45,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"The variable ~S is defined but never used.\" {10070CDB13}>\n",
+      "\n",
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"The variable ~S is defined but never used.\" {10070CE1E3}>\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "[K164 Morphism (degree 0): K160 -> K117]"
+      ]
+     },
+     "execution_count": 45,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def aug (build-mrph\n",
+    "             :sorc d-twtp-gd\n",
+    "             :trgt gd\n",
+    "             :degr 0\n",
+    "             :intr #'(lambda (degr tnpr)\n",
+    "                        (with-tnpr (degr1 gmsm1 degr2 loop2) tnpr\n",
+    "                            (if (zerop degr1)\n",
+    "                                (term-cmbn degr 1 loop2)\n",
+    "                                (zero-cmbn degr))))\n",
+    "             :strt :gnrt\n",
+    "             :orgn '(aug. of d-twtp-gd)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 46,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<9 * <<Loop[15]>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 46,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? aug (cmbn 2 9 (tnpr 0 1 2 (loop3 0 15 1))\n",
+    "               3 (tnpr 1 3 1 (loop3 0 7 1))))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The morphism {\\tt cochain} is simply a composition of morphisms. We apply it upon the simplices `(0)`, `(0 1 2)`, `(0 1 2 3)`, ..., coded as the integers $1,7,15,\\ldots$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 47,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K166 Morphism (degree -1): K112 -> K117]"
+      ]
+     },
+     "execution_count": 47,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def cochain (i-cmps aug (dffr d-twtp-gd) coaug))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 48,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN -1}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 48,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? cochain 0 1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 49,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<1 * <<Loop[0 3][7]>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 49,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? cochain 2 7)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 50,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<1 * <<Loop[1-0 3][0 7][1 13]>>>\n",
+       "<-1 * <<Loop[1-0 3][1 7][15]>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 50,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? cochain 3 15)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 51,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<-1 * <<Loop[2-1-0 3][1-0 7][2-0 13][2-1 25]>>>\n",
+       "<1 * <<Loop[2-1-0 3][1-0 7][2-1 13][2 29]>>>\n",
+       "<1 * <<Loop[2-1-0 3][2-0 7][0 15][2-1 25]>>>\n",
+       "<-1 * <<Loop[2-1-0 3][2-0 7][2-1 13][1 29]>>>\n",
+       "<-1 * <<Loop[2-1-0 3][2-1 7][1 15][2 27]>>>\n",
+       "<1 * <<Loop[2-1-0 3][2-1 7][2 15][31]>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 51,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? cochain 4 31)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 52,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<-1 * <<Loop[3-2-1-0 3][2-1-0 7][3-1-0 13][3-2-0 25][3-2-1 49]>>>\n",
+       "<1 * <<Loop[3-2-1-0 3][2-1-0 7][3-1-0 13][3-2-1 25][3-2 57]>>>\n",
+       "<1 * <<Loop[3-2-1-0 3][2-1-0 7][3-2-0 13][3-0 29][3-2-1 49]>>>\n",
+       "<-1 * <<Loop[3-2-1-0 3][2-1-0 7][3-2-0 13][3-2-1 25][3-1 57]>>>\n",
+       "<-1 * <<Loop[3-2-1-0 3][2-1-0 7][3-2-1 13][3-1 29][3-2 53]>>>\n",
+       "<1 * <<Loop[3-2-1-0 3][2-1-0 7][3-2-1 13][3-2 29][3 61]>>>\n",
+       "<1 * <<Loop[3-2-1-0 3][3-1-0 7][1-0 15][3-2-0 25][3-2-1 49]>>>\n",
+       "<-1 * <<Loop[3-2-1-0 3][3-1-0 7][1-0 15][3-2-1 25][3-2 57]>>>\n",
+       "<-1 * <<Loop[3-2-1-0 3][3-1-0 7][3-2-0 13][2-0 29][3-2-1 49]>>>\n",
+       "<1 * <<Loop[3-2-1-0 3][3-1-0 7][3-2-0 13][3-2-1 25][2-1 57]>>>\n",
+       "... ...\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 52,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? cochain 5 63)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We verify, that for a simplex in dimension $n$, the length of the resulting combination is $(n-1)!$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 53,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "24"
+      ]
+     },
+     "execution_count": 53,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(length(cmbn-list *))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 54,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 5}\n",
+       "<1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-2-1-0 13][4-3-1-0 25][4-3-2-0 49][4-3-2-1 97]>>>\n",
+       "<-1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-2-1-0 13][4-3-1-0 25][4-3-2-1 49][4-3-2 113]>>>\n",
+       "<-1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-2-1-0 13][4-3-2-0 25][4-3-0 57][4-3-2-1 97]>>>\n",
+       "<1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-2-1-0 13][4-3-2-0 25][4-3-2-1 49][4-3-1 113]>>>\n",
+       "<1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-2-1-0 13][4-3-2-1 25][4-3-1 57][4-3-2 105]>>>\n",
+       "<-1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-2-1-0 13][4-3-2-1 25][4-3-2 57][4-3 121]>>>\n",
+       "<-1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-3-1-0 13][4-1-0 29][4-3-2-0 49][4-3-2-1 97]>>>\n",
+       "<1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-3-1-0 13][4-1-0 29][4-3-2-1 49][4-3-2 113]>>>\n",
+       "<1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-3-1-0 13][4-3-2-0 25][4-2-0 57][4-3-2-1 97]>>>\n",
+       "<-1 * <<Loop[4-3-2-1-0 3][3-2-1-0 7][4-3-1-0 13][4-3-2-0 25][4-3-2-1 49][4-2-1 113]>>>\n",
+       "... ...\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 54,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? cochain 6 127)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 55,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "120"
+      ]
+     },
+     "execution_count": 55,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(length(cmbn-list *))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Expression of the differential $d_{\\otimes t}$\n",
+    "\n",
+    "The differential $d_{\\otimes t}$ of the twisted tensor product $B \\otimes_t GB$ may be expressed as the sum of the differential of the simple tensor product $B \\otimes GB$ and the following morphism of degree -1,\n",
+    "$$\\nabla: B \\otimes GB \\longrightarrow B \\otimes GB,$$\n",
+    "where $\\nabla$ is the composition of the morphisms of the following sequence:\n",
+    "$$B\\otimes GB \\buildrel {\\Delta\\otimes 1_{GB}} \\over \\longrightarrow  (B\\otimes B) \\otimes GB\n",
+    "              \\buildrel {assoc} \\over \\longrightarrow  B \\otimes (B \\otimes GB)\n",
+    "              \\buildrel {1_B \\otimes (t \\otimes 1_{GB})} \\over \\longrightarrow  B \\otimes (GB \\otimes GB)\n",
+    "              \\buildrel {1_B \\otimes \\varpi} \\over \\longrightarrow B \\otimes GB.$$\n",
+    "In the previous formula, $\\Delta$ is the coproduct in the coalgebra $B$, $\\varpi$ is the product in the algebra $GB$, $1_B$ is the identity morphism in $B$ and $1_{GB}$ the identity morphism in $GB$. We need also, as in a previous example, an \"assoc\" morphism for the compound tensor product. The user will note the easiness for coding in `Kenzo` such a long composition of morphisms."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 56,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K167 Chain-Complex]"
+      ]
+     },
+     "execution_count": 56,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def 3-left (tnsr-prdc (tnsr-prdc d d) gd))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 57,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K169 Chain-Complex]"
+      ]
+     },
+     "execution_count": 57,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def 3-right (tnsr-prdc d (tnsr-prdc d gd)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 58,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"The variable ~S is defined but never used.\" {10074A1213}>\n",
+      "\n",
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"The variable ~S is defined but never used.\" {10074A18E3}>\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "[K171 Morphism (degree 0): K167 -> K169]"
+      ]
+     },
+     "execution_count": 58,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def assoc1 (build-mrph\n",
+    "               :sorc 3-left\n",
+    "               :trgt 3-right\n",
+    "               :degr 0\n",
+    "               :intr #'(lambda (degr a2-a) \n",
+    "                         (with-tnpr (degra2 gnrta2 degra gnrta) a2-a\n",
+    "                            (with-tnpr (degr1 gnrt1 degr2 gnrt2) gnrta2\n",
+    "                               (cmbn (+ degr1 degr2 degra) \n",
+    "                                     1 (tnpr degr1 \n",
+    "                                             gnrt1 \n",
+    "                                             (+ degr2 degra) \n",
+    "                                             (tnpr degr2 gnrt2 degra gnrta))))))\n",
+    "               :strt :gnrt\n",
+    "               :orgn '(assoc-double-tensor-product)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 59,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K172 Morphism (degree 0): K112 -> K112]"
+      ]
+     },
+     "execution_count": 59,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def id-d (idnt-mrph d))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 60,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K173 Morphism (degree 0): K117 -> K117]"
+      ]
+     },
+     "execution_count": 60,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def id-gd (idnt-mrph gd))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 61,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K185 Morphism (degree -1): K139 -> K139]"
+      ]
+     },
+     "execution_count": 61,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def nabla \n",
+    " (add (dffr (tnsr-prdc d gd))\n",
+    "      (i-cmps (tnsr-prdc id-d (aprd gd))\n",
+    "              (tnsr-prdc id-d (tnsr-prdc cochain id-gd))\n",
+    "              assoc1\n",
+    "              (tnsr-prdc (cprd d) id-gd))))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We verify that the morphism `nabla` and the differential of the  twisted tensor product (accessible through the symbol `d-twtp-gd`) return the same results:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 62,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<1 * <TnPr 1 <<Loop[7]>>>>\n",
+       "<-1 * <TnPr 1 <<Loop[0 3][7]>>>>\n",
+       "<1 * <TnPr 3 <<Loop[3\\-1][5]>>>>\n",
+       "<-1 * <TnPr 3 <<Loop[6]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 62,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? (dffr d-twtp-gd) 2 (tnpr 1 3 1 (loop3 0 7 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 63,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<1 * <TnPr 1 <<Loop[7]>>>>\n",
+       "<-1 * <TnPr 1 <<Loop[0 3][7]>>>>\n",
+       "<1 * <TnPr 3 <<Loop[3\\-1][5]>>>>\n",
+       "<-1 * <TnPr 3 <<Loop[6]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 63,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? nabla 2 (tnpr 1 3 1 (loop3 0 7 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 64,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K186 Morphism (degree -1): K139 -> K139]"
+      ]
+     },
+     "execution_count": 64,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def zero (sbtr nabla (dffr d-twtp-gd)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 65,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 65,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero 2 (tnpr 1 3 1 (loop3 0 7 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 66,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 66,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero 5 (tnpr  3 15 2 (loop3 0 15 1)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### The cup product of the twisting cochain\n",
+    "\n",
+    "The cup product $t \\sqcup t$ of the twisting cochain $t$,\n",
+    "$$t\\sqcup t: B \\longrightarrow GB$$\n",
+    "is defined by the following composition of morphisms:\n",
+    "$$B \\buildrel {\\Delta} \\over \\longrightarrow B \\otimes B\n",
+    "    \\buildrel {t \\otimes t} \\over \\longrightarrow GB \\otimes GB \n",
+    "    \\buildrel {\\varpi} \\over \\longrightarrow GB.$$\n",
+    "We may now verify the formula\n",
+    "$$ d_{GB} \\circ t + t \\circ d_B + t\\sqcup t =0,$$\n",
+    "where $d_B$ and $d_{GB}$ are  the respective differential morphisms in $B$ and $GB$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 67,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K189 Morphism (degree -2): K112 -> K117]"
+      ]
+     },
+     "execution_count": 67,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def t-cup-t (i-cmps (aprd gd) (tnsr-prdc cochain cochain) (cprd d)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 68,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K190 Morphism (degree -2): K112 -> K117]"
+      ]
+     },
+     "execution_count": 68,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def dt (cmps (dffr gd) cochain))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 69,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K191 Morphism (degree -2): K112 -> K117]"
+      ]
+     },
+     "execution_count": 69,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def td (cmps cochain (dffr d)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 70,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K193 Morphism (degree -2): K112 -> K117]"
+      ]
+     },
+     "execution_count": 70,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def zero-2 (i-add dt td t-cup-t))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 71,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 71,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero-2 3 15)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 72,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 72,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero-2 4 31)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 73,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 73,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero-2 5 63)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 74,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 74,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero-2 6 127)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "So the user has easily verified the *cochain condition* for the twisting cochain $t$.\n",
+    "\n",
+    "## The essential contraction\n",
+    "\n",
+    "It is known that if $B$ is a reduced simplicial set, the space $B \\times_\\tau GB$ is contractible. So it is possible to build a reduction of this space over $\\mathbb{Z}$. This reduction depends of the important contraction\n",
+    "$$\\chi_{\\times\\tau}:  {{\\cal C}_*(B) \\times_\\tau {\\cal C}_*(GB)} \\longrightarrow \n",
+    "                      {{\\cal C}_*(B) \\times_\\tau {\\cal C}_*(GB)}, $$\n",
+    "which is a homotopy operator (degree: $+1$) satisfying the relation\n",
+    "$$d \\circ \\chi_{\\times\\tau} + \\chi_{\\times\\tau} \\circ d =1.$$\n",
+    "Now, from the reduction\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "{{\\cal C}_*(X \\times_\\tau GX)} \\quad& \\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(X \\times_\\tau GX)} \\cr\n",
+    " {\\scriptstyle f} \\downarrow \\uparrow {\\scriptstyle g} \\qquad \\cr\n",
+    " {{\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)} \\cr\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "there is also  an induced contraction \n",
+    "$$\\chi_{\\otimes t}:  {{\\cal C}_*(B) \\otimes_t {\\cal C}_*(GB)} \\longrightarrow \n",
+    "                      {{\\cal C}_*(B) \\otimes_t {\\cal C}_*(GB)}, $$\n",
+    "defined by $\\chi_{\\otimes t}= f \\circ \\chi_{\\times\\tau} \\circ g$.\n",
+    "This implies that the twisted tensor product is also contractible over $\\mathbb{Z}$.\n",
+    "\n",
+    "The two functions of `Kenzo` for this construction are:\n",
+    "\n",
+    "`crts-contraction` *space* *[Function]*\n",
+    "> Return the homotopy morphism corresponding to the contraction $\\chi_{\\times\\tau}$. The lisp definition is:\n",
+    "```commonlisp\n",
+    "          (defun CRTS-CONTRACTION \n",
+    "             (space\n",
+    "                &aux (twisted-crts-prdc (twisted-crts-prdc space)))\n",
+    "             (the morphism\n",
+    "                (build-mrph\n",
+    "                   :sorc twisted-crts-prdc \n",
+    "                   :trgt twisted-crts-prdc \n",
+    "                   :degr +1\n",
+    "                   :intr (crts-contraction-intr \n",
+    "                               (cmpr space)\n",
+    "                               (bspn space) \n",
+    "                               (face space) \n",
+    "                               (cmpr twisted-crts-prdc))\n",
+    "                   :strt :gnrt\n",
+    "                   :orgn `(crts-contraction ,space))))\n",
+    "```\n",
+    "At execution time, the work is essentialy done by the function put in the slot `:intr`.\n",
+    "\n",
+    "`tnpr-contraction` *space* *[Function]*\n",
+    "> Return the induced morphism $\\chi_{\\otimes t}= f \\circ \\chi_{\\times\\tau} \\circ g$, as shown by the lisp definition:\n",
+    "```commonlisp\n",
+    "          (defun TNPR-CONTRACTION \n",
+    "             (space\n",
+    "                &aux (szczarba (szczarba space))\n",
+    "                     (f (f szczarba))\n",
+    "                     (g (g szczarba))\n",
+    "                     (crts-contraction (crts-contraction space)))\n",
+    "             (the morphism\n",
+    "                (i-cmps f crts-contraction g)))\n",
+    "```\n",
+    "\n",
+    "### Examples\n",
+    "\n",
+    "Let us use again our familiar simplicial set $\\bar\\Delta$. First we apply  the homotopy operator $\\chi_{\\times t}$   upon various generators. Note that the binary coding of the simplices allows normal arithmetic operators to generate them."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 75,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K112 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 75,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def delta (deltab))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 76,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K194 Morphism (degree 1): K134 -> K134]"
+      ]
+     },
+     "execution_count": 76,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def chi-x (crts-contraction delta))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 77,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 77,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 0 (crpr 0 1 0 +null-loop+))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 78,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <CrPr - 96 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 78,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 0 (crpr 0 1 0 (loop3 0 96 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 79,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <CrPr - 96 0 <<Loop[384]>>>>\n",
+       "<-1 * <CrPr - 384 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 79,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 80,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <CrPr - 96 0 <<Loop[384][1536]>>>>\n",
+       "<-1 * <CrPr - 384 0 <<Loop[1536]>>>>\n",
+       "<-1 * <CrPr - 1536 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 80,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1 0 (+ 512 1024) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 81,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <CrPr - 96 0 <<Loop>>>>\n",
+       "<-1 * <CrPr - 96 0 <<Loop[96]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 81,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 0 (crpr 0 1 0 (loop3 0 96 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 82,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<1 * <CrPr - 224 1-0 <<Loop>>>>\n",
+       "<-1 * <CrPr 1 3 0 <<Loop[224]>>>>\n",
+       "<1 * <CrPr 1 96 0 <<Loop[224]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 82,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 1 (crpr 0 3 0 (loop3 0 (+ 32 64 128) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 83,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<-1 * <CrPr - 480 2-1-0 <<Loop>>>>\n",
+       "<1 * <CrPr 2 7 1 <<Loop[480]>>>>\n",
+       "<-1 * <CrPr 2 224 1 <<Loop[480]>>>>\n",
+       "<-1 * <CrPr 2-1 5 0 <<Loop[480]>>>>\n",
+       "<1 * <CrPr 2-1 160 0 <<Loop[480]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 83,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 2 (crpr 0 7 0 (loop3 0 (+ 32 64 128 256) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 84,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<-1 * <CrPr 3-1 7 2-0 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
+       "<1 * <CrPr 3-2 7 1-0 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 84,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 85,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 85,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 86,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<-1 * <CrPr 3-1 7 2-0 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
+       "<1 * <CrPr 3-2 7 1-0 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 86,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 87,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 87,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 88,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<-1 * <CrPr 3-0 7 2-1 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 88,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 89,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 89,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 90,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<-1 * <CrPr 3-0 7 2-1 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 90,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 91,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 91,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-x 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Then, for testing  $\\chi_{\\otimes t}$ we use the image of all those generators by the function `f` of the Brown reduction.\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "    {{\\cal C}_*(X \\times_\\tau GX)} \\quad& \\stackrel{h}{\\longrightarrow} & {}^s{{\\cal C}_*(X \\times_\\tau GX)} \\cr\n",
+    " {\\scriptstyle f} \\downarrow \\uparrow {\\scriptstyle g}\\qquad  \\cr\n",
+    " {{\\cal C}_*(X) \\otimes_t {\\cal C}_*(GX)} \\cr\n",
+    "\\end{aligned}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 92,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "T"
+      ]
+     },
+     "execution_count": 92,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(setf *tnpr-with-degrees* t)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 93,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K196 Morphism (degree 1): K160 -> K160]"
+      ]
+     },
+     "execution_count": 93,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def chi-t (tnpr-contraction delta))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 94,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K162 Reduction K134 => K160]"
+      ]
+     },
+     "execution_count": 94,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def reduc (szczarba delta))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 95,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K152 Morphism (degree 0): K134 -> K160]"
+      ]
+     },
+     "execution_count": 95,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def f (f reduc))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 96,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * <TnPr 0 1 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 96,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 0 (crpr 0 1 0 +null-loop+))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 97,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 97,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 98,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * <TnPr 0 32 0 <<Loop[96]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 98,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 99,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <TnPr 1 96 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 99,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 100,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * <TnPr 0 1 0 <<Loop[96][384]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 100,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 101,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <TnPr 1 96 0 <<Loop[384]>>>>\n",
+       "<-1 * <TnPr 1 384 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 101,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 102,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * <TnPr 0 1 0 <<Loop[96][384][1536]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 102,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 1 0 (+ 256 128) 1 0 (+ 512 1024) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 103,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <TnPr 1 96 0 <<Loop[384][1536]>>>>\n",
+       "<-1 * <TnPr 1 384 0 <<Loop[1536]>>>>\n",
+       "<-1 * <TnPr 1 1536 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 103,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 104,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * <TnPr 0 1 0 <<Loop[96\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 104,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 0 (crpr 0 1 0 (loop3 0 96 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 105,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<-1 * <TnPr 1 96 0 <<Loop>>>>\n",
+       "<-1 * <TnPr 1 96 0 <<Loop[96]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 105,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 106,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 1}\n",
+       "<1 * <TnPr 0 1 1 <<Loop[0 3][224]>>>>\n",
+       "<1 * <TnPr 1 3 0 <<Loop[192]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 106,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 1 (crpr 0 3 0 (loop3 0 (+ 32 64 128) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 107,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<-1 * <TnPr 1 3 1 <<Loop[224]>>>>\n",
+       "<1 * <TnPr 1 96 1 <<Loop[224]>>>>\n",
+       "<1 * <TnPr 2 224 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 107,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 108,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 2}\n",
+       "<-1 * <TnPr 0 1 2 <<Loop[1-0 3][0 7][1 416]>>>>\n",
+       "<1 * <TnPr 0 1 2 <<Loop[1-0 3][1 7][480]>>>>\n",
+       "<1 * <TnPr 1 3 1 <<Loop[0 6][448]>>>>\n",
+       "<1 * <TnPr 2 7 0 <<Loop[384]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 108,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 2 (crpr 0 7 0 (loop3 0 (+ 32 64 128 256) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 109,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2 480]>>>>\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][1 480]>>>>\n",
+       "<-1 * <TnPr 0 32 3 <<Loop[2-1-0 96][1-0 224][2 480]>>>>\n",
+       "<1 * <TnPr 0 32 3 <<Loop[2-1-0 96][2-0 224][1 480]>>>>\n",
+       "<-1 * <TnPr 1 5 2 <<Loop[480]>>>>\n",
+       "<1 * <TnPr 1 160 2 <<Loop[480]>>>>\n",
+       "<1 * <TnPr 2 7 1 <<Loop[448]>>>>\n",
+       "<-1 * <TnPr 2 224 1 <<Loop[448]>>>>\n",
+       "<-1 * <TnPr 3 480 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 109,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 110,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-1 7][2-0 224\\2][3-0 224\\-2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 110,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 111,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<1 * <TnPr 2 7 2 <<Loop[1 224\\2][2 224\\-2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 111,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 112,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 112,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 113,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 113,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 114,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-1 7][2-0 224\\-2][3-0 224\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 114,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 2 7 1 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 115,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "<1 * <TnPr 2 7 2 <<Loop[1 224\\-2][2 224\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 115,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 116,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 116,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 2 7 4 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 117,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 117,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 118,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\2][3-2 224\\-2]>>>>\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\2][3-1 224\\-2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 118,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) 2 4 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 119,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 119,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 120,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\2][2-0 224\\-2]>>>>\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\2][1-0 224\\-2]>>>>\n",
+       "<1 * <TnPr 2 7 1 <<Loop[0 192\\2][224\\-2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 120,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) 2 1 (+ 32 64 128) -2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 121,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 121,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 122,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\-2][3-2 224\\2]>>>>\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\-2][3-1 224\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 122,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 1 7 2 (loop3 2 (+ 32 64 128) -2 4 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 123,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 123,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 124,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 0 1 3 <<Loop[2-1-0 3][1-0 7][2-1 224\\-2][2-0 224\\2]>>>>\n",
+       "<-1 * <TnPr 0 1 3 <<Loop[2-1-0 3][2-0 7][2-1 224\\-2][1-0 224\\2]>>>>\n",
+       "<1 * <TnPr 2 7 1 <<Loop[0 192\\-2][224\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 124,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? f 3 (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 125,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 4}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 125,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? chi-t *)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We verify now the required property of the homotopy operator $\\chi_{\\times\\tau}$. As usual, we build by composition a morphism `zero`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 126,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K135 Morphism (degree -1): K134 -> K134]"
+      ]
+     },
+     "execution_count": 126,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def dfr (bndr (twisted-crts-prdc delta)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 127,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K201 Morphism (degree 0): K134 -> K134]"
+      ]
+     },
+     "execution_count": 127,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def zero (i-sbtr (idnt-mrph (sorc dfr)) (cmps dfr chi-x) \n",
+    "                   (cmps chi-x dfr)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 128,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<CrPr 2 7 1 <<Loop[1 224\\-2][0 224\\2]>>>"
+      ]
+     },
+     "execution_count": 128,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def simplx (crpr 4 7 2 (loop3 2 (+ 32 64 128) -2 1 (+ 32 64 128) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 129,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 129,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero 3 simplx)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Then, we do the same test for the homotopy operator $\\chi_{\\otimes t}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 130,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K160 Chain-Complex]"
+      ]
+     },
+     "execution_count": 130,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def tw (twisted-tnsr-prdc d))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 131,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K206 Morphism (degree 0): K160 -> K160]"
+      ]
+     },
+     "execution_count": 131,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def zero-2  (i-sbtr (idnt-mrph tw) (cmps tw chi-t) (cmps chi-t tw)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 132,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<TnPr 0 1 3 <<Loop[31\\2][992\\-1]>>>"
+      ]
+     },
+     "execution_count": 132,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def simpx2 (tnpr 0 1 3 (loop3 0 (mask 5) 2 0 (* 32 (mask 5)) -1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 133,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 133,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(time (? zero-2 3 simpx2))"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Note nevertheless that the relation is not satisfied for the base generator $*\\otimes *$, namely the tensor product of both base generators. This is due to the fact that both contractions,  $\\chi_{\\times t}$ and  $\\chi_{\\otimes t}$, correspond to  reductions over  $\\mathbb{Z}$ and not on 0."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 134,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<TnPr 0 1 0 <<Loop>>>"
+      ]
+     },
+     "execution_count": 134,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def gnrt-0 (tnpr 0 1 0 +null-loop+))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 135,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * <TnPr 0 1 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 135,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? zero-2 0 gnrt-0)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Let us build, for instance, the reduction of the twisted tensor product over $\\mathbb{Z}$ and check it. In that case, the two homomorphisms $f$ and $g$ are respectively the *augmentation* and *coaugmentation* morphisms.\n",
+    "$$\n",
+    "\\begin{aligned}\n",
+    "{{\\cal C}_*(B) \\otimes_t {\\cal C}_*(GB)} & \\qquad\\stackrel{\\chi_{\\otimes t}}{\\longrightarrow} &\n",
+    "  {}^s{{\\cal C}_*(B) \\otimes_t {\\cal C}_*(GB)} \\cr\n",
+    " {\\scriptstyle aug} \\downarrow \\uparrow {\\scriptstyle coaug}\\quad & &  \\cr\n",
+    " {{\\cal C}_*(\\mathbb{Z})} \\qquad& & \\cr\n",
+    "\\end{aligned}\n",
+    "$$\n",
+    "We recall that, the unit chain complex corresponding to $\\mathbb{Z}$ is built by the function `z-chcm`. The unique generator in degree 0 is `:z-gnrt`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 136,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"The variable ~S is defined but never used.\" {10052B6D53}>\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "[K209 Cohomology-Class on K160 of degree 0]"
+      ]
+     },
+     "execution_count": 136,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def aug (build-mrph\n",
+    "            :sorc tw \n",
+    "            :trgt (z-chcm) \n",
+    "            :degr 0\n",
+    "            :intr #'(lambda (degr gnrt)\n",
+    "                       (if (zerop degr)\n",
+    "                           (term-cmbn 0 1 :z-gnrt)\n",
+    "                           (zero-cmbn degr)))\n",
+    "            :strt :gnrt \n",
+    "            :orgn '(aug tw)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 137,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "SIMPLE-STYLE-WARNING: \n",
+      "  #<SB-INT:SIMPLE-STYLE-WARNING \"The variable ~S is defined but never used.\" {10053C9373}>\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "[K210 Morphism (degree 0): K207 -> K160]"
+      ]
+     },
+     "execution_count": 137,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def coaug (build-mrph\n",
+    "              :sorc (z-chcm) \n",
+    "              :trgt tw \n",
+    "              :degr 0\n",
+    "              :intr #'(lambda (degr gnrt)\n",
+    "                         (if (zerop degr)\n",
+    "                             (term-cmbn 0 1 (tnpr 0 1 0 +null-loop+))\n",
+    "                             (zero-cmbn degr)))\n",
+    "              :strt :gnrt\n",
+    "              :orgn '(coaug tw)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 138,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K211 Reduction K160 => K207]"
+      ]
+     },
+     "execution_count": 138,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def rdct (build-rdct :f aug \n",
+    "                       :g coaug \n",
+    "                       :h Chi-t \n",
+    "                       :orgn '(reduction-tw-Z)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 139,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "---done---"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 139,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(pre-check-rdct rdct)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 140,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<1 * <TnPr 1 3 2 <<Loop[15\\2]>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 140,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(setf *tc* (cmbn 3 1 (tnpr 1 3 2 (loop3 0 15 2))))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 141,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 0}\n",
+       "<1 * Z-GNRT>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 141,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(setf *bc* (cmbn 0 1 :z-gnrt))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 142,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "\n",
+      "*TC* => \n",
+      "----------------------------------------------------------------------{CMBN 3}\n",
+      "<1 * <TnPr 1 3 2 <<Loop[15\\2]>>>>\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "*BC* => \n",
+      "----------------------------------------------------------------------{CMBN 0}\n",
+      "<1 * Z-GNRT>\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *TDD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 1}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *BDD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN -2}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *DF-FD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 2}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *DG-GD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN -1}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *ID-FG* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 0}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *ID-GF-DH-HD* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 3}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *HH* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 5}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *FH* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 4}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "Checking *HG* = 0\n",
+      "Result: \n",
+      "----------------------------------------------------------------------{CMBN 1}\n",
+      "------------------------------------------------------------------------------\n",
+      "\n",
+      "---done---"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 142,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(check-rdct-no-wait)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## An unproved property of the algebraic Kan contraction\n",
+    "\n",
+    "Since the early version of the software (1990), and through a host of examples, an amazing algebraic property of the Kan contraction $\\chi_{\\otimes t}$ has been experimentally discovered. Many computations have been done of $\\chi_{\\otimes t}(b \\otimes g)$, for various choices of $b \\otimes g$ and in all these cases,  as soon as $degree(b) > 0$, then the result is null.\n",
+    "In other words, $\\chi_{\\otimes t}$ *is null outside the base fiber.* Up to now, no actual proof of this result has yet  been given.\n",
+    "\n",
+    "We show that, using the \"soft\" version of $\\bar{\\Delta}$."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 143,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K1 Simplicial-Set]"
+      ]
+     },
+     "execution_count": 143,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def db (soft-deltab))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 144,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "[K231 Morphism (degree 1): K88 -> K88]"
+      ]
+     },
+     "execution_count": 144,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def tt (tnpr-contraction db))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 145,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<TnPr 3 (DELT . 31) 4 <<Loop[(DELT . 63)\\2]>>>"
+      ]
+     },
+     "execution_count": 145,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def gnrt1 (tnpr 3 (code '(0 1 2 3 4)) \n",
+    "                 4 (loop3 0 (code '(0 1 2 3 4 5)) 2)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 146,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 8}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 146,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? tt 7 gnrt1)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 147,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "<TnPr 4 (DELT . 5460) 5 <<Loop[(DELT . 10922)\\5]>>>"
+      ]
+     },
+     "execution_count": 147,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(def gnrt2 (tnpr 4 (code '(2 4 6 8 10 12))  \n",
+    "                  5 (loop3 0 (code '(1 3 5 7 9 11 13)) 5)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 148,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 10}\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 148,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? tt 9 gnrt2)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "But if $degree(b) = 0$, then the result is non-null in general."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 149,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 3}\n",
+       "<-1 * <TnPr 0 (DELT . 1) 3 <<Loop[2-1-0 (DELT . 3)][1-0 (DELT . 7)][2 (DELT . 15)]>>>>\n",
+       "<1 * <TnPr 0 (DELT . 1) 3 <<Loop[2-1-0 (DELT . 3)][2-0 (DELT . 7)][1 (DELT . 15)]>>>>\n",
+       "<1 * <TnPr 1 (DELT . 5) 2 <<Loop[(DELT . 15)]>>>>\n",
+       "<-1 * <TnPr 2 (DELT . 7) 1 <<Loop[(DELT . 14)]>>>>\n",
+       "<-1 * <TnPr 3 (DELT . 15) 0 <<Loop>>>>\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 149,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? tt 2 (tnpr 0 (code '(0)) 2 (loop3 0 (code '(0 1 2 3)) 1)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 150,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "data": {
+      "text/plain": [
+       "\n",
+       "----------------------------------------------------------------------{CMBN 5}\n",
+       "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-1-0 (DELT . 50)][4-3-2 (DELT . 114)]>>>>\n",
+       "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-1-0 (DELT . 50)][4-3-2 (DELT . 114)\\2]>>>>\n",
+       "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-1-0 (DELT . 50)][4-3-2 (DELT . 114)\\3]>>>>\n",
+       "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-1-0 (DELT . 50)][4-3-2-0 (DELT . 98)][4-3-2-1 (DELT . 98)]>>>>\n",
+       "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-1-0 (DELT . 50)][4-3-2-0 (DELT . 98)][4-3-2-1 (DELT . 98)\\2]>>>>\n",
+       "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-1-0 (DELT . 82)][4-3-2 (DELT . 114)]>>>>\n",
+       "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-1-0 (DELT . 82)][4-3-2 (DELT . 114)\\2]>>>>\n",
+       "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-2-0 (DELT . 50)][4-3-0 (DELT . 114)][4-3-2-1 (DELT . 98)]>>>>\n",
+       "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-2-0 (DELT . 50)][4-3-0 (DELT . 114)][4-3-2-1 (DELT . 98)\\2]>>>>\n",
+       "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT . 26)][4-3-2-0 (DELT . 50)][4-3-1 (DELT . 114)]>>>>\n",
+       "... ...\n",
+       "------------------------------------------------------------------------------\n"
+      ]
+     },
+     "execution_count": 150,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(? tt 4 (tnpr 0 (code '(0)) 4 (loop3 0 (code '(1 2 3 4 5 6)) 3)))"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 151,
+   "metadata": {
+    "collapsed": false
+   },
+   "outputs": [
+    {
+     "name": "stderr",
+     "output_type": "stream",
+     "text": [
+      "TYPE-ERROR: \n",
+      "  #<TYPE-ERROR expected-type: SEQUENCE datum: \n",
+      "----------------------------------------------------------------------{CMBN 5}\n",
+      "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-1-0 (DELT ..)][4-3-2 (DELT ..)]>>>>\n",
+      "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-1-0 (DELT ..)][4-3-2 (DELT ..)\\2]>>>>\n",
+      "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-1-0 (DELT ..)][4-3-2 (DELT ..)\\3]>>>>\n",
+      "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-1-0 (DELT ..)][4-3-2-0 (DELT ..)][4-3-2-1 (DELT ..)]>>>>\n",
+      "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-1-0 (DELT ..)][4-3-2-0 (DELT ..)][4-3-2-1 (DELT ..)\\2]>>>>\n",
+      "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-1-0 (DELT ..)][4-3-2 (DELT ..)]>>>>\n",
+      "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-1-0 (DELT ..)][4-3-2 (DELT ..)\\2]>>>>\n",
+      "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-2-0 (DELT ..)][4-3-0 (DELT ..)][4-3-2-1 (DELT ..)]>>>>\n",
+      "<1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-2-0 (DELT ..)][4-3-0 (DELT ..)][4-3-2-1 (DELT ..)\\2]>>>>\n",
+      "<-1 * <TnPr 0 (DELT . 2) 5 <<Loop[4-3-2-1-0 (DELT . 6)][3-2-1-0 (DELT . 14)][4-2-1-0 (DELT ..)][4-3-2-0 (DELT ..)][4-3-1 (DELT ..)]>>>>\n",
+      "... ...\n",
+      "------------------------------------------------------------------------------\n",
+      ">\n"
+     ]
+    },
+    {
+     "data": {
+      "text/plain": [
+       "NIL"
+      ]
+     },
+     "execution_count": 151,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "(length *)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Lisp file concerned in this chapter\n",
+    "\n",
+    "`ls-twisted-products.lisp`."
+   ]
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SBCL Lisp",
+   "language": "lisp",
+   "name": "lisp"
+  },
+  "language_info": {
+   "codemirror_mode": "text/x-common-lisp",
+   "mimetype": "text/x-common-lisp",
+   "name": "common-lisp",
+   "pygments_lexer": "common-lisp",
+   "version": "X3J13"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 0
+}