First draft of Chapter 10.
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examples/Chapter10.ipynb
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examples/Chapter10.ipynb
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"cells": [
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Programming the Kan theory\n",
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"\n",
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"## Introduction\n",
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"\n",
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"Let us recall some definitions in relation with the Kan simplicial sets.\n",
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"\n",
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"**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",
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"$$ \\partial_j\\sigma_k=\\partial_k\\sigma_{j+1},\\quad k \\not= i,\\quad j+1 \\not= i, \\quad 0 \\leq j \\leq n-1, \\quad 0 \\leq k \\leq n-1, \\quad k \\leq j. $$\n",
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"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",
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"\n",
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"### Example\n",
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"\n",
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"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",
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"\n",
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"**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",
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"\n",
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"### Example\n",
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"\n",
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"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",
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"\n",
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"**Definition 3.** A simplicial set is a Kan simplicial set if for every hat there exists a filling.\n",
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"\n",
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"### Representation of a Kan simplicial set\n",
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"\n",
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"A Kan simplicial set is implemented as an instance of the class ``KAN``, subclass of the class ``SIMPLICIAL-SET``.\n",
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"\n",
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"```lisp\n",
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"(DEFCLASS KAN (simplicial-set)\n",
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" ((kfll :type kfll :initarg :kfll :reader kfll1)))\n",
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"```\n",
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"\n",
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"We recall that this new class inherits also from the class ``CHAIN-COMPLEX``. It has one slot of its own:\n",
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"\n",
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"``kfll``\n",
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"> 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",
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"\n",
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"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",
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"\n",
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"### Helpful functions on Kan simplicial sets\n",
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"\n",
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"``cat-init`` *[Function]*\n",
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"> Clear in particular ``*Kan-list*``, the list of user created Kan simplicial sets and reset the global counter to 1.\n",
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"\n",
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"``Kan`` *n* *[Function]*\n",
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"> Retrieve in the list ``*Kan-list*`` the Kan object instance whose the Kenzo identification is n. If it does not exist, return ``NIL``.\n",
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"\n",
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"``kfll &rest`` *args* *[Macro]*\n",
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"> With only one argument (a Kan instance) return the slot ``kfll`` of this instance. With 4 arguments\n",
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"like ``(kfll`` *kan indx dmns hat*``)``, return a filling of the Kan hat *hat*, of dimension *dmns*\n",
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"and of index *indx* by applying the filling function value of the slot ``kfll`` of the Kan instance *kan*.\n",
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"\n",
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"``smst-kan`` *smst kfll* *[Function]*\n",
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"> 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",
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"\n",
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"``check-hat`` *kan indx dmns hat* *[Function]*\n",
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"> 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",
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"\n",
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"``check-kan`` *kan indx dmns hat* *[Function]*\n",
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"> 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",
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"\n",
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"### Examples\n",
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"\n",
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"Let us take again the small examples of the introduction. First we define a function ``dkfll``, \n",
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"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."
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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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(defun dkfll (indx dmns hat)\n",
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" (cond ((= 1 dmns)\n",
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" (absm 1 (gmsm (first hat))))\n",
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" ((= 0 indx)\n",
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" (let ((del-1 (absm-int-ext (first hat)))\n",
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" (del-2 (absm-int-ext (second hat))))\n",
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" (absm-ext-int\n",
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" (cons (first del-1)\n",
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" (cons (second del-2) (rest del-1))))))\n",
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" ((= 1 indx)\n",
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" (let ((del-0 (absm-int-ext (first hat)))\n",
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" (del-2 (absm-int-ext (second hat))))\n",
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" (absm-ext-int\n",
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" (cons (first del-2) del-0))))\n",
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" (t\n",
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" (let ((del-0 (absm-int-ext (first hat)))\n",
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" (del-1 (absm-int-ext (second hat))))\n",
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" (absm-ext-int\n",
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" (cons (first del-1) del-0))))))"
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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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(def d (delta-infinity))"
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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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(smst-kan d #'dkfll)"
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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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(kfll d 0 1 (list (absm 0 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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(kfll d 0 2 (list (absm 0 5) (absm 0 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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(kfll d 1 2 (list (absm 0 6) (absm 0 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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(kfll d 2 2 (list (absm 0 6) (absm 0 5)))"
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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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(kfll d 0 2 (list (absm 0 3) (absm 0 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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(kfll d 1 2 (list (absm 1 2) (absm 0 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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(check-hat d 1 2 (list (absm 1 2) (absm 0 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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(check-hat d 0 1 (list (absm 0 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": null,
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"metadata": {
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"collapsed": false
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},
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"outputs": [],
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"source": [
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"(check-kan d 0 1 (list (absm 0 1)))"
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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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"More elaborate examples with Kan simplicial sets will be given later in the loop spaces chapter.\n",
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"\n",
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"### Lisp files concerned in this chapter\n",
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"\n",
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"``kan.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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