282 lines
9.4 KiB
Plaintext
282 lines
9.4 KiB
Plaintext
{
|
|
"cells": [
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"# Simplicial morphisms\n",
|
|
"\n",
|
|
"The software ``Kenzo`` implements simplicial morphisms in a way analogous to chain complex morphisms.\n",
|
|
"\n",
|
|
"## Representation of a simplicial morphism\n",
|
|
"\n",
|
|
"A simplicial morphism is an instance of the class ``SIMPLICIAL-MRPH``, subclass of the class ``MORPHISM``.\n",
|
|
"\n",
|
|
"```lisp\n",
|
|
"(DEFCLASS SIMPLICIAL-MRPH (morphism)\n",
|
|
" ((sintr :type sintr :initarg :sintr :reader sintr)))\n",
|
|
"```\n",
|
|
"\n",
|
|
"It has one slot of its own:\n",
|
|
"> ``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",
|
|
"\n",
|
|
"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",
|
|
"\n",
|
|
"## The function ``build-smmr``\n",
|
|
"\n",
|
|
"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",
|
|
"\n",
|
|
"> ``build-smmr :sorc`` *sorc* ``:trgt`` *trgt* ``:degr`` *degr* ``:sintr`` *sintr* ``:intr`` *intr* ``:strt`` *strt* ``:orgn`` *orgn*\n",
|
|
"\n",
|
|
"defined with keyword parameters and returning an instance of the class ``SIMPLICIAL-MRPH``. The keyword arguments of ``build-smmr`` are:\n",
|
|
"\n",
|
|
"- *sorc*, the source object, an object of type ``SIMPLICIAL-SET``.\n",
|
|
"- *trgt*, the target object, an object of type ``SIMPLICIAL-SET``.\n",
|
|
"- *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",
|
|
"- *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",
|
|
"- ``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",
|
|
"- *strt*, the strategy, i.e. ``:gnrt`` or ``:cmbn`` attached to the previous function.\n",
|
|
"- *orgn*, a relevant comment list.\n",
|
|
"\n",
|
|
"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",
|
|
"\n",
|
|
"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",
|
|
"\n",
|
|
"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",
|
|
"\n",
|
|
"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",
|
|
"\n",
|
|
"``? &rest`` *args* *[Macro]*\n",
|
|
"> 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",
|
|
"\n",
|
|
"### Examples\n",
|
|
"\n",
|
|
"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."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(def d3 (delta 3))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(def sm1 (build-smmr\n",
|
|
" :sorc d3 :trgt d3 :degr 0\n",
|
|
" :sintr #'(lambda (dmns gmsm)\n",
|
|
" (declare (ignore dmns))\n",
|
|
" (absm 0 gmsm))\n",
|
|
" :orgn '(identity delta-3)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm1 2 7)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm1 1 (absm 1 1))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm1 (cmbn 2 1 7))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"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."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(def sm2 (build-smmr\n",
|
|
" :sorc d3 :trgt d3 :degr 0\n",
|
|
" :sintr #'(lambda (dmns gmsm)\n",
|
|
" (declare (ignore gmsm))\n",
|
|
" (absm (mask dmns) 1)) ;;; mask(n)=2^n - 1 \n",
|
|
" :orgn '(null delta-3)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm2 0 4)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm2 2 7)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm2 1 (absm 1 1))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm2 3 15)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm2 (cmbn 3 2 15))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"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."
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(def sm3 (build-smmr\n",
|
|
" :sorc d3 :trgt d3 :degr 0\n",
|
|
" :sintr #'(lambda (dmns gmsm)\n",
|
|
" (declare (ignore gmsm))\n",
|
|
" (absm (mask dmns) 1))\n",
|
|
" :intr #'cmbn-opps\n",
|
|
" :strt :cmbn\n",
|
|
" :orgn '(null and opposite delta-3)))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm3 3 15)"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm3 (cmbn 3 7 15))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "code",
|
|
"execution_count": null,
|
|
"metadata": {
|
|
"collapsed": false
|
|
},
|
|
"outputs": [],
|
|
"source": [
|
|
"(? sm3 (cmbn 3 11 (dlop-ext-int '(0 1 2 3))))"
|
|
]
|
|
},
|
|
{
|
|
"cell_type": "markdown",
|
|
"metadata": {},
|
|
"source": [
|
|
"### Lisp files concerned in this chapter\n",
|
|
"\n",
|
|
"``simplicial-mrphs.lisp``.\n",
|
|
"\n",
|
|
"[``classes.lisp``, ``macros.lisp``, ``various.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
|
|
}
|