{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Programming the Kan theory\n", "\n", "## Introduction\n", "\n", "Let us recall some definitions in relation with the Kan simplicial sets.\n", "\n", "**Definition 1.** A Kan \"hat\" of dimension $n$ and index $i$ is a collection of $n$ simplices of dimension $(n-1)$, noted $(\\sigma_0, \\ldots, \\sigma_{i-1}, \\sigma_{i+1}, \\ldots, \\sigma_n)$, such that the following conditions are fulfilled:\n", "$$ \\partial_j\\sigma_k=\\partial_k\\sigma_{j+1},\\quad k \\not= i,\\quad j+1 \\not= i, \\quad 0 \\leq j \\leq n-1, \\quad 0 \\leq k \\leq n-1, \\quad k \\leq j. $$\n", "Note that there is no $\\sigma_i$ term in the list above; in a sense, the Kan process consists in constructing the missing $\\sigma_i$ to be considered as a \"composition\" of the given $\\sigma_j$'s.\n", "\n", "### Example\n", "\n", "We are going to work with the simplices of $\\Delta^\\mathbb{N}$. The set $(\\sigma_1, \\sigma_2) = $ `((0 2), (0 1))` is a Kan hat of dimension 2 and index 0. In effect, $\\delta_1\\sigma_1=\\delta_1\\sigma_2 =$ `(0)`. Likewise, the two sets `((1 2), (0 1))` and `((1 2), (0 2))` are Kan hats of dimension 2 and of respective index 1 and 2. Of course, this definition applies equally if the simplices are degenerate.\n", "\n", "**Definition 2.** A *filling* of a Kan hat $(\\sigma_0, \\ldots, \\sigma_{i-1}, \\sigma_{i+1}, \\ldots, \\sigma_n)$, is an $n$-simplex $\\tilde{\\sigma}$, such that for every $j \\not= i$, $\\partial_j\\tilde{\\sigma}=\\sigma_j.$\n", "\n", "### Example\n", "\n", "A filling of `((0 2), (0 1))` is the 2-simplex `(0 1 2)`. A filling of the Kan hat `((1 1), (0 1))` is the degenerate 2-simplex `(0 1 1)`. A filling of the vertex `(0)` is the degenerate 1-simplex `(0 0)`.\n", "\n", "**Definition 3.** A simplicial set is a Kan simplicial set if for every hat there exists a filling.\n", "\n", "### Representation of a Kan simplicial set\n", "\n", "A Kan simplicial set is implemented as an instance of the class `KAN`, subclass of the class `SIMPLICIAL-SET`.\n", "\n", "```commonlisp\n", "(DEFCLASS KAN (simplicial-set)\n", " ((kfll :type kfll :initarg :kfll :reader kfll1)))\n", "```\n", "\n", "We recall that this new class inherits also from the class `CHAIN-COMPLEX`. It has one slot of its own:\n", "\n", "`kfll`\n", "> an object of type `KFLL`, in fact a lisp function with 3 parameters describing a Kan hat: an index (an integer), a dimension (an integer) and the Kan hat (a list of abstract simplices). This function must return a filling of the Kan hat argument, i.e. an abstract simplex satisfying the theoritical definition.\n", "\n", "A printing method has been associated to the class `KAN` and the external representation of an instance is a string like `[K`*n* `Kan-Simplicial-Set]`, where *n* is the number plate of the Kenzo object.\n", "\n", "### Helpful functions on Kan simplicial sets\n", "\n", "`cat-init` *[Function]*\n", "> Clear in particular `*Kan-list*`, the list of user created Kan simplicial sets and reset the global counter to 1.\n", "\n", "`Kan` *n* *[Function]*\n", "> Retrieve in the list `*Kan-list*` the Kan object instance whose the Kenzo identification is n. If it does not exist, return `NIL`.\n", "\n", "`kfll &rest` *args* *[Macro]*\n", "> With only one argument (a Kan instance) return the slot `kfll` of this instance. With 4 arguments\n", "like `(kfll` *kan indx dmns hat*`)`, return a filling of the Kan hat *hat*, of dimension *dmns*\n", "and of index *indx* by applying the filling function value of the slot `kfll` of the Kan instance *kan*.\n", "\n", "`smst-kan` *smst kfll* *[Function]*\n", "> With the filling lisp function *kfll*, **transform** the simplicial set *smst* in an object of type `KAN` (in other word, *smst* is `modified`). This is the easiest way to build a Kan simplicial set.\n", "\n", "`check-hat` *kan indx dmns hat* *[Function]*\n", "> Useful verification function to check if the collection of simplices `hat` is a valid Kan hat of dimension *dmns* and of index *indx* in the Kan simplicial set *kan*. In fact this works equally if *kan* is a general simplicial set.\n", "\n", "`check-kan` *kan indx dmns hat* *[Function]*\n", "> Useful verification function to check if the collection of simplices *hat* is a valid Kan hat of dimension *dmns* and of index *indx* in the Kan simplicial set *kan*. This verification function applies the filling function of the instance *kan* to the argument *hat* and perform the verification of the faces relations upon the resulting $dmns$-simplex.\n", "\n", "### Examples\n", "\n", "Let us take again the small examples of the introduction. First we define a function `dkfll`, \n", "a filling function suitable for a Kan hat in $\\Delta^\\mathbb{N}$. The user will note that in the abstract simplices the degeneracy operators and the geometric simplices are coded in binary." ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "collapsed": false }, "outputs": [ { "name": "stderr", "output_type": "stream", "text": [ "SIMPLE-STYLE-WARNING: \n", " #\n" ] }, { "data": { "text/plain": [ "DKFLL" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(defun dkfll (indx dmns hat)\n", " (cond ((= 1 dmns)\n", " (absm 1 (gmsm (first hat))))\n", " ((= 0 indx)\n", " (let ((del-1 (absm-int-ext (first hat)))\n", " (del-2 (absm-int-ext (second hat))))\n", " (absm-ext-int\n", " (cons (first del-1)\n", " (cons (second del-2) (rest del-1))))))\n", " ((= 1 indx)\n", " (let ((del-0 (absm-int-ext (first hat)))\n", " (del-2 (absm-int-ext (second hat))))\n", " (absm-ext-int\n", " (cons (first del-2) del-0))))\n", " (t\n", " (let ((del-0 (absm-int-ext (first hat)))\n", " (del-1 (absm-int-ext (second hat))))\n", " (absm-ext-int\n", " (cons (first del-1) del-0))))))" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[K1 Simplicial-Set]" ] }, "execution_count": 2, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(def d (delta-infinity))" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "[K1 Kan-Simplicial-Set]" ] }, "execution_count": 3, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(smst-kan d #'dkfll)" ] }, { "cell_type": "code", "execution_count": 4, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(kfll d 0 1 (list (absm 0 1)))" ] }, { "cell_type": "code", "execution_count": 5, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(kfll d 0 2 (list (absm 0 5) (absm 0 3)))" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(kfll d 1 2 (list (absm 0 6) (absm 0 3)))" ] }, { "cell_type": "code", "execution_count": 7, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(kfll d 2 2 (list (absm 0 6) (absm 0 5)))" ] }, { "cell_type": "code", "execution_count": 8, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 8, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(kfll d 0 2 (list (absm 0 3) (absm 0 3)))" ] }, { "cell_type": "code", "execution_count": 9, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "" ] }, "execution_count": 9, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(kfll d 1 2 (list (absm 1 2) (absm 0 3)))" ] }, { "cell_type": "code", "execution_count": 10, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "T" ] }, "execution_count": 10, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(check-hat d 1 2 (list (absm 1 2) (absm 0 3)))" ] }, { "cell_type": "code", "execution_count": 11, "metadata": { "collapsed": false }, "outputs": [ { "data": { "text/plain": [ "T" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(check-hat d 0 1 (list (absm 0 1)))" ] }, { "cell_type": "code", "execution_count": 12, "metadata": { "collapsed": false }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "---done---" ] }, { "data": { "text/plain": [ "NIL" ] }, "execution_count": 12, "metadata": {}, "output_type": "execute_result" } ], "source": [ "(check-kan d 0 1 (list (absm 0 1)))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "More elaborate examples with Kan simplicial sets will be given later in the loop spaces chapter.\n", "\n", "### Lisp files concerned in this chapter\n", "\n", "`kan.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 }