mmarco/kenzo
1
0
Fork 0
Code Issues Pull Requests Releases Wiki Activity

Formatting updates.

Browse Source
This commit is contained in:
gheber 2016-02-23 19:04:46 -06:00
parent 5d8815af23
commit c435129a73
1 changed files with 28 additions and 29 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

57
examples/Overview.ipynb
View File

@ -11,13 +11,13 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"In this overview we try to show, without entering in detailed explanations, various possibilities of $\\texttt{Kenzo}$ in algebraic topology.\n",
"In this overview we try to show, without entering in detailed explanations, various possibilities of ``Kenzo`` in algebraic topology.\n",
"\n",
"The material is arranged in a sequence of cells. Cells marked with $\\texttt{In [*]}$ represent (editable) input expressions, which in a traditional REPL (Read-Evaluate-Print Loop) would be typed by the user. Cells marked with $\\texttt{Out [*]}$ represent the value of the correpsponding input expression. Note that if an input expression was not evaluated previously, the ouput cell might be missing. Finally, there might be additional, non-editable, diagnostic output printed between an input and output cell. This is a side-effect of the input form evaluation and can be ignored on first reading.\n",
"The material is arranged in a sequence of cells. Cells marked with ``In [*]`` represent (editable) input expressions, which in a traditional REPL (Read-Evaluate-Print Loop) would be typed by the user. Cells marked with ``Out [*]`` represent the value of the correpsponding input expression. Note that if an input expression was not evaluated previously, the ouput cell might be missing. Finally, there might be additional, non-editable, diagnostic output printed between an input and output cell. This is a side-effect of the input form evaluation and can be ignored on first reading.\n",
"\n",
"### Getting off the Ground\n",
"\n",
"Load the $\\texttt{Kenzo}$ software by evaluating the follwing expression. When loaded successfully, the value of this form is a single string atom list $(\\texttt{\"kenzo\"})$."
"Load the ``Kenzo`` software by evaluating the follwing expression. When loaded successfully, the value of this form is a single string atom list ``\"kenzo\"``."
]
},
{
@ -35,7 +35,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The main functionality of the $\\texttt{Kenzo}$ software is defined in a Lisp package called $\\texttt{cat}$. Initialize the software by calling the $\\texttt{cat-init}$ function of the $\\texttt{cat}$ package. Upon successful initialization this form evaluates to $\\texttt{NIL}$."
"The main functionality of the ``Kenzo`` software is defined in a Lisp package called ``cat``. Initialize the software by calling the ``cat-init`` function of the ``cat`` package. Upon successful initialization this form evaluates to ``NIL``."
]
},
{
@ -64,7 +64,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"You can print the $\\texttt{Kenzo}$ version by calling the $\\texttt{kenzo-version}$ function:"
"You can print the ``Kenzo`` version by calling the ``kenzo-version`` function:"
]
},
{
@ -82,9 +82,9 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"### $\\texttt{Kenzo}$ in Action\n",
"### ``Kenzo`` in Action\n",
"\n",
"Let us begin by the space $\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z}, 3)$ described as a simplicial set having only three non-degenerate simplices, namely in dimension $0$, $3$ and $4$. In the representation created by the software, the $0$-simplex (base point), the $3$-simplex and the $4$-simplex are respectively labelled $\\texttt{\"*\"}$, $\\texttt{M3}$ and $\\texttt{N4}$. Two faces of the $4$-simplex $\\texttt{N4}$ are identified with the $3$-simplex $\\texttt{M3}$, the others being contracted on the base point. To create the simplicial set one types simply: "
"Let us begin by the space $\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z}, 3)$ described as a simplicial set having only three non-degenerate simplices, namely in dimension $0$, $3$ and $4$. In the representation created by the software, the 0-simplex (base point), the 3-simplex and the 4-simplex are respectively labelled ``\"*\"``, ``M3`` and ``N4``. Two faces of the 4-simplex ``N4`` are identified with the 3-simplex ``M3``, the others being contracted on the base point. To create the simplicial set one types simply: "
]
},
{
@ -102,7 +102,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The system answers that a $\\texttt{Kenzo}$ object has been created, with number $1$ and type $\\texttt{SIMPLICIAL SET}$. This object may be referenced by the symbol $\\texttt{m23}$."
"The system answers that a ``Kenzo`` object has been created, with number 1 and type ``SIMPLICIAL SET``. This object may be referenced by the symbol ``m23``."
]
},
{
@ -120,7 +120,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"We may compute the homology groups of this space, using the underlying chain complex induced by the simplicial set description. Here we compute the $H_i$ from $0$ to $4$ included. When in the answer the component part is void, it means that the corresponding homology group is null."
"We may compute the homology groups of this space, using the underlying chain complex induced by the simplicial set description. Here we compute the $H_i$ from 0 to 4 included. When in the answer the component part is void, it means that the corresponding homology group is null."
]
},
{
@ -138,7 +138,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"As $\\texttt{m23}$ is a simplicial set, it is possible to create the cartesian product $\\texttt{m23}\\times\\texttt{m23}$ by the function $\\texttt{crts-prdc}$. This is a new simplicial set."
"As ``m23`` is a simplicial set, it is possible to create the cartesian product $\\texttt{m23}\\times\\texttt{m23}$ by the function ``crts-prdc``. This is a new simplicial set."
]
},
{
@ -185,9 +185,9 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"As shown by the last command, the number of elements of the basis is quite large (230). The user will note that the basis elements are formed by cartesian products of *degenerated simplices*. In the list, an element like $\\texttt{<CrPr 1-0 N4 5-3-2 M3>}$ means $\\eta_1\\eta_0\\mathrm{N4}\\times\\eta_5\\eta_3\\eta_2\\mathrm{M3}$.\n",
"As shown by the last command, the number of elements of the basis is quite large (230). The user will note that the basis elements are formed by cartesian products of *degenerated simplices*. In the list, an element like ``<CrPr 1-0 N4 5-3-2 M3>`` means $\\eta_1\\eta_0\\mathrm{N4}\\times\\eta_5\\eta_3\\eta_2\\mathrm{M3}$.\n",
"\n",
"We may construct also the tensor product $\\mathrm{m23}\\otimes\\mathrm{m23}$ from the underlying chain complex of the simplicial set $\\texttt{m23}$. This tensor product is a new chain complex and we see that the basis in dimension $6$ has only one element:"
"We may construct also the tensor product $\\mathrm{m23}\\otimes\\mathrm{m23}$ from the underlying chain complex of the simplicial set ``m23``. This tensor product is a new chain complex and we see that the basis in dimension 6 has only one element:"
]
},
{
@ -216,7 +216,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian product space: as chain complexes, $\\texttt{m23xm23}$ and $\\texttt{t2m23}$ have the same homology groups, but the computations using the tensor product are considerably faster."
"The Eilenberg-Zilber theorem is used to compute the homology groups of the cartesian product space: as chain complexes, $\\texttt{m23xm23}$ and ``t2m23`` have the same homology groups, but the computations using the tensor product are considerably faster."
]
},
{
@ -234,9 +234,8 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us consider now the space $\\mathrm{K}(\\mathbb{Z}, 1)$. This is an Abelian simplicial group created in $\\texttt{Kenzo}$ by the function $\\texttt{k-z}$. In this simplicial group, a simplex in dimension $n$ is mathematically represented by a sequence of integers, known as a *bar* object: $$ [a_1 \\mid a_2 \\mid \\ldots \\mid a_n].$$\n",
"In $\\texttt{Kenzo}$, a non-degenerate simplex of $\\mathrm{K}(\\mathbb{Z}, 1)$ in dimension $n$ will be simply a list of $n$ non-null integers, for instance: $\\texttt{(2 3 4 5)}$. In dimension $0$, the only \n",
"simplex is $\\texttt{NIL}$ (the base point)."
"Let us consider now the space $\\mathrm{K}(\\mathbb{Z}, 1)$. This is an Abelian simplicial group created in ``Kenzo`` by the function ``k-z``. In this simplicial group, a simplex in dimension $n$ is mathematically represented by a sequence of integers, known as a *bar* object: $$ [a_1 \\mid a_2 \\mid \\ldots \\mid a_n].$$\n",
"In ``Kenzo``, a non-degenerate simplex of $\\mathrm{K}(\\mathbb{Z}, 1)$ in dimension $n$ will be simply a list of $n$ non-null integers, for instance: ``(2 3 4 5)``. In dimension 0, the only simplex is ``NIL`` (the base point)."
]
},
{
@ -284,7 +283,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The printed results are the printed representation of *combinations*, i.e., integer linear combinations of generators resulting from the application of the morphisms. The degree of the combination is indicated by the information: $\\texttt{CMBN n}$. \n",
"The printed results are the printed representation of *combinations*, i.e., integer linear combinations of generators resulting from the application of the morphisms. The degree of the combination is indicated by the information: ``CMBN n``. \n",
"\n",
"In the same way, we may create the Abelian simplicial groups $\\mathrm{K}(\\mathbb{Z}/2\\mathbb{Z}, n)$:"
]
@ -316,7 +315,7 @@
"metadata": {},
"source": [
"Let us play now with the sphere $S^3$ and its loop spaces. $S^3$ and $\\Omega^2 S^3$ are created by \n",
"respective calls to the functions $\\texttt{sphere}$ and $\\texttt{loop-space}$. Then we compute the $H_4$ and $H_5$ of $\\Omega^2 S^3$:"
"respective calls to the functions ``sphere`` and ``loop-space``. Then we compute the $H_4$ and $H_5$ of $\\Omega^2 S^3$:"
]
},
{
@ -374,7 +373,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"In the following instruction, we locate in the symbol $\\texttt{L1}$ the canonical generator of $\\pi_2 (\\Omega^1S^3)$, that is the $2$-simplex coming from the original sphere. In fact, the object created by the command $\\texttt{(loop3 0 's3 1)}$ is the \"word\" $S3^1$ belonging to the Kan simplicial version $G(S^3)$ (a simplicial group) of the loop space $\\Omega S^3$."
"In the following instruction, we locate in the symbol ``L1`` the canonical generator of $\\pi_2 (\\Omega^1S^3)$, that is the 2-simplex coming from the original sphere. In fact, the object created by the command $\\texttt{(loop3 0 's3 1)}$ is the \"word\" $S3^1$ belonging to the Kan simplicial version $G(S^3)$ (a simplicial group) of the loop space $\\Omega S^3$."
]
},
{
@ -392,7 +391,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us consider also the $2$-degeneracy of the base point of the loop space. In the printed result, the user will recognize the degeneracy $\\eta_1\\eta_0$ of the null loop, base point of $\\Omega^1 S^3$: "
"Let us consider also the 2-degeneracy of the base point of the loop space. In the printed result, the user will recognize the degeneracy $\\eta_1\\eta_0$ of the null loop, base point of $\\Omega^1 S^3$: "
]
},
{
@ -410,7 +409,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"We may build now a new space by pasting a disk $D3$ as indicated by the following call. It means that we \"paste\" to the space $\\texttt{os3}$ a $3$-simplex named $\\texttt{D3}$, the attaching map being described by the list of its faces in dimension $2$."
"We may build now a new space by pasting a disk $D3$ as indicated by the following call. It means that we \"paste\" to the space ``os3`` a 3-simplex named ``D3``, the attaching map being described by the list of its faces in dimension 2."
]
},
{
@ -429,7 +428,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us compute a few homology groups of the new space $\\texttt{dos3}$:"
"Let us compute a few homology groups of the new space ``dos3``:"
]
},
{
@ -447,7 +446,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"But more interesting, let us build the loop space of the object $\\texttt{dos3}$ and let us compute the homology in dimension $5$:"
"But more interesting, let us build the loop space of the object ``dos3`` and let us compute the homology in dimension 5:"
]
},
{
@ -556,7 +555,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"We see that the result is a linear combination of words composed from degeneracies of $\\texttt{L2}$. The following instruction selects the generator part of the \n",
"We see that the result is a linear combination of words composed from degeneracies of ``L2``. The following instruction selects the generator part of the \n",
"second element of the previous combination."
]
},
@ -575,7 +574,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us use the lisp function $\\texttt{mapcar}$ (one among the various iteration functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object $\\texttt{L4}$, this list is a \"Kan hat\"."
"Let us use the lisp function ``mapcar`` (one among the various iteration functions of Lisp) to create the list of the faces 1, 2, 3 and 4 of the object ``L4``, this list is a \"Kan hat\"."
]
},
{
@ -593,7 +592,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"The function $\\texttt{kfll}$ tries to find a filling of this \"Kan hat\", and we see that the face $2$ of the resulting simplex (which is very different from $\\texttt{L4}$) is the same as the face $2$ of $\\texttt{L4}$."
"The function ``kfll`` tries to find a filling of this \"Kan hat\", and we see that the face 2 of the resulting simplex (which is very different from ``L4``) is the same as the face 2 of ``L4``."
]
},
{
@ -633,7 +632,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Let ${\\cal G}$ be a simplicial group $0$-reduced. $\\Omega S^3$ is such a group. The software $\\texttt{Kenzo}$ allows the construction of the universal bundle $\\bar{\\cal W}{\\cal G}$, i.e. the classifying space of ${\\cal G}$. In our case, as $\\Omega S^3$ in non-Abelian, the result is not a simplicial group but only a simplicial set. We verify that the $H_4$ is null."
"Let ${\\cal G}$ be a simplicial group 0-reduced. $\\Omega S^3$ is such a group. The software ``Kenzo`` allows the construction of the universal bundle $\\bar{\\cal W}{\\cal G}$, i.e. the classifying space of ${\\cal G}$. In our case, as $\\Omega S^3$ in non-Abelian, the result is not a simplicial group but only a simplicial set. We verify that the $H_4$ is null."
]
},
{
@ -673,7 +672,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Let us end this short overview with an example of computation of homotopy groups. The method used in $\\texttt{Kenzo}$ is the Whitehead tower. In this current version only the case where the first non-null homology group (in non-null dimension) is $\\mathbb{Z}$ or $\\mathbb{Z}/{2\\mathbb{Z}}$ can be processed; however if this homology group is a direct sum of several copies of $\\mathbb{Z}$ or $\\mathbb{Z}/{2 \\mathbb{Z}}$, then the corresponding stage of the Whitehead tower may also be constructed step by step.\n",
"Let us end this short overview with an example of computation of homotopy groups. The method used in ``Kenzo`` is the Whitehead tower. In this current version only the case where the first non-null homology group (in non-null dimension) is $\\mathbb{Z}$ or $\\mathbb{Z}/{2\\mathbb{Z}}$ can be processed; however if this homology group is a direct sum of several copies of $\\mathbb{Z}$ or $\\mathbb{Z}/{2 \\mathbb{Z}}$, then the corresponding stage of the Whitehead tower may also be constructed step by step.\n",
"\n",
"We take again $\\mathrm{Moore}(\\mathbb{Z}/2\\mathbb{Z}, 3)$ whose $H_3$ is $\\mathbb{Z}/2\\mathbb{Z}$. First the fundamental cohomology class is constructed:"
]
@ -693,7 +692,7 @@
"cell_type": "markdown",
"metadata": {},
"source": [
"Then the function $\\texttt{z2-whitehead}$ is called to build a fibration over the simplicial set $\\texttt{m23}$ canonically associated to the cohomology class $\\texttt{ch3}$."
"Then the function ``z2-whitehead`` is called to build a fibration over the simplicial set ``m23`` canonically associated to the cohomology class ``ch3``."
]
},
{