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gheber 2016-05-05 09:15:50 -05:00
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examples/12_Fibrations.ipynb
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examples/13_Loop_Spaces.ipynb
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@ -14,15 +14,15 @@
"\n", "\n",
"**Definition.** A word is a *finite* sequence of letters, possibly repeated, without any pair of the form $a_ja_j^{-1}$ and $a_k^{-1}a_k$. The word is said *reduced*.\n", "**Definition.** A word is a *finite* sequence of letters, possibly repeated, without any pair of the form $a_ja_j^{-1}$ and $a_k^{-1}a_k$. The word is said *reduced*.\n",
"\n", "\n",
"Now, let $\\mathbb{Z}^{*{\\cal A}}$ the set of *words* on $\\cal A$. It is easy to show that if we define in $\\mathbb{Z}^{*{\\cal A}}$ an internal law of composition as the formal *concatenation* of two words, on which the cancelation rule is applied, i.e. any pair of the form $a_ja_j^{-1}$ and $a_k^{-1}a_k$ is replaced by the *void sequence*, then $\\mathbb{Z}^{*{\\cal A}}$ is a group (non-commutative in general). The void word is the neutral element for that law. If we consider the concatenation as a *product*, then, as usual in group theory, a sequence of $k$ times the same letter $a$ will be written $a^k$ and a sequence of $k$ times the same letter $a^{-1}$, $a^{-k}$. So a word formed from letters by the concatenation product together with the cancellation rule, will be written under the reduced form:\n", "Now, let $F(\\cal A)$ the set of *words* on $\\cal A$. It is easy to show that if we define in $F(\\cal A)$ an internal law of composition as the formal *concatenation* of two words, on which the cancelation rule is applied, i.e. any pair of the form $a_ja_j^{-1}$ and $a_k^{-1}a_k$ is replaced by the *void sequence*, then $F(\\cal A)$ is a group (non-commutative in general). The void word is the neutral element for that law. If we consider the concatenation as a *product*, then, as usual in group theory, a sequence of $k$ times the same letter $a$ will be written $a^k$ and a sequence of $k$ times the same letter $a^{-1}$, $a^{-k}$. So a word formed from letters by the concatenation product together with the cancellation rule, will be written under the reduced form:\n",
"$$a_1^{n_1}a_2^{n_2}\\ldots a_k^{n_k},\\ n_i\\in \\mathbb{Z}^*,\\ a_i\\in {\\cal A},\\ a_i \\not= a_{i+1}$$\n", "$$a_1^{n_1}a_2^{n_2}\\ldots a_k^{n_k},\\ n_i\\in \\mathbb{Z}^*,\\ a_i\\in {\\cal A},\\ a_i \\not= a_{i+1}$$\n",
"and its inverse is\n", "and its inverse is\n",
"$$a_k^{-n_k}\\ldots a_2^{-n_2}a_1^{-n_1}.$$\n", "$$a_k^{-n_k}\\ldots a_2^{-n_2}a_1^{-n_1}.$$\n",
"After this resume, let $X$ be a reduced simplicial set (i.e. having only one 0-simplex, namely its base point). The Kan simplicial version $GX$ of the loop space $\\Omega(|X|)$ is defined as follows. The set of $n$-simplices $GX_n$ is the free group generated by the $(n+1)$-simplices $\\sigma \\in X_{n+1}$ except those which are $n$-degenerate. In other words:\n", "Now let $X$ be a reduced simplicial set (i.e. having only one 0-simplex, namely its base point). The Kan simplicial version $GX$ of the loop space $\\Omega(|X|)$ is defined as follows. The set of $n$-simplices $GX_n$ is the free group generated by the $(n+1)$-simplices $\\sigma \\in X_{n+1}$ except those which are $n$-degenerate. In other words:\n",
"$$ GX_n=\\mathbb{Z}^{*X^+_{n+1}},\\ X^+_{n+1}=X_{n+1}-\\eta_n X_n. $$\n", "$$ GX_n=F(X^+_{n+1}),\\ X^+_{n+1}=X_{n+1}-\\eta_n X_n. $$\n",
"In fact, from a strictly mathematical point of view, the right definition is \n", "In fact, from a strictly mathematical point of view, the right definition is \n",
"$$ GX_n = \\mathbb{Z}^{*X^+_{n+1}} / {\\overline{\\eta_n X_n}},$$\n", "$$ GX_n = F(X^+_{n+1}) / {\\overline{\\eta_n X_n}},$$\n",
"i.e. the quotient group of $\\mathbb{Z}^{*X^+_{n+1}}$, the free group generated by **all** the $n+1$--simplices, the $n$-degenerate included, by the normal subgroup ${\\overline{\\eta_n X_n}}$ generated by the $n$-degenerate simplices. Both definitions are equivalent, the second is mathematicaly correct but only the first one is visible in the program.\n", "i.e. the quotient group of $F(X^+_{n+1})$, the free group generated by **all** the $n+1$-simplices, the $n$-degenerate ones included, by the normal subgroup ${\\overline{\\eta_n X_n}}$ generated by the $n$-degenerate simplices. Both definitions are equivalent, the second is mathematicaly correct but only the first one is visible in the program.\n",
"\n", "\n",
"For a better understanding, we may distinguish carefully a generator $\\sigma$ belonging to $X_{n+1}$ and the corresponding letter $\\tau (\\sigma) \\in GX_n$. So a word, element of $GX_n$ will be written \n", "For a better understanding, we may distinguish carefully a generator $\\sigma$ belonging to $X_{n+1}$ and the corresponding letter $\\tau (\\sigma) \\in GX_n$. So a word, element of $GX_n$ will be written \n",
"$${\\tau(\\sigma_1)}^{\\epsilon_1} {\\tau(\\sigma_2)}^{\\epsilon_2} \\ldots {\\tau(\\sigma_p)}^{\\epsilon_p}.$$\n", "$${\\tau(\\sigma_1)}^{\\epsilon_1} {\\tau(\\sigma_2)}^{\\epsilon_2} \\ldots {\\tau(\\sigma_p)}^{\\epsilon_p}.$$\n",
@ -36,19 +36,16 @@
"\\overline{\\eta}_i(\\tau(\\sigma)) & = & \\tau(\\eta_i\\sigma),\\qquad 0 \\leq i \\leq n.\n", "\\overline{\\eta}_i(\\tau(\\sigma)) & = & \\tau(\\eta_i\\sigma),\\qquad 0 \\leq i \\leq n.\n",
"\\end{aligned}\n", "\\end{aligned}\n",
"$$\n", "$$\n",
"When applied to words, the two homomorphisms, $\\overline{\\partial}$ and $\\overline{\\eta}$, satisfy (for clarity, we omit here the operator $\\tau$):\n", "\n",
"These definitions for the generators of $GX_n$ extend to arbitrary elements in a unique way, for example:\n",
"$$\n", "$$\n",
"\\begin{aligned}\n", "\\overline{\\partial}_i(a_1^{n_1} \\cdots a_k^{n_k}) = (\\partial_i(a_1))^{n_1} \\cdots (\\partial_i(a_k))^{n_k}\n",
"\\overline{\\partial}(a_1^{n_1}\\ldots a_k^{n_k}) &=& (\\overline{\\partial}(a_1))^{n_1}\\ldots \n",
" (\\overline{\\partial}(a_k))^{n_k}, \\\\\n",
"\\overline{\\eta}(a_1^{n_1}\\ldots a_k^{n_k}) &=& (\\overline{\\eta}(a_1))^{n_1}\\ldots (\\overline{\\eta}(a_k))^{n_k}. \n",
"\\end{aligned}\n",
"$$\n", "$$\n",
"It is known that the $\\overline{\\partial}_i$ and the $\\overline{\\eta}_i$ satisfy the fundamental relations between face and degeneracy operators (see the chapter on simplicial sets). So, $GX$ is itself a simplicial set; each $GX_n$ is a group, each face and degeneracy operator is an homomorphism, so that $GX$ is a *simplicial group*. The importance of the simplicial group $GX$ is emphasized by the Kan theorem, stating that if we consider the realization $|X|$ of $X$ and the loop space $\\Omega(|X|, x_0)$ at the base point $x_0 \\in |X|$, then $\\Omega(|X|, x_0)$ and the realization $|GX|$ of $GX$ have the same homotopy type.\n", "It is known that the $\\overline{\\partial}_i$ and the $\\overline{\\eta}_i$ satisfy the fundamental relations between face and degeneracy operators (see the chapter on simplicial sets). So, $GX$ is itself a simplicial set; each $GX_n$ is a group, each face and degeneracy operator is a homomorphism, so that $GX$ is a *simplicial group*. The importance of the simplicial group $GX$ is emphasized by the Kan theorem, stating that if we consider the realization $|X|$ of $X$ and the loop space $\\Omega(|X|, x_0)$ at the base point $x_0 \\in |X|$, then $\\Omega(|X|, x_0)$ and the realization $|GX|$ of $GX$ have the same homotopy type.\n",
"\n", "\n",
"## Representation of letters and words\n", "## Representation of letters and words\n",
"\n", "\n",
"In the system, the chosen representation for letters and words follows the very definition of the free group $\\mathbb{Z}^{*X_{n+1}^+}$. Here, a letter is a generator of $GX_n$ i.e. an **abstract simplex** $\\sigma$ of $X^+_{n+1}$. A word will be implemented as a sequence of such letters to a certain power (including 1). The chosen terminology is the following:\n", "In the system, the chosen representation for letters and words follows the very definition of the free group $F(X_{n+1}^+)$. Here, a letter is a generator of $GX_n$ i.e. an **abstract simplex** $\\sigma$ of $X^+_{n+1}$. A word will be implemented as a sequence of such letters to a certain power (including 1). The chosen terminology is the following:\n",
"\n", "\n",
"- A term like $\\sigma^p$ is a **power**.\n", "- A term like $\\sigma^p$ is a **power**.\n",
"- A word ${\\sigma_1}^{\\epsilon_1} {\\sigma_2}^{\\epsilon_2} \\ldots {\\sigma_p}^{\\epsilon_p}$ is a **loop**. \n", "- A word ${\\sigma_1}^{\\epsilon_1} {\\sigma_2}^{\\epsilon_2} \\ldots {\\sigma_p}^{\\epsilon_p}$ is a **loop**. \n",
@ -57,14 +54,14 @@
"\n", "\n",
"### Representation of a power\n", "### Representation of a power\n",
"\n", "\n",
"A power is internally represented by lisp object of type `apowr`. This has the form:\n", "A power is internally represented by a lisp object of type `apowr`. This has the form:\n",
"\n", "\n",
"`(`*dgop* *gmsm* `.` *expn*`)`\n", "`(`*dgop* *gmsm* `.` *expn*`)`\n",
"\n", "\n",
"in which we may recognize:\n", "in which we may recognize:\n",
"\n", "\n",
"- the components of the abstract simplex: *dgop* and *gmsm*,\n", "- the components of the abstract simplex: *dgop* and *gmsm*,\n",
"- the exponent of the letter: *expn*, an integer.\n", "- the exponent of the letter: *expn*, a non-zero integer.\n",
"\n", "\n",
"The associated constructor is the macro `apowr`.\n", "The associated constructor is the macro `apowr`.\n",
"\n", "\n",
@ -75,9 +72,9 @@
"\n", "\n",
"A loop is represented by a lisp object of the form:\n", "A loop is represented by a lisp object of the form:\n",
"\n", "\n",
"`(:loop` $apower_1$ ... $apower_k$`)`\n", "`#S(LOOP` $apower_1$ ... $apower_k$`)`\n",
"\n", "\n",
"The corresponding type is `LOOP`. The constructor is the macro `make-loop` but the pratical one is `loop3` and the associated\n", "The corresponding type is `LOOP`. The constructor is the macro `make-loop` but the practical one is `loop3` and the associated\n",
"printing function prints the object under the form:\n", "printing function prints the object under the form:\n",
"\n", "\n",
"`<<Loop [`*ext-dgop1 gmsm1* *expn1*`] ... [`*ext-dgopk gmsmk* *expnk*`]>>`\n", "`<<Loop [`*ext-dgop1 gmsm1* *expn1*`] ... [`*ext-dgopk gmsmk* *expnk*`]>>`\n",
@ -472,9 +469,9 @@
"output_type": "stream", "output_type": "stream",
"text": [ "text": [
"\n", "\n",
"The object is a STANDARD-OBJECT of type CAT::SIMPLICIAL-SET.\n", "The object is a STANDARD-OBJECT of type SIMPLICIAL-SET.\n",
"0. CMPR: #<FUNCTION S-CMPR>\n", "0. CMPR: #<FUNCTION S-CMPR>\n",
"1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {1005901ECB}>\n", "1. BASIS: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-BASIS) {10057E9E6B}>\n",
"2. BSGN: *\n", "2. BSGN: *\n",
"3. DFFR: [K19 Morphism (degree -1): K18 -> K18]\n", "3. DFFR: [K19 Morphism (degree -1): K18 -> K18]\n",
"4. GRMD: [K18 Simplicial-Set]\n", "4. GRMD: [K18 Simplicial-Set]\n",
@ -482,7 +479,7 @@
"6. IDNM: 18\n", "6. IDNM: 18\n",
"7. ORGN: (BUILD-FINITE-SS (* 2 S2 (* * *) 3 S3 (S2 S2 S2 S2)))\n", "7. ORGN: (BUILD-FINITE-SS (* 2 S2 (* * *) 3 S3 (S2 S2 S2 S2)))\n",
"8. CPRD: [K22 Morphism (degree 0): K18 -> K20]\n", "8. CPRD: [K22 Morphism (degree 0): K18 -> K20]\n",
"9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {1005901EEB}>\n", "9. FACE: #<CLOSURE (FLET CAT::RSLT :IN CAT::FINITE-SS-FACE) {10057E9E8B}>\n",
"\n", "\n",
"> \n", "> \n",
"Checking the 0-simplices...\n", "Checking the 0-simplices...\n",