First draft of Chapter2
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"cell_type": "markdown",
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"### 2.5.2 Useful functions and macros for bicones\n",
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"### Useful functions and macros for bicones\n",
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"\n",
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"``(bcnb gnrt)`` *[Macro]*\n",
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"> Build the representation of the generator *gnrt* belonging to the chain complex $B$.\n",
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"\n",
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"``(bcnc gnrt)`` *[Macro]*\n",
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"> Build the representation of the generator *gnrt* belonging to the chain complex $C$.\n",
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"\n",
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"``(bcnd gnrt)`` *[Macro]*\n",
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"> Build the representation of the generator *gnrt* belonging to the chain complex $D$.\n",
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"\n",
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"``(make-bicn-cmbn cmbnb cmbnc cmbnd)`` *[Function]*\n",
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"> Build a bicone combination from the 3 combinations *cmbnb*, *cmbnc* and *cmbnd* belonging respectively and in that order to $B$, $C$ and $D$. The degrees of the combinations *cmbnb* and *cmbnd* must be the same, say $n$. The degree of *cmbnc* must be $n+1$. The degree of the new created combination is $n$.\n",
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"\n",
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"``(bicn-cmbn-cmbnb cmbn)`` *[Function]*\n",
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"> Extract from the bicone combination of degree $n$, *cmbn*, the combination relative to $B$ as a legal combination of degree $n$ in $B$. If there is no $B$-component, return the null combination of degree $n$.\n",
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"\n",
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"``(bicn-cmbn-cmbnc cmbn)`` *[Function]*\n",
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"> Extract from the bicone combination of degree $n$, *cmbn*, the combination relative to $C$ as a legal combination of degree $n+1$ in $C$. If there is no $C$-component, return the null combination of degree $n+1$.\n",
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"\n",
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"``(bicn-cmbn-cmbnd cmbn)`` *[Function]*\n",
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"> Extract from the bicone combination of degree $n$, *cmbn*, the combination relative to $D$ as a legal combination of degree $n$ in $D$. If there is no $D$-component, return the null combination of degree $n$\n",
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"\n",
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"``(dispatch-bicn-cmbn cmbn)`` *[Function]*\n",
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"> Give the 3-values result constitued by the 3 combinations components of the bicone combination *cmbn*. These combinations are valid combinations in their respective chain complexes.\n",
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"\n",
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"#### Examples"
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"cell_type": "markdown",
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"source": [
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"## 2.6 Construction of a bicone from 2 reductions\n",
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"#### Example"
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"## Construction of a bicone from 2 reductions\n",
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"\n",
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"From the slots of the respective chain complexes $B$, $C$ and $D$, it is possible to build the bicone chain complex. The 3 essential functions are the following:\n",
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"\n",
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"``(bicone-cmpr cmprb cmprc cmprd)`` *[Function]*\n",
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"> From the 3 comparison functions *cmprb*, *cmprc* and *cmprd*, build a comparison function adequate to compare the generators as represented in the bicone.\n",
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"\n",
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"``(bicone-basis basisb basisc basisd)`` *[Function]*\n",
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"> From the 3 basis function *basisb*, *basisc* and *basisd*, build a basis function for the bicone. If at least one of the chain complex component of the bicone is *locally effective*, the function returns the symbol ``:locally-effective``.\n",
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"\n",
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"``(bicone-intr-dffr cmprc dffrb dffrc dffrd f1 f2)`` *[Function]*\n",
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"Define the differential in the bicone according to the formula:\n",
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"$$\n",
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"d(cb, cc, cd)= (d_B(cb), f_1(cb) + f_2(cd) - d_C(cc), d_D(cd)),\n",
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"$$\n",
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"where the notation is self explanatory. One sees that one needs the comparison operator *cmprc* of $C$, to order correctly the second combination of the resulting triple.\n",
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"\n",
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"Let us consider now 2 reductions $(\\hat C_1, C_1, f1, g1, h1)$ and $(\\hat C_2, C_2, f2, g2, h2)$\n",
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"in which $C_1 = C_2$. It is possible to define a bicone, with the identification $\\hat C_1=B, C_1=C_2=C, \\hat C_2=D$. This is realised by the following function:\n",
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"\n",
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"``(bicone rdct1 rdct2)`` *[Function]*\n",
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"> Build the bicone chain complex from the objects ot type ``reduction`` *rdct1* and *rdct2*.\n",
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"The bicone is built with the following identification: $B$ is the top chain complex of *rdct1*,\n",
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"$D$ is the top chain complex of *rdct2*. The bottom chain complexes of both *rdct1* and *rdct2*\n",
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"must be the same and $C$ is that chain complex. Of course, both chain morphisms $f_1$ and\n",
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"$f_2$ are respectively the morphism $f$ of the reductions. The construction of the bicone is realised\n",
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"by a call to the building function ``build-chcm`` using the 3 functions ``bicone-cmpr``, ``bicone-basis`` and ``bicone-intr-dffr`` with arguments coming from *rdct1* an *rdct2*. \n",
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"The slot ``bsgn`` (base point) is let undefined and the strategy is by combination.\n",
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"\n",
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"#### Example\n",
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"\n",
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"Let us take again the example with the standard simplex $\\Delta^n$. The following function defines an effective version of ${\\cal C}_*(\\Delta^n)$, so we may ask for the basis in some dimensions."
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@ -1010,6 +1065,14 @@
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"(basis delta3 4)"
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"cell_type": "markdown",
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"source": [
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"Using the function ``make-rdct`` defined in previous example which builds a reduction between \n",
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"$\\Delta^m$ and $\\Delta^n$, we may create a bicone chain complex with the function ``bicone`` and ask for basis in some dimensions."
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"(basis bic 4)"
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"cell_type": "markdown",
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"source": [
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"We verify the validity of the differential in the bicone ``bic``."
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"(? bic (? bic (cmbn 2 3 (bcnb '(0 1 3)) 4 (bcnc '(0 1 2 3)) 5 (bcnd '(0 1 4)))))"
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"(? bic *)"
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"## 2.7 Composition of homotopy equivalences\n",
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"#### Example"
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"## Composition of homotopy equivalences\n",
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"\n",
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"Let us consider 2 homotopy equivalences, as in the following diagram.\n",
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"\n",
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"$$\n",
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"\\begin{align}\n",
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"& \\qquad B & & & & \\qquad D \\\\\n",
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" \\rho_1 \\swarrow\\nearrow & & \\nwarrow\\searrow \\rho_2 & \n",
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"& \\rho'_1\\swarrow\\nearrow & & \\nwarrow\\searrow\\rho'_2 \\\\\n",
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"A \\quad & & C & & C \\quad & & E \\quad\n",
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"\\end{align}\n",
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"$$\n",
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"\n",
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"where the right bottom complex of the first one is the same as the left bottom reduction\n",
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"of the second. We may use then the bicone concept to build a bicone chain complex with $B$, $C$ and $D$, say $BCN$, and finally build a new homotopy equivalence between $A$, $BCN$ and $E$.\n",
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"\n",
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"$$\n",
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"\\begin{align}\n",
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" & \\qquad BCN & \\\\\n",
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" \\rho''_1 \\swarrow\\nearrow & & \\nwarrow\\searrow \\rho''_2 \\\\\n",
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"A \\quad & & E\\quad\n",
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"\\end{align}\n",
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"$$\n",
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"\n",
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"This is realized by the method ``cmps``.\n",
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"\n",
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"``(cmps hmeq1 hmeq2)`` *[Method]*\n",
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"> Build a homotopy equivalence by composition of both homotopy equivalences *hmeq1*, *hmeq2*.\n",
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"The system controls the validity of the composition.\n",
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"\n",
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"### Example\n",
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"\n",
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"Starting with a very simple chain complex and the trivial homotopy equivalence on the chain complex, we verify the correctness of various compositions. The chain complex ``c`` has only one vertex ``(a)`` in every dimension and the differential is the null morphism."
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" :orgn '(c)))"
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"cell_type": "markdown",
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"source": [
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"We build the trivial homotopy equivalence on ``c`` and we compose it with itself."
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"(def h2 (cmps h1 h1))"
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"We verify the coherency of the morphisms in the left and right reduction of ``h2``. The combination ``*tc*`` must belong to a bicone whereas ``*bc*`` belongs to ``c``."
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"(check-rdct-no-wait)"
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"We compose now ``h2`` with itself. We let ``*bc*`` unchanged, but ``*tc*`` is a little more complicated because the chain complexes $B$ and $D$ are themselves bicones."
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"outputs": [],
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"source": [
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"(setf *tc* (cmbn 3 1 (bcnb (bcnb 'a)) 10 (bcnb (bcnc 'a)) 100 (bcnb (bcnd 'a))\n",
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" 1000 (bcnc 'a) 10000 (bcnd (bcnb 'a)) 5234 (bcnd (bcnc 'a))\n",
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" 223 (bcnd (bcnd 'a))))"
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" 1000 (bcnc 'a) 10000 (bcnd (bcnb 'a)) 5234 (bcnd (bcnc 'a))\n",
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" 223 (bcnd (bcnd 'a))))"
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]
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{
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