From bd3b3e8a510b1b2f0a738636de8a1b2f1b379702 Mon Sep 17 00:00:00 2001 From: gheber Date: Sun, 10 Apr 2016 16:57:08 -0500 Subject: [PATCH] Added Francis' changes. --- examples/07_Simplicial_Sets.ipynb | 6 +- examples/08_Simplicial_morphisms.ipynb | 140 ++----------------------- 2 files changed, 15 insertions(+), 131 deletions(-) diff --git a/examples/07_Simplicial_Sets.ipynb b/examples/07_Simplicial_Sets.ipynb index b16f2d1..0489d7a 100644 --- a/examples/07_Simplicial_Sets.ipynb +++ b/examples/07_Simplicial_Sets.ipynb @@ -84,7 +84,9 @@ "For instance, if $\\sigma$ is a non-degenerate simplex, and $\\sigma'$ is the degenerate simplex $\\eta_3\\eta_1\\sigma$, the corresponding abstract simplices are respectively $\\lbrace \\emptyset, \\sigma \\rbrace$ and $\\lbrace \\eta_3\\eta_1, \\sigma \\rbrace$.\n", "\n", "An abstract simplex is represented internally in the system by the following lisp object:\n", - "$$\\texttt{#S(ABSM}\\quad\\texttt{:DGOP}\\quad\\mathit{n}\\quad\\texttt{:GMSM}\\quad\\mathit{any}\\texttt{)}$$\n", + "```commonlisp\n", + "#S(ABSM :DGOP n :GMSM any)\n", + "```\n", "where,\n", "\n", "1. *dgop* is a non-negative integer coding a *strictly decreasing integer list*. This strictly decreasing list of integers represents a sequence of $\\eta$ operators and is coded as a **unique** integer as follows. The iterated degeneracy operator $\\eta_{i_k} \\ldots \\eta_{i_1}$ is coded as the integer $\\sum_{j = 1}^k 2^{i_j}$; in other words we use a binary number where the binary digits 1 correspond to the indices of the $\\eta_i$'s. This is made possible by the \\emph{strictly} decreasing property of the indices. For example the degeneracy operator $\\eta_6 \\eta 3 \\eta_0$ is coded by the integer 73, for $73 = 2^6 + 2^3 + 2^0$. This leads to a compact coding, replacing the numerous operations about degeneracy operators by efficient handlings of bits of fixnums. In particular, 0 codes the void composition of degeneracy operator, that is, the identity operator. Two functions are provided to help the user for the transformation in both directions.\n", @@ -381,7 +383,7 @@ "\n", "```commonlisp\n", "(build-smst :cmpr cmpr :basis basis :bspn bspn :face face\n", - " :face* face* :intr-bndr intr-bndr bndr-strt bndr-strt\n", + " :face* face* :intr-bndr intr-bndr :bndr-strt bndr-strt\n", " :intr-dgnl intr-dgnl :dgnl-strt dgnl-strt :orgn orgn\n", "```\n", "\n", diff --git a/examples/08_Simplicial_morphisms.ipynb b/examples/08_Simplicial_morphisms.ipynb index f50cf55..081a8b8 100644 --- a/examples/08_Simplicial_morphisms.ipynb +++ b/examples/08_Simplicial_morphisms.ipynb @@ -6,7 +6,7 @@ "source": [ "# Simplicial morphisms\n", "\n", - "The software `Kenzo` implements simplicial morphisms in a way analogous to chain complex morphisms.\n", + "The program `Kenzo` implements simplicial morphisms in a way analogous to chain complex morphisms.\n", "\n", "## Representation of a simplicial morphism\n", "\n", @@ -20,20 +20,22 @@ "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", + "A printing method has been associated to the class `SIMPLICIAL-MRPH`. A string like `[Simplicial-Morphism K`*s* `-> K`*t*`]` is the external representation of an instance of such a class, where *s* (resp. *t*) is the identification number of the source (resp. target) simplicial set. \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", + "To facilitate the construction of instances of the class `SIMPLICIAL-MRPH` and to free the user to call the standard constructor `make-instance`, the program provides the function\n", "\n", - "> `build-smmr :sorc` *sorc* `:trgt` *trgt* `:degr` *degr* `:sintr` *sintr* `:intr` *intr* `:strt` *strt* `:orgn` *orgn*\n", + "```commonlisp\n", + "build-smmr :sorc sorc :trgt trgt :degr degr :sintr sintr :intr intr :strt strt :orgn orgn\n", + "```\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", + "- *sintr*, the internal lisp function defining the effective mapping between simplicial sets. This argument is mandatory. 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", @@ -42,16 +44,12 @@ "\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." + "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." ] }, { @@ -88,7 +86,7 @@ "output_type": "stream", "text": [ "SIMPLE-STYLE-WARNING: \n", - " #\n" + " #\n" ] }, { @@ -184,7 +182,7 @@ "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." + "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, except in degree 0." ] }, { @@ -199,7 +197,7 @@ "output_type": "stream", "text": [ "SIMPLE-STYLE-WARNING: \n", - " #\n" + " #\n" ] }, { @@ -334,122 +332,6 @@ "(? 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": 12, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "name": "stderr", - "output_type": "stream", - "text": [ - "SIMPLE-STYLE-WARNING: \n", - " #\n" - ] - }, - { - "data": { - "text/plain": [ - "[K8 Simplicial-Morphism K1 -> K1]" - ] - }, - "execution_count": 12, - "metadata": {}, - "output_type": "execute_result" - } - ], - "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": 13, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "" - ] - }, - "execution_count": 13, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(? sm3 3 15)" - ] - }, - { - "cell_type": "code", - "execution_count": 14, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "\n", - "----------------------------------------------------------------------{CMBN 3}\n", - "<-7 * 15>\n", - "------------------------------------------------------------------------------\n" - ] - }, - "execution_count": 14, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(? sm3 (cmbn 3 7 15))" - ] - }, - { - "cell_type": "code", - "execution_count": 15, - "metadata": { - "collapsed": false - }, - "outputs": [ - { - "data": { - "text/plain": [ - "\n", - "----------------------------------------------------------------------{CMBN 3}\n", - "<-11 * 15>\n", - "------------------------------------------------------------------------------\n" - ] - }, - "execution_count": 15, - "metadata": {}, - "output_type": "execute_result" - } - ], - "source": [ - "(? sm3 (cmbn 3 11 (dlop-ext-int '(0 1 2 3))))" - ] - }, { "cell_type": "markdown", "metadata": {},