diff --git a/examples/Overview.ipynb b/examples/Overview.ipynb index 90a3c25..1371701 100644 --- a/examples/Overview.ipynb +++ b/examples/Overview.ipynb @@ -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 $\\texttt{}$ 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 ```` 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``." ] }, {