From b70f0e14061f20be21ef9bb2d2bb76c829b33ea0 Mon Sep 17 00:00:00 2001 From: gheber Date: Sun, 13 Mar 2016 04:37:29 -0500 Subject: [PATCH] First draft of Chapter 11. --- examples/Chapter11.ipynb | 1119 ++++++++++++++++++++++++++++++++------ 1 file changed, 966 insertions(+), 153 deletions(-) diff --git a/examples/Chapter11.ipynb b/examples/Chapter11.ipynb index 28df6a9..81a7ee0 100644 --- a/examples/Chapter11.ipynb +++ b/examples/Chapter11.ipynb @@ -236,7 +236,7 @@ }, { "cell_type": "code", - "execution_count": 6, + "execution_count": 5, "metadata": { "collapsed": false }, @@ -258,7 +258,7 @@ "NIL" ] }, - "execution_count": 6, + "execution_count": 5, "metadata": {}, "output_type": "execute_result" } @@ -268,34 +268,34 @@ " (check-kan KZ1 i 3 (remove (nth i hat) hat)))" ] }, + { + "cell_type": "code", + "execution_count": 6, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "( )" + ] + }, + "execution_count": 6, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def hat1 (remove (nth 1 hat) hat))" + ] + }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, - "outputs": [ - { - "data": { - "text/plain": [ - "( )" - ] - }, - "execution_count": 7, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(def hat1 (remove (nth 1 hat) hat))" - ] - }, - { - "cell_type": "code", - "execution_count": 8, - "metadata": { - "collapsed": false - }, "outputs": [ { "data": { @@ -303,7 +303,7 @@ "" ] }, - "execution_count": 8, + "execution_count": 7, "metadata": {}, "output_type": "execute_result" } @@ -321,7 +321,7 @@ }, { "cell_type": "code", - "execution_count": 9, + "execution_count": 8, "metadata": { "collapsed": false }, @@ -332,7 +332,7 @@ "" ] }, - "execution_count": 9, + "execution_count": 8, "metadata": {}, "output_type": "execute_result" } @@ -341,6 +341,28 @@ "(grml KZ1 3 (crpr 0 simplex 0 simplex))" ] }, + { + "cell_type": "code", + "execution_count": 9, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "" + ] + }, + "execution_count": 9, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def invsimplex (grin KZ1 3 simplex))" + ] + }, { "cell_type": "code", "execution_count": 10, @@ -351,7 +373,7 @@ { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 10, @@ -360,7 +382,7 @@ } ], "source": [ - "(def invsimplex (grin KZ1 3 simplex))" + "(grin KZ1 3 *) " ] }, { @@ -373,7 +395,7 @@ { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 11, @@ -382,7 +404,29 @@ } ], "source": [ - "(grin KZ1 3 *) " + "(grml KZ1 3 (crpr 0 simplex 0 (gmsm invsimplex)))" + ] + }, + { + "cell_type": "code", + "execution_count": 12, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + ">" + ] + }, + "execution_count": 12, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(2absm-acrpr (absm 0 simplex ) (grin KZ1 3 simplex))" ] }, { @@ -404,7 +448,7 @@ } ], "source": [ - "(grml KZ1 3 (crpr 0 simplex 0 (gmsm invsimplex)))" + "(grml KZ1 3 *) " ] }, { @@ -417,7 +461,7 @@ { "data": { "text/plain": [ - ">" + "[K13 Abelian-Simplicial-Group]" ] }, "execution_count": 14, @@ -426,7 +470,7 @@ } ], "source": [ - "(2absm-acrpr (absm 0 simplex ) (grin KZ1 3 simplex))" + "(def KZ2 (k-z2-1))" ] }, { @@ -439,7 +483,7 @@ { "data": { "text/plain": [ - "" + "" ] }, "execution_count": 15, @@ -448,7 +492,7 @@ } ], "source": [ - "(grml KZ1 3 *) " + "(grml KZ2 3 (crpr 0 3 0 3))" ] }, { @@ -461,7 +505,7 @@ { "data": { "text/plain": [ - "[K13 Abelian-Simplicial-Group]" + "" ] }, "execution_count": 16, @@ -470,7 +514,7 @@ } ], "source": [ - "(def KZ2 (k-z2-1))" + "(grin KZ2 4 4)" ] }, { @@ -479,50 +523,6 @@ "metadata": { "collapsed": false }, - "outputs": [ - { - "data": { - "text/plain": [ - "" - ] - }, - "execution_count": 17, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(grml KZ2 3 (crpr 0 3 0 3))" - ] - }, - { - "cell_type": "code", - "execution_count": 18, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "" - ] - }, - "execution_count": 18, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(grin KZ2 4 4)" - ] - }, - { - "cell_type": "code", - "execution_count": 19, - "metadata": { - "collapsed": false - }, "outputs": [ { "data": { @@ -530,7 +530,7 @@ "" ] }, - "execution_count": 19, + "execution_count": 17, "metadata": {}, "output_type": "execute_result" } @@ -548,7 +548,7 @@ }, { "cell_type": "code", - "execution_count": 20, + "execution_count": 18, "metadata": { "collapsed": false }, @@ -559,7 +559,7 @@ "NIL" ] }, - "execution_count": 20, + "execution_count": 18, "metadata": {}, "output_type": "execute_result" } @@ -568,6 +568,50 @@ "(z-absm-bar (absm 0 '()))" ] }, + { + "cell_type": "code", + "execution_count": 19, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "(0)" + ] + }, + "execution_count": 19, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(z-absm-bar (absm 1 '()))" + ] + }, + { + "cell_type": "code", + "execution_count": 20, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "(2)" + ] + }, + "execution_count": 20, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(z-absm-bar (absm 0 '(2)))" + ] + }, { "cell_type": "code", "execution_count": 21, @@ -578,7 +622,7 @@ { "data": { "text/plain": [ - "(0)" + "(0 0 0 1 1 1 1 1 1 1)" ] }, "execution_count": 21, @@ -587,7 +631,7 @@ } ], "source": [ - "(z-absm-bar (absm 1 '()))" + "(z2-absm-bar (absm 7 7))" ] }, { @@ -596,50 +640,6 @@ "metadata": { "collapsed": false }, - "outputs": [ - { - "data": { - "text/plain": [ - "(2)" - ] - }, - "execution_count": 22, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(z-absm-bar (absm 0 '(2)))" - ] - }, - { - "cell_type": "code", - "execution_count": 23, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "(0 0 0 1 1 1 1 1 1 1)" - ] - }, - "execution_count": 23, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(z2-absm-bar (absm 7 7))" - ] - }, - { - "cell_type": "code", - "execution_count": 24, - "metadata": { - "collapsed": false - }, "outputs": [ { "data": { @@ -647,7 +647,7 @@ "(1 1 1 1 1 1 1)" ] }, - "execution_count": 24, + "execution_count": 22, "metadata": {}, "output_type": "execute_result" } @@ -665,11 +665,29 @@ }, { "cell_type": "code", - "execution_count": null, + "execution_count": 23, "metadata": { - "collapsed": true + "collapsed": false }, - "outputs": [], + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "Warning: no evaluable input\n" + ] + }, + { + "data": { + "text/plain": [ + "NIL" + ] + }, + "execution_count": 23, + "metadata": {}, + "output_type": "execute_result" + } + ], "source": [ "#|\n", "labsms ==>\n", @@ -723,7 +741,7 @@ }, { "cell_type": "code", - "execution_count": 25, + "execution_count": 23, "metadata": { "collapsed": false }, @@ -734,7 +752,7 @@ "[K1 Abelian-Simplicial-Group]" ] }, - "execution_count": 25, + "execution_count": 23, "metadata": {}, "output_type": "execute_result" } @@ -745,7 +763,7 @@ }, { "cell_type": "code", - "execution_count": 26, + "execution_count": 24, "metadata": { "collapsed": false }, @@ -767,7 +785,7 @@ "8. APRD: \"unbound\"\n", "9. CPRD: [K5 Morphism (degree 0): K1 -> K3]\n", "10. FACE: #\n", - "11. KFLL: #\n", + "11. KFLL: #\n", "12. GRML: [K11 Simplicial-Morphism K6 -> K1]\n", "13. GRIN: [K12 Simplicial-Morphism K1 -> K1]\n", "\n", @@ -780,7 +798,7 @@ "NIL" ] }, - "execution_count": 26, + "execution_count": 24, "metadata": {}, "output_type": "execute_result" } @@ -801,7 +819,7 @@ }, { "cell_type": "code", - "execution_count": 27, + "execution_count": 25, "metadata": { "collapsed": false }, @@ -812,7 +830,7 @@ "" ] }, - "execution_count": 27, + "execution_count": 25, "metadata": {}, "output_type": "execute_result" } @@ -823,7 +841,7 @@ }, { "cell_type": "code", - "execution_count": 28, + "execution_count": 26, "metadata": { "collapsed": false }, @@ -846,7 +864,7 @@ "------------------------------------------------------------------------------\n" ] }, - "execution_count": 28, + "execution_count": 26, "metadata": {}, "output_type": "execute_result" } @@ -864,7 +882,7 @@ }, { "cell_type": "code", - "execution_count": 29, + "execution_count": 27, "metadata": { "collapsed": false }, @@ -886,7 +904,7 @@ "8. APRD: [K26 Morphism (degree 0): K3 -> K1]\n", "9. CPRD: [K5 Morphism (degree 0): K1 -> K3]\n", "10. FACE: #\n", - "11. KFLL: #\n", + "11. KFLL: #\n", "12. GRML: [K11 Simplicial-Morphism K6 -> K1]\n", "13. GRIN: [K12 Simplicial-Morphism K1 -> K1]\n", "\n", @@ -899,7 +917,7 @@ "NIL" ] }, - "execution_count": 29, + "execution_count": 27, "metadata": {}, "output_type": "execute_result" } @@ -920,14 +938,809 @@ "\n", "$$\n", "\\begin{aligned}\n", - "& ({\\cal A}\\otimes {\\cal A}) \\otimes {\\cal A} & \\buildrel {assoc} \\over \\longrightarrow &\n", + "({\\cal A}\\otimes {\\cal A}) \\otimes {\\cal A} & \\quad\\buildrel {assoc} \\over \\longrightarrow &\n", " {\\cal A} \\otimes ({\\cal A}\\otimes {\\cal A}) \\cr\n", - "& {\\scriptstyle \\nabla \\otimes 1}\\downarrow & & \\downarrow {\\scriptstyle 1 \\otimes \\nabla} \\cr\n", - "& {\\cal A}\\otimes {\\cal A} & & {\\cal A}\\otimes {\\cal A} \\cr\n", - "& {\\scriptstyle \\nabla}\\searrow & & \\swarrow {\\scriptstyle \\nabla} \\cr\n", - "& & {\\cal A} & \\cr\n", + "{\\scriptstyle \\nabla \\otimes 1}\\downarrow \\qquad& & \\downarrow {\\scriptstyle 1 \\otimes \\nabla}\\quad \\cr\n", + "{\\cal A}\\otimes {\\cal A}\\quad & & {\\cal A}\\otimes {\\cal A}\\quad \\cr\n", + "{\\scriptstyle \\nabla}\\searrow \\quad & & \\swarrow {\\scriptstyle \\nabla} \\qquad \\cr\n", + " & \\qquad{\\cal A} & \\cr\n", "\\end{aligned}\n", - "$$" + "$$\n", + "\n", + "Let us define two chain complexes with the two ways to compose two tensorial products:" + ] + }, + { + "cell_type": "code", + "execution_count": 28, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K1 Abelian-Simplicial-Group]" + ] + }, + "execution_count": 28, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def kz1 (k-z-1))" + ] + }, + { + "cell_type": "code", + "execution_count": 29, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K27 Chain-Complex]" + ] + }, + "execution_count": 29, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def 3-left (tnsr-prdc (tnsr-prdc kz1 kz1) kz1))" + ] + }, + { + "cell_type": "code", + "execution_count": 30, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K29 Chain-Complex]" + ] + }, + "execution_count": 30, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def 3-right (tnsr-prdc kz1 (tnsr-prdc kz1 kz1)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Then we define the morphism `assoc` and test it:" + ] + }, + { + "cell_type": "code", + "execution_count": 31, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "name": "stderr", + "output_type": "stream", + "text": [ + "SIMPLE-STYLE-WARNING: \n", + " #\n" + ] + }, + { + "data": { + "text/plain": [ + "[K31 Morphism (degree 0): K27 -> K29]" + ] + }, + "execution_count": 31, + "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", + " (declare (ignore degr))\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": 32, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "T" + ] + }, + "execution_count": 32, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(setf *tnpr-with-degrees* t)" + ] + }, + { + "cell_type": "code", + "execution_count": 33, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 7}\n", + "<1 * >>\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 33, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? assoc1 7 (tnpr 4 (tnpr 2 '(1 2) 2 '(2 3)) 3 '(4 5 6)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We define now the other morphisms shown in the diagram:" + ] + }, + { + "cell_type": "code", + "execution_count": 34, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K26 Morphism (degree 0): K3 -> K1]" + ] + }, + "execution_count": 34, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def nabla (aprd kz1))" + ] + }, + { + "cell_type": "code", + "execution_count": 35, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K32 Morphism (degree 0): K1 -> K1]" + ] + }, + "execution_count": 35, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def idnt (idnt-mrph kz1))" + ] + }, + { + "cell_type": "code", + "execution_count": 36, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K33 Morphism (degree 0): K29 -> K3]" + ] + }, + "execution_count": 36, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def 1-t-nabla (tnsr-prdc idnt nabla))" + ] + }, + { + "cell_type": "code", + "execution_count": 37, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K34 Morphism (degree 0): K27 -> K3]" + ] + }, + "execution_count": 37, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def nabla-t-1 (tnsr-prdc nabla idnt)) " + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Now, if the diagram is commutative, the difference morphism, noted `zero`, between the following morphisms `right` and `left`, must give the null combination if applied to an element of $K(\\mathbb{Z}, 1) \\otimes K(\\mathbb{Z}, 1)) \\otimes K(\\mathbb{Z},1).$" + ] + }, + { + "cell_type": "code", + "execution_count": 38, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K35 Morphism (degree 0): K27 -> K1]" + ] + }, + "execution_count": 38, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def left (cmps nabla nabla-t-1))" + ] + }, + { + "cell_type": "code", + "execution_count": 39, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K37 Morphism (degree 0): K27 -> K1]" + ] + }, + "execution_count": 39, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def right (i-cmps nabla 1-t-nabla assoc1))" + ] + }, + { + "cell_type": "code", + "execution_count": 40, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K38 Morphism (degree 0): K27 -> K1]" + ] + }, + "execution_count": 40, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def zero (sbtr left right))" + ] + }, + { + "cell_type": "code", + "execution_count": 41, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 7}\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 41, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? zero 7 (tnpr 4 (tnpr 2 '(1 2) 2 '(2 3)) 3 '(4 5 6)))" + ] + }, + { + "cell_type": "code", + "execution_count": 42, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 9}\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 42, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? zero 9 (tnpr 7 (tnpr 3 '(2 3 4) 4 '(5 6 7 8)) 2 '(11 12)))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "The resulting combination of the rather simple double tensor product has yet 140 terms:" + ] + }, + { + "cell_type": "code", + "execution_count": 43, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 7}\n", + "<1 * (1 2 3 2 4 5 6)>\n", + "<-1 * (1 2 3 4 2 5 6)>\n", + "<1 * (1 2 3 4 5 2 6)>\n", + "<-1 * (1 2 3 4 5 6 2)>\n", + "<1 * (1 2 4 3 2 5 6)>\n", + "<-1 * (1 2 4 3 5 2 6)>\n", + "<1 * (1 2 4 3 5 6 2)>\n", + "<1 * (1 2 4 5 3 2 6)>\n", + "<-1 * (1 2 4 5 3 6 2)>\n", + "<1 * (1 2 4 5 6 3 2)>\n", + "... ...\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 43, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? right 7 (tnpr 4 (tnpr 2 '(1 2) 2 '(2 3)) 3 '(4 5 6)))" + ] + }, + { + "cell_type": "code", + "execution_count": 44, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "140" + ] + }, + "execution_count": 44, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(length (cmbn-list *))" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Coming back to the Bar of an algebra\n", + "\n", + "The existence of a canonical product for the underlying algebra of a simplicial group allows us now to generate the bar of this algebra. Let us use again $K(\\mathbb{Z}, 1).$ We begin to test separately the vertical and the horizontal differential." + ] + }, + { + "cell_type": "code", + "execution_count": 45, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K1 Abelian-Simplicial-Group]" + ] + }, + "execution_count": 45, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def kz1 (k-z-1))" + ] + }, + { + "cell_type": "code", + "execution_count": 46, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN -1}\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 46, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def d-vert (bar-intr-vrtc-dffr (dffr kz1)))\n", + "(funcall d-vert 0 (abar))" + ] + }, + { + "cell_type": "code", + "execution_count": 47, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "<>" + ] + }, + "execution_count": 47, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def abar0 (abar 3 '(1 2) 3 '(-1 -2) 3 '(3 4)))" + ] + }, + { + "cell_type": "code", + "execution_count": 48, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 8}\n", + "<-1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 48, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(funcall d-vert 9 abar0)" + ] + }, + { + "cell_type": "code", + "execution_count": 49, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 8}\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<-1 * <>>\n", + "... ...\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 49, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def d-hrz (bar-intr-hrzn-dffr (aprd kz1)))\n", + "(funcall d-hrz 9 abar0)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "We may now construct the bar chain complex of the algebra `kz1` and test the differential, sum of both previous ones:" + ] + }, + { + "cell_type": "code", + "execution_count": 50, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "[K42 Chain-Complex]" + ] + }, + "execution_count": 50, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def bara (bar kz1))" + ] + }, + { + "cell_type": "code", + "execution_count": 51, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 8}\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<-1 * <>>\n", + "... ...\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 51, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? bara 9 abar0)" + ] + }, + { + "cell_type": "code", + "execution_count": 52, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 7}\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 52, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? bara *)" + ] + }, + { + "cell_type": "code", + "execution_count": 53, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "<>" + ] + }, + "execution_count": 53, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(def abar1 (abar '(5 (7 11 4 -9) 4 (8 3 2) 5 (-1 -2 -3 -4))))" + ] + }, + { + "cell_type": "code", + "execution_count": 54, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 13}\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "<-1 * <>>\n", + "<1 * <>>\n", + "<-1 * <>>\n", + "... ...\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 54, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? bara 14 abar1)" + ] + }, + { + "cell_type": "code", + "execution_count": 55, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "84" + ] + }, + "execution_count": 55, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(length (cmbn-list *))" + ] + }, + { + "cell_type": "code", + "execution_count": 56, + "metadata": { + "collapsed": false + }, + "outputs": [ + { + "data": { + "text/plain": [ + "\n", + "----------------------------------------------------------------------{CMBN 12}\n", + "------------------------------------------------------------------------------\n" + ] + }, + "execution_count": 56, + "metadata": {}, + "output_type": "execute_result" + } + ], + "source": [ + "(? bara **)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "### Lisp files concerned in this chapter\n", + "\n", + "`simplicial-groups.lisp`, `k-pi-n.lisp`.\n", + "\n", + "`classes.lisp`, `macros.lisp`, `various.lisp`]." ] } ],