482 lines
13 KiB
Plaintext
482 lines
13 KiB
Plaintext
{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Simplicial morphisms\n",
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"\n",
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"The software `Kenzo` implements simplicial morphisms in a way analogous to chain complex morphisms.\n",
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"\n",
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"## Representation of a simplicial morphism\n",
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"\n",
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"A simplicial morphism is an instance of the class `SIMPLICIAL-MRPH`, subclass of the class `MORPHISM`.\n",
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"\n",
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"```commonlisp\n",
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"(DEFCLASS SIMPLICIAL-MRPH (morphism)\n",
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" ((sintr :type sintr :initarg :sintr :reader sintr)))\n",
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"```\n",
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"\n",
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"It has one slot of its own:\n",
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"> `sintr`, an object of type `SINTR`, in fact a lisp function defining the morphism between the source and target simplicial sets. It must have 2 parameters: a dimension (an integer) and a geometric simplex of this dimension (a generator of any type) . It must return an abstract simplex, image in the target simplicial set of this geometric simplex.\n",
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"\n",
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"A printing method has been associated to the class `SIMPLICIAL-MRPH`. A string like `[K`*n* `Simplicial-Morphism]` is the external representation of an instance of such a class, where *n* is the number plate of the `Kenzo` object.\n",
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"\n",
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"## The function `build-smmr`\n",
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"\n",
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"To facilitate the construction of instances of the class `SIMPLICIAL-MRPH` and to free the user to call the standard constructor `make-instance`, the software provides the function\n",
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"\n",
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"> `build-smmr :sorc` *sorc* `:trgt` *trgt* `:degr` *degr* `:sintr` *sintr* `:intr` *intr* `:strt` *strt* `:orgn` *orgn*\n",
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"\n",
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"defined with keyword parameters and returning an instance of the class `SIMPLICIAL-MRPH`. The keyword arguments of `build-smmr` are:\n",
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"\n",
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"- *sorc*, the source object, an object of type `SIMPLICIAL-SET`.\n",
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"- *trgt*, the target object, an object of type `SIMPLICIAL-SET`.\n",
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"- *degr*, the degree of the morphism, an integer. In this chapter, we consider only the 0 degree case (the usual one). The case -1 is particularly important: it allows to implement the notion of twisting operator defining a fibration (See the chapter Fibration).\n",
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"- *sintr*, the internal lisp function defining the effective mapping between simplicial sets. If the integer *degr* is 0 and if the following keyword argument *intr* is omitted, then the function `build-smmr` builds a lisp function implementing the induced mapping between the underlying source and target chain complexes. This function is installed in the slot `intr`. The strategy is then set to `:gnrt`.\n",
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"- `intr`, a user defined morphism for the underlying chain complexes. This argument is optional and taken in account only if the degree is 0 and in this case, supersedes the previous derived mapping. The strategy is then mandatory. If the degree is not null, the implementor has decided to set the corresponding slot to `NIL`.\n",
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"- *strt*, the strategy, i.e. `:gnrt` or `:cmbn` attached to the previous function.\n",
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"- *orgn*, a relevant comment list.\n",
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"\n",
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"After a call to `build-smmr`, the simplicial morphism instance is added to a list of previously constructed ones (`*smmr-list*`). As the other similar lists, the list `*smmr-list*` may be cleared by the function `cat-init`.\n",
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"\n",
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"The effective application of a simplicial morphism upon arguments, is realized with the macro `?` which calls the adequate method defined for this kind of objects.\n",
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"\n",
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"After a call to `build-smmr`, the simplicial morphism instance is added to a list of previously constructed ones (`*smmr-list*`). As the other similar lists, the list `*smmr-list*` may be cleared by the function `cat-init`.\n",
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"\n",
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"The effective application of a simplicial morphism upon arguments, is realized with the macro `?` which calls the adequate method defined for this kind of objects.\n",
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"\n",
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"`? &rest` *args* *[Macro]*\n",
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"> Versatile macro for applying a simplicial morphism indifferently as `(?` *smmr dmns absm-or-gmsm*`)` or `(?` *smmr cmbn*`)`. In the first case, the third argument is either an abstract simplex or a geometric one. In the second case, if the second argument is a combination, *smmr* is then considered as a chain complex morphism and it is the function in the slot `intr` which is applied, so that it makes sense only if the degree of the mapping is 0.\n",
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"\n",
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"### Examples\n",
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"\n",
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"In the following examples, we work with $\\Delta_3$. We define three simplicial morphisms, `sm1`, `sm2` and `sm3`. In `sm1`, the mapping is the identity mapping and we can see that this identity mapping has been propagated on the underlying chain complex."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[K1 Simplicial-Set]"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(def d3 (delta 3))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"SIMPLE-STYLE-WARNING: \n",
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" #<SB-INT:SIMPLE-STYLE-WARNING \"undefined ~(~A~): ~S\" {10053D8EA3}>\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"[K6 Simplicial-Morphism K1 -> K1]"
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]
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},
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"execution_count": 2,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(def sm1 (build-smmr\n",
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" :sorc d3 :trgt d3 :degr 0\n",
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" :sintr #'(lambda (dmns gmsm)\n",
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" (declare (ignore dmns))\n",
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" (absm 0 gmsm))\n",
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" :orgn '(identity delta-3)))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<AbSm - 7>"
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]
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},
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"execution_count": 3,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm1 2 7)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<AbSm 0 1>"
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]
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},
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm1 1 (absm 1 1))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\n",
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"----------------------------------------------------------------------{CMBN 2}\n",
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"<1 * 7>\n",
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"------------------------------------------------------------------------------\n"
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]
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},
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"execution_count": 5,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm1 (cmbn 2 1 7))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"In the simplicial morphism `sm2`, the mapping is the \"null\" mapping. In fact, it consists in applying any abstract simplex in dimension $n$ upon the $n$-degenerate base point. Of course, in terms of chain complex, the corresponding mapping applies any chain complex element upon the null combination of same degree."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"SIMPLE-STYLE-WARNING: \n",
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" #<SB-INT:SIMPLE-STYLE-WARNING \"undefined ~(~A~): ~S\" {1005794883}>\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"[K7 Simplicial-Morphism K1 -> K1]"
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]
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},
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(def sm2 (build-smmr\n",
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" :sorc d3 :trgt d3 :degr 0\n",
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" :sintr #'(lambda (dmns gmsm)\n",
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" (declare (ignore gmsm))\n",
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" (absm (mask dmns) 1)) ;;; mask(n)=2^n - 1 \n",
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" :orgn '(null delta-3)))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<AbSm - 1>"
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]
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},
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm2 0 4)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<AbSm 1-0 1>"
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]
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},
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"execution_count": 8,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm2 2 7)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<AbSm 0 1>"
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]
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},
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"execution_count": 9,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm2 1 (absm 1 1))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<AbSm 2-1-0 1>"
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]
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},
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm2 3 15)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\n",
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"----------------------------------------------------------------------{CMBN 3}\n",
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"------------------------------------------------------------------------------\n"
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]
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},
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm2 (cmbn 3 2 15))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"In the simplicial morphism `sm3`, we keep the same \"null\" mapping as in `sm2` but we choose as mapping for the underlying chain complex the opposite of any chain complex element."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"name": "stderr",
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"output_type": "stream",
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"text": [
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"SIMPLE-STYLE-WARNING: \n",
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" #<SB-INT:SIMPLE-STYLE-WARNING \"undefined ~(~A~): ~S\" {1005A399D3}>\n"
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]
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},
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{
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"data": {
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"text/plain": [
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"[K8 Simplicial-Morphism K1 -> K1]"
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]
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},
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"execution_count": 12,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(def sm3 (build-smmr\n",
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" :sorc d3 :trgt d3 :degr 0\n",
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" :sintr #'(lambda (dmns gmsm)\n",
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" (declare (ignore gmsm))\n",
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" (absm (mask dmns) 1))\n",
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" :intr #'cmbn-opps\n",
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" :strt :cmbn\n",
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" :orgn '(null and opposite delta-3)))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"<AbSm 2-1-0 1>"
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]
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},
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"execution_count": 13,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm3 3 15)"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\n",
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"----------------------------------------------------------------------{CMBN 3}\n",
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"<-7 * 15>\n",
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"------------------------------------------------------------------------------\n"
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]
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},
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"execution_count": 14,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm3 (cmbn 3 7 15))"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {
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"collapsed": false
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},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"\n",
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"----------------------------------------------------------------------{CMBN 3}\n",
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"<-11 * 15>\n",
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"------------------------------------------------------------------------------\n"
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]
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},
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"(? sm3 (cmbn 3 11 (dlop-ext-int '(0 1 2 3))))"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"### Lisp files concerned in this chapter\n",
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"\n",
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"`simplicial-mrphs.lisp`.\n",
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"\n",
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"[`classes.lisp`, `macros.lisp`, `various.lisp`]."
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]
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "SBCL Lisp",
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"language": "lisp",
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"name": "lisp"
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},
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"language_info": {
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"codemirror_mode": "text/x-common-lisp",
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"mimetype": "text/x-common-lisp",
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"name": "common-lisp",
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"pygments_lexer": "common-lisp",
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"version": "X3J13"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 0
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}
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