Ticket #368: testpdf.tex

File testpdf.tex, 61.8 KB (added by shmuel, 8 months ago)

LaTeX source for the PDF

Line 
1% This is pdfTeX, Version 3.14159265-2.6-1.40.18 (MiKTeX 2.9.6350)
2% LaTeX2e <2017-04-15>
3
4% Package: amsbsy 1999/11/29 v1.2d Bold Symbols
5% Package: amsfonts 2013/01/14 v3.01 Basic AMSFonts support
6% Package: amsmath 2017/09/02 v2.17a AMS math features
7% Package: amsopn 2016/03/08 v2.02 operator names
8% Package: amstext 2000/06/29 v2.01 AMS text
9% Package: amsthm 2017/10/31 v2.20.4
10% Package: bm 2017/01/16 v1.2c Bold Symbol Support (DPC/FMi)
11% Package: cleveref 2018/03/27 v0.21.4 Intelligent cross-referencing
12% Package: enumitem 2011/09/28 v3.5.2 Customized lists
13% Package: expl3 2018-06-01 L3 programming layer (code)
14% Package: expl3 2018-06-01 L3 programming layer (loader)
15% Package: graphics 2017/06/25 v1.2c Standard LaTeX Graphics (DPC,SPQR)
16% Package: graphicx 2017/06/01 v1.1a Enhanced LaTeX Graphics (DPC,SPQR)
17% Package: hyperref 2018/02/06 v6.86b Hypertext links for LaTeX
18% Package: ifpdf 2017/03/15 v3.2 Provides the ifpdf switch
19% Package: ifthen 2014/09/29 v1.1c Standard LaTeX ifthen package (DPC)
20% Package: keyval 2014/10/28 v1.15 key=value parser (DPC)
21% Package: mathtools 2018/01/08 v1.21 mathematical typesetting tools
22% Package: nameref 2016/05/21 v2.44 Cross-referencing by name of section
23% Package: pgf 2015/08/07 v3.0.1a (rcs-revision 1.15)
24% Package: scalerel 2016/12/29 v1.8 Routines for constrained scaling and stretchi
25% Package: showlabels 2015/12/08 v1.7
26% Package: stix2 2018/04/02 v2.0.0-latex STIX Two fonts support package
27% Package: thmtools 2014/04/21 v66
28% Package: tikz 2015/08/07 v3.0.1a (rcs-revision 1.151)
29% Package: tikz-cd 2014/10/30 v0.9e Commutative diagrams with tikzx
30
31% Package: xparse 2018-05-12 L3 Experimental document command parser
32% Package: xstring 2013/10/13  v1.7c  String manipulations (C Tellechea)
33% Package: xy 2013/10/06 Xy-pic version 3.8.9
34% Version 2
35% 1.  Define \into
36% 2.  Define m-chart at a point and subchart at a point
37% 3.  Change definition of M-atlas morphism
38% 3.  Change definition of Ck-atlas morphism
39
40\documentclass{article}
41\usepackage{amsmath}
42%\usepackage{amssymb}
43\usepackage{amsthm}
44\usepackage{bm}
45\usepackage{enumitem}
46\usepackage{ifthen}
47%\usepackage{mathrsfs}
48\usepackage{mathtools}
49\usepackage{scalerel}
50\usepackage{stix2}[notext,not1] %reqires XeTeX or luaTeX
51\usepackage{thmtools}
52\usepackage{tikz-cd}
53\usepackage{xparse} % loads expl3
54%See interface3.pdf
55\usepackage{xstring}
56
57\usepackage[cmtip,all,barr]{xy}
58%Remove when xybarr.tex bug fixed
59\newdir_{ (}{{ }*!/-.5em/@_{(}}
60
61%Morphisms of category
62\newcommand \Ar {\mathrm{Ar}}
63
64\DeclareMathOperator \arin {\stackrel{\Ar}{\in}}
65
66%Category of atlases from E to C
67%\DeclareMathOperator \Atl {\mathscr{A\!t\!l}}
68\newcommand \Atl [1] []
69   { \ifthenelse {\equal{#1}{}}
70     {\mathscr{A\!t\!l}}
71     {\mathscr{A\!t\!l}^{\mathrm{#1}}}
72   }
73
74\DeclareMathOperator \Atlas {\mathrm{Atlas}}
75
76\newcommand \Bun {\mathrm{Bun}}
77
78\newcommand \BunProd {\mathrm{Bun}-\mathrm{prod}}
79
80\newcommand \C {\mathrm{C}}
81
82% Select font for standard categories
83\newcommand \Cat [1] {\mathbf{#1}}
84
85% Select font for categories and sequences of categories
86\newcommand \catname [1] {\mathscr{#1}}
87\newcommand \catseqname [1] {\bm{\mathscr{#1}}}
88% Can't use \bm without stix and XeTeX
89
90\newcommand \Ck {\mathrm{C^k}}
91
92\newcommand \Classic {\mathrm{Classic}}
93
94\DeclareMathOperator \codomain {\mathrm{codomain}}
95
96% Extended composition operators for function sequences
97\newcommand {\compose} [1] [def]
98   { \ifthenelse {\equal{#1}{def}}
99       {\mathbin \circ}
100       {\mathbin{\overset{#1} {\circ}}}
101   }
102
103% Compose head
104\newcommand {\composeh} [1] [def]
105   { \ifthenelse {\equal{#1}{def}}
106       {\mathbin \odot}
107       {\mathbin{\overset{#1} {\odot}}}
108   }
109
110% Compose tail
111\newcommand {\composet} [1] [def]
112   { \ifthenelse {\equal{#1}{def}}
113       {\mathbin \cdot}
114       {\mathbin{\overset{#1} {\cdot}}}
115   }
116
117\DeclareMathOperator \defeq {\stackrel{\mathrm{def}}{=}}
118
119\DeclareMathOperator \domain {\mathrm{domain}}
120
121\DeclareMathOperator \Domain {\seqname{domain}}
122
123\newcommand \false {\mathrm{False}}
124
125\newcommand \Fib   {\mathrm{Fib}}
126
127\newcommand \full [2] [] {\underset{\mathrm {{#1}full}}{#2}}
128
129\newcommand \fullcref [1]
130   { \ifthenelse {\equal{\nameref{#1}}{}}
131       {\cref{#1}}
132       {\cref{#1} (\nameref{#1})}
133   }
134
135% Select font for functions and sequences of functions
136% \newcommand \funcname [1] {\mathit{#1}}
137% \newcommand \funcseqname [1] {\bm{#1}}
138
139\DeclareMathOperator \Functor {\mathop{\mathscr{F}}}
140
141\DeclareMathOperator \head {\mathrm{head}}
142
143\DeclareMathOperator \Hom {\mathrm{Hom}}
144
145\DeclareMathOperator \Id {\mathrm{Id}}
146
147\DeclareMathOperator \ID {\mathbf{Id}}
148
149\newcommand \into {\negthickspace\mon}
150
151%Propositional function isCk_{f,A}(x,y,...)
152\DeclareMathOperator \isCk {\mathrm{isCk}}
153
154%Propositional function isAtl(A,E,C)
155\newcommand \isAtl [1] []
156   { \ifthenelse {\equal{#1}{}}
157     {\mathrm{isAtl}}
158     {\mathrm{isAtl}^{\mathrm{#1}}}
159   }
160
161%Propositional function isLCS(L, M, A, F, Sigma)
162\DeclareMathOperator \isLCS {\mathrm{isLCS}}
163
164\DeclareMathOperator \iso {\stackrel{\sim}{=}}
165
166% Join two tuples
167\DeclareMathOperator \join {\mathrm{join}}
168
169%Category of M-Sigma local coordinate spaces
170\newcommand \LCS {\mathrm{LCS}}
171
172\DeclareMathOperator \length {\mathrm{length}}
173
174\DeclareMathOperator \lengtho {\mathrm length0}
175
176\newcommand \M {\mathrm{M}}
177
178\newcommand \Man {\mathrm{Man}}
179
180% Adjust : spacing for f maps a to b
181\newcommand \maps {\!\colon}
182
183%Set long names upright, short names italic
184\newcommand{\mathvarname}[1]{%
185  \begingroup\noexpandarg
186  \StrLen{#1}[\temp]%
187  \ifnum\temp>1
188    \mathrm{#1}%
189  \else
190    #1%
191  \fi
192  \endgroup
193}
194
195\newcommand \maxfull [2] [] {\underset{\mathrm {{#1}max-full}}{#2}}
196\newcommand \maximal [2] [] {\underset{\mathrm {{#1}max}}{#2}}
197
198\newcommand \minimal [2] [] {\underset{{#1}\mathrm {min}}{#2}}
199
200%Model category
201\newcommand \Mod {\mathrm{Mod}}
202
203%Morphisms between two objects
204\newcommand \Mor {\mathrm{Mor}}
205
206% Morphism of category
207\DeclareMathOperator \morphin {\stackrel{\Ar}{\in}}
208
209%Object of category
210\newcommand \Ob {\mathrm{Ob}}
211\DeclareMathOperator \objin {\stackrel{\Ob}{\in}}
212
213\newcommand \onto {\epi}
214
215% All non-null open sets
216\newcommand \op [2] [] {\underset{{#1}\mathrm {op}}{#2}}
217
218\newcommand \optriv [2] [] {\underset{{#1}\mathrm {op-triv}}{#2}}
219
220\newcommand \pagecref [1]
221   { \ifthenelse {\equal{\nameref{#1}}{}}
222       {\cref{#1} on \cpageref{#1}}
223       {\cref{#1} (\nameref{#1}) on \cpageref{#1}\!}
224   }
225
226\newcommand \Pagecref [1]
227   { \ifthenelse {\equal{\nameref{#1}}{}}
228       {\Cref{#1} on \cpageref{#1}}
229       {\Cref{#1} (\nameref{#1}) on \cpageref{#1}}
230   }
231
232\DeclareMathOperator \range {\mathrm{range}}
233
234\DeclareMathOperator \Range {\seqname{range}}
235
236\newcommand \restrictto {\!\restriction}
237
238\DeclareMathOperator \seqeq {\stackrel{()}{=}}
239
240\DeclareMathOperator \seqin {\stackrel{()}{\in}}
241
242%Font for sequence
243%\newcommand \seqname [1] {\bm{\mathit{\mathsf{#1}}}}
244
245\newcommand \Set {\mathbf{Set}}
246%
247%\newcommand \sing [2] [] {\underset{{#1}\mathrm{Sing}{#2}}
248%\newcommand \Sing [2] [] {\underset{\bm{{#1}\mathrm{Sing}}{#2}}
249
250\newcommand \singcat [2] [] {\underset{{#1}\mathscr{S\!i\!n\!g}}{#2}}
251\newcommand \Singcat [2] [] {\underset{{#1}\mathscr{S\!i\!n\!g}}{#2}}
252% Can't use \bm without stix and XeTeX
253
254\newcommand \strict [2] [] {\underset{\mathrm {{#1}strict}}{#2}}
255
256\newcommand \subcat [1] [] {\overset{\mathrm{{#1}cat}}{\subseteq}}
257\newcommand \SUBCAT [1] [] {\overset{\mathbf{{#1}cat}}{\subseteq}}
258
259\newcommand \submod [1] [] {\overset{\mathrm{{#1}mod}}{\subseteq}}
260
261\DeclareMathOperator \SUBSETEQ {\stackrel{()}{\subseteq}}
262
263\DeclareMathOperator \tail {\mathrm{tail}}
264
265\newcommand \toiso {\,\to/{>}->>/^{\iso}}
266
267\newcommand \Top {\mathrm{Top}}
268\newcommand \Topcat {\mathscr{T\!o\!p}}
269
270% Font for topology
271\newcommand \topname [1] {\mathfrak{#1}}
272
273\newcommand \Topology {\mathfrak{Top}}
274
275\newcommand \triv [2] [] {\underset{{#1}\mathrm {triv}}{#2}}
276\newcommand \Triv [2] [] {\underset{{#1}\mathbf {triv}}{#2}}
277
278\newcommand \trivcat [2] [] {\underset{{#1}\mathscr{T\!r\!i\!v}}{#2}}
279\newcommand \Trivcat [2] [] {\underset{{#1}\bm{\mathscr{T\!r\!i\!v}}}{#2}}
280% Can't use \bm without stix and XeTeX
281
282\newcommand \true {\mathrm{True}}
283% Can't use long name because \mathscr doesn't support lower case.
284\newcommand \truthcat {\mathscr{T}}
285\newcommand \truthset {\mathbb{T}}
286\newcommand \truthspace {\mathrm{Truthspace}}
287\newcommand \truthtop {\mathfrak{Truthtop}}
288
289\newcommand \unioncat [1] [] {\overset{\mathrm{{#1}cat}}{\cup}}
290\newcommand \UNIONCAT [1] [] {\overset{\mathbf{{#1}cat}}{\cup}}
291
292% Existential and universal quamtifiers
293% Set former
294% Union and intersection
295
296% --------------------------------------------------------------------
297% | From here to closing --- belongs in package                      |
298% |                                                                  |
299
300% Copyright 2016 Shmuel (Seymour J.) Metz
301% I grant permission to the AMS, arXiv.org, the LATEX3 project and the
302% TeX users group to incorporate these commands in any LaTeX package
303% that may be freely redistributed, provided that they attribute the
304% source.
305
306% These commands are intended to allow semantic markup for some
307% common mathermtical constructs:
308%   \equant{variables}{proposition}     Existential qunatifier
309%   \uquant{variables}{proposition}     Universal   quantifier
310%   \union[indices]{set}                Union
311%   \intersection[indices]{set}         Intersection
312%   \set {elements}[propositions]       Set of
313%   \set{elements}[propositions]*       Set of, split
314%   \setupquant{optionstring}           Style of \equant, \uquant
315%   \setupset{optionstring}             Style of \set, \union, \intersection
316%   \seqname{name}                      Render sequence/set/tuple name
317
318% Because LaTeX has problems with parameters containg \\, the \set command
319% has code to split the line between the elements and the proposition if
320% the invocation is \set* and the environment is multline.
321
322% Surround individual indices in \intersection, individual variables in
323% quantifiers, individual propositions in \set and individual indices in
324% \union with braces, e.g., \equant {{x \in X},{y \in Y}}{P(x,y)},
325% \set{x}[{P(x)}, {Q(x)}], \union[{i \in I},{j \in J}]{O(i.j)}.
326
327% Setup keywords for \equant, \uquant
328% subscript  =none            Default
329%                             \exists var1 ... \exists varn prop
330%             stacked         \exists_\substack{var1 \\ ... varn} prop
331%             multiple        \exists_{var1,...,varn} prop
332% parentheses=none            Default
333%             single          (\exists var1 ... \exists varn) prop
334%             multiple        (\exists var1) ... (\exists varn) prop
335% separator=                  Default {}
336
337% Setup keywords for \intersection, \set, \union
338% subscript  =none            \bigcap ix1, ..., ixn set
339%             stacked         Default
340%                             \bigcap__\substack{ix1 \\ ... ixn} set
341%             multiple        \bigcap_{ix1,...,ixn} set
342% separator=                  Default {\mid}
343%                             {elements separator prop1 ^ ... propn}
344
345\ExplSyntaxOn
346
347\int_gzero_new:N \g_style_quant_parens_int
348\int_gzero_new:N \g_style_quant_subscr_int
349\int_gzero_new:N \g_style_set_subscr_int
350
351\NewDocumentCommand{\equant}{mm}
352  {
353    \quant:nnn {\exists} {#1} {#2}
354  }
355
356\NewDocumentCommand{\uquant}{mm}
357  {
358    \quant:nnn {\forall} {#1} {#2}
359  }
360
361\NewDocumentCommand \setupquant {m}
362  {
363    \keys_set:nn {shmuel / quant} {#1}
364  }
365
366\keys_define:nn {shmuel / quant}
367 {
368  subscript            .choices:nn =
369    {
370      {
371        none,
372        stacked,
373        multiple
374      }
375      {
376        \int_gset:Nn \g_style_quant_subscr_int {\l_keys_choice_int-1}
377      }
378    },
379  subscript            .default:n = multiple,
380  subscript            .initial:n = none,
381  parentheses          .choices:nn =
382    {
383      {
384        none,
385        single,
386        multiple
387      }
388      {
389        \int_gset:Nn \g_style_quant_parens_int {\l_keys_choice_int - 1}
390      }
391    },
392  parentheses          .default:n = multiple,
393  parentheses          .initial:n = none,
394  separater            .tl_set:N = \g_style_quant_sep_tl,
395  separater            .default:n = {.},
396  separater            .initial:n = {}
397 }
398
399\cs_new:Npn \quant:nnn #1 #2 #3
400  {
401    %\int_show:N \g_style_quant_parens_int
402    %\int_show:N \g_style_quant_subscr_int
403    % g_style_quant_parens_int \ \int_use:N \g_style_quant_parens_int \
404    % g_style_quant_subscr_int \ \int_use:N \g_style_quant_subscr_int \
405    \clist_set:Nn \l_tmpa_clist {#2}
406    \int_case:nn
407      {\g_style_quant_subscr_int}
408      {
409        {0}
410        {
411          % No subscript
412          % Set separater to ) ( quantifier or just quantifier
413          \int_compare:nTF {\g_style_quant_parens_int = 2}
414            {\tl_set:Nn \l_tmpa_tl {\right ) \left ( #1}}
415            {\tl_set:Nn \l_tmpa_tl {#1}}
416          \int_compare:nT {\g_style_quant_parens_int > 0} {\left (}
417          #1
418          \clist_use:Nn \l_tmpa_clist {\l_tmpa_tl}
419          \int_compare:nT {\g_style_quant_parens_int > 0} {\right )}
420          \g_style_quant_sep_tl #3
421        }
422        {1}
423        {
424          % Stacked subscript on single quantifier
425          \fp_set:Nn
426            \l_tmpa_fp
427            {ceil{\clist_count:N{\l_tmpa_clist} - 1} * .1 + 1}
428          \int_compare:nT {\g_style_quant_parens_int > 0} {\left (}
429          \scaleobj{\fp_to_decimal:N \l_tmpa_fp}{#1} \sb
430             { \substack { \clist_use:Nn \l_tmpa_clist { \\ } } }
431          \int_compare:nT {\g_style_quant_parens_int > 0} {\right )}
432          #3
433        }
434        {2}
435        {
436          % Subscripts on separate quantifiers
437          \clist_map_inline:Nn
438            \l_tmpa_clist
439            {
440              % (quantifier \sb predicate) or quantifier \sb predicate
441              {
442                \int_compare:nT
443                  {\g_style_quant_parens_int > 0}
444                  {\left (}
445                \scaleobj{1.2}{#1} \sb
446                {##1}
447                \int_compare:nT
448                  {\g_style_quant_parens_int > 0}
449                  {\right )}
450              }
451            }
452          #3
453        }
454      }
455  }
456
457\NewDocumentCommand{\set}{mos}
458  {
459    \IfBooleanTF {#3}
460    {
461%      \msg_term:n {set with star}
462%      \msg_term:n{{P1 #1}}
463%      \msg_term:n{{P2 #2}}
464      \bool_set_true:N \l_tmpa_bool
465%      \bool_show:N \l_tmpa_bool
466    }
467    {
468      \bool_set_false:N \l_tmpa_bool
469    }
470    \set_of:nnn {#1} {#2} {\l_tmpa_bool}
471  }
472
473%\tl_new:N \g_style_set_sep_tl
474%\tl_gset:Nn \g_style_set_sep_tl {\mid}
475
476\NewDocumentCommand \setupset {m}
477  {
478    \keys_set:nn {shmuel / set} {#1}
479  }
480
481\keys_define:nn {shmuel / set}
482  {
483    separater .tl_set:N = \g_style_set_sep_tl,
484    subscript            .choices:nn =
485      {
486        {
487          stacked,
488          multiple
489        }
490        {
491          \int_gset:Nn \g_style_set_subscr_int {\l_keys_choice_int-1}
492        }
493      },
494    separater .initial:n = {\mid},
495    subscript .initial:n = stacked
496  }
497
498\cs_new:Npn \set_of:nnn #1 #2 #3
499  {
500    \IfValueTF {#2}
501    {
502%      \msg_term:n {set_of:nn \ has \ predicates \ #2}
503    }
504    {
505%      \msg_term:n {set_of:nn \ has \ no \ predicates}
506    }
507    \clist_set:Nn \l_tmpa_clist {#2}
508%    \msg_term:n {l_tmpa_clist \ set}
509    \tl_gset:Nn \g_tmpa_tl {\clist_use:Nn \l_tmpa_clist {\land}}
510%    \msg_term:n {g_tmpa_tl \ set \ to \ \g_tmpa_tl}
511    \IfValueTF {#2}
512    {
513      \bool_if:nTF {#3}
514      {
515        \scalerel*{\{}{#1\g_tmpa_tl}
516        #1
517%        \msg_term:n  {scalerel returns \scalerel{\g_style_set_sep_tl}{\g_tmpa_tl}}
518        \clist_set:Nn \l_tmpa_clist {#2}
519        \scalerel*{\mathbin{\g_style_set_sep_tl}}{#1\g_tmpa_tl}
520        \\
521        \clist_set:Nn \l_tmpa_clist {#2}
522        \g_tmpa_tl
523%         \tl_show:N \g_tmpa_tl
524        \clist_set:Nn \l_tmpa_clist {#2}
525        \scalerel*{\}}{#1\g_tmpa_tl}
526      }
527      {
528        \left \{
529        #1
530        \clist_set:Nn \l_tmpa_clist {#2}
531        \scalerel*{\mathbin{\g_style_set_sep_tl}}{#1\g_tmpa_tl}
532        \g_tmpa_tl
533        \right \}
534      }
535    }
536    {
537      \left \{ #1 \right \}
538    }
539  }
540
541\NewDocumentCommand{\funcname}{m}
542  {
543    \funcname:n {#1}
544  }
545
546\cs_new:Npn \funcname:n #1
547  {
548    %code here \tl_count:n
549    \int_compare:nTF {\tl_count:n{#1} > 1}
550      {
551        {\mathrm{#1}}
552      }
553      {
554        {\mathit{#1}}
555      }
556  }
557
558\NewDocumentCommand{\funcseqname}{m}
559  {
560    \funcseqname:n {#1}
561  }
562
563\cs_new:Npn \funcseqname:n #1
564  {
565    %code here \tl_count:n
566    \int_compare:nTF {\tl_count:n{#1} > 1}
567      {
568        {\bm{\mathrm{#1}}}
569      }
570      {
571        {\bm {\mathit{#1}}}
572      }
573  }
574
575\NewDocumentCommand{\seqname}{m}
576  {
577    \seqname:n {#1}
578  }
579
580\cs_new:Npn \seqname:n #1
581  {
582    %code here \tl_count:n
583    \int_compare:nTF {\tl_count:n{#1} > 1}
584      {
585        {\mathbf{#1}}
586      }
587      {
588        {\mathbfit{#1}}
589      }
590  }
591
592\NewDocumentCommand{\intersection}{om}
593  {
594    \unint_of:nnn \bigcap {#1} {#2}
595  }
596
597\NewDocumentCommand{\union}{om}
598  {
599    \unint_of:nnn \bigcup {#1} {#2}
600  }
601
602\cs_new:Npn \unint_of:nnn #1 #2 #3
603  {
604    \IfValueTF {#2}
605    {
606%      \int_show:N \g_style_set_subscr_int
607      \clist_set:Nn \l_tmpa_clist {#2}
608      \int_case:nn
609        {\g_style_set_subscr_int}
610        {
611          {0}
612          {
613            % Stacked subscript
614%            \msg_term:n {\clist_count:N{\l_tmpa_clist} \ tokens \ stacked}
615%            \clist_show:N \l_tmpa_clist
616            \fp_set:Nn
617              \l_tmpa_fp
618              {ceil{\clist_count:N{\l_tmpa_clist} - 1} * .1 + 1}
619            \scaleobj{\fp_to_decimal:N \l_tmpa_fp}{#1} \sb
620            {\substack { \clist_use:Nn \l_tmpa_clist { \\ } }}
621            #3
622          }
623          {1}
624          {
625            % Subscripts comma separated
626%            \msg_term:n {\clist_count:N{\l_tmpa_clist} \ tokens \ comma \ separated}
627%            \clist_show:N \l_tmpa_clist
628            #1 \sb
629            {\clist_use:Nn \l_tmpa_clist {,}}
630            #3
631          }
632        }
633    }
634    {
635      % No subscript
636%      \msg_term:n {No~subscript}
637%      \clist_show:N \l_tmpa_clist
638      #1 #3
639    }
640  }
641
642\ExplSyntaxOff
643
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645% | From opening to here --- belongs in package                      |
646% --------------------------------------------------------------------
647
648
649\newtheorem{theorem}{Theorem}[section]
650\newtheorem{lemma}[theorem]{Lemma}
651\def\lemmaautorefname{Lemma}          % Needed for \autoref
652\newtheorem{corollary}[theorem]{Corollary}
653\def\corollaryautorefname{Corollary}  % Needed for \autoref
654
655\theoremstyle{definition}
656\newtheorem{definition}[theorem]{Definition}
657\def\definitionutorefname{Definition} % Needed for \autoref
658\newtheorem{example}[theorem]{Example}
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660
661\theoremstyle{remark}
662\newtheorem{remark}[theorem]{Remark}
663
664\numberwithin{equation}{section}
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679
680\usepackage[colorlinks,hidelinks,draft=false]{hyperref}
681
682\usepackage{cleveref}
683\def\corollaryautorefname{Corollary}  % Needed for \autoref
684\def\definitionutorefname{Definition} % Needed for \autoref
685\def\lemmaautorefname{Lemma}          % Needed for \autoref
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687   bookmarksnumbered=true,
688   colorlinks,
689   pdfinfo={
690      Author={Shmuel (Seymour J.) Metz},
691      Keywords={fiber bundles,manifolds},
692      Subject={Topology},
693      Title={Local Coordinate Spaces: a proposed unification of manifolds with fiber bundles, and associated machinery}
694   }
695}
696\def\corollaryautorefname{Corollary}  % Needed for \autoref
697\def\definitionutorefname{Definition} % Needed for \autoref
698\def\lemmaautorefname{Lemma}          % Needed for \autoref
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700\usepackage[draft]{showlabels}
701\showlabels{cite}
702\showlabels{cref}
703\showlabels{crefrange}
704
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716
717\begin{document}
718\hyphenation{near pres-en-ta-tion}
719
720\title%
721{%
722   Local Coordinate Spaces:
723  a proposed unification of manifolds and fiber bundles,
724  and associated machinery\thanks%
725  {
726  I wish to gratefully thank Walter Hoffman (z"l),
727  Milton Parnes, Dr. Stanley H. Levy, the Mathematics department of
728  Wayne State University, the Mathematics department of the State
729  University of New York at Buffalo and others who guided my education.
730  }
731}
732\author{Shmuel (Seymour J.) Metz
733}
734\maketitle
735
736% \address{4963 Oriskany Drive\\Annandale, VA  22003-5141}
737% \email{smetz3@gmu.edu}
738% \urladdr{http://mason.gmu.edu/~smetz3}
739
740% \subjclass[2010]{Primary 18F15; Secondary 55R65,57N99,58A05}
741% 18F15  Abstract manifolds and fiber bundles
742% 32Qxx  Complex manifolds
743% 53-    Differential geometry
744% 53Axx  Classical differential geometry
745% 53A99  None of the above, but in this section
746% 55Rxx  Fiber spaces and bundles
747% 55R10  Fiber bundles
748% 55R65  Generalizations of fiber spaces and bundles
749% 57Nxx  Topological manifolds
750% 57N99  None of the above, but in this section
751% 57Rxx  Differential topology
752% 58Axx  General theory of differentiable manifolds
753% 58A05  Differentiable manifolds, foundations
754% 58Bxx  Infinite-dimensional manifolds
755
756% Primary 18F15  Abstract manifolds and fiber bundles
757% Secondary 55R65,57N99,58A05
758% \keywords{fiber bundles,manifolds}
759
760\begin{abstract}
761This paper presents a unified view of manifolds and fiber bundles,
762which, while superficially different, have strong parallels. It
763introduces the notions of an m-atlas and of a local coordinate space,
764and shows that special cases are equivalent to fiber bundles and
765manifolds. Along the way it defines some convenient notation, defines
766categories of atlases, and constructs potentially useful functors.
767\end{abstract}
768
769\setupquant {parentheses=multiple,subscript=stacked}
770\setupset   {}
771
772\tikzset{
773    rot90/.style={anchor=south, rotate=90, inner sep=.2mm}
774}
775
776\part {Introduction}
777\label{part;intro}
778
779Historically, the concept of pseudo-groups allowed unifying manifolds
780and manifolds with boundary. The definitions of fiber bundles and
781manifolds have strong parallels, and can be unified in a similar fashion;
782there are several ways to do so. The central part of this paper,
783\pagecref{part;lcs}\negmedspace, defines an approach using categories and
784commutative diagrams that is designed for easy exposition at the possible
785expense of abstractness and generality. In particular, I have chosen to
786assume the Axiom of Choice (AOC).
787
788This paper treats atlases as objects of interest in their own right,
789although it does not give them primacy. It introduces notions that
790are convenient for use here and others that, while not used here, may be
791useful for future work. It defines the new notions of
792\hyperref[def:model]{model space}\negmedspace
793\footnote{The phrase has been used before, but with a different
794meaning.},
795\hyperref[def:m-atlas]{m-atlas}
796and of \hyperref[def:M-ATLmorph]{m-atlas morphism}.
797Informally, a model space is a topological space with a category
798specifying a family of open sets and functions satisfying specified
799conditions.
800
801Although this paper incidentally defines partial equivalents to
802manifolds and fiber bundles using model spaces and model atlases, it
803proposes the more general
804\hyperref[def:LCS]{Local Coordinate Space (LCS)} in order to explicitly
805reflect the role of the group in fiber bundles.
806
807A local coordinate space (LCS) is a space (total space) with some
808additional structure, including a coordinate model space and an atlas
809whose transition functions are restricted to morphisms of the coordinate
810model space; one can impose, e.g., differentiability restrictions, by
811appropriate choice of the coordinate category. There is an equivalent
812paradigm that avoids explicit mention of the total space by imposing
813compatibility conditions on the transition functions, but that approach
814is beyond the scope of this paper.
815
816This paper defines functors among categories of atlases, categories of
817model spaces, categories of local coordinate spaces, categories of
818manifolds and categories of fiber bundles; it constructs more machinery
819than is customary in order to facilitate the presentation of those
820categories and functors.
821
822\Crefrange{part;conv}{part;pre} present nomenclature and give basic
823results.
824\Cref{part;m-charts} defines m-atlases, m-atlas morphisms and categories
825of them; \cref{lem:ATLiscat} proves that the defined categories are
826indeed categories.
827\Cref{part;lcs} defines local coordinate spaces and
828categories of them; \pagecref{the:LCSiscat} proves that the defined
829categories are indeed categories.
830
831\Pagecref{Examples} gives some examples of structures that can be
832represented as local coordinate spaces; \pagecref{part;man} and
833\pagecref{part;bun} present two of the examples in detail, showing the
834equivalence of manifolds and fiber bundles with special cases of local
835coordinate spaces by explicitly exhibiting functors to and from local
836coordinate spaces.
837\begin{remark}
838The unconventional definitions of manifold and fiber bundle are intended
839to make their relationship to local coordinate spaces more natural.
840\end{remark}
841
842Most of the lemmata, theorems and corollaries in this paper should be
843substantially identical to results that are familiar to the reader. What
844is novel is the perspective and the material directly related to local
845coordinate spaces. The presentation assumes only a basic knowledge of
846Category Theory, such as may be found in the first chapter of
847\cite{CftWM} or
848{
849  \showlabelsinline
850  \cite{JoyCat}.
851}
852
853\section{New concepts and notation}
854\label{sec:new}
855This paper introduces a significant number of new concepts and some
856modifications of the definitions for some conventional concepts. It also
857introduces some notation of lesser importance.  The following are the
858most important.
859
860\begin{enumerate}
861\item \hyperref[def:NCD]{Nearly commutative diagram (NCD)}, NCD at a point,
862locally NCD and special cases with related nomenclature
863\item \hyperref[def:model]{Model space} and related concepts
864\item \hyperref[def:ModTop]{Model topology} and
865\hyperref[def:M-para]{M-paracompactness}
866\item \hyperref[sec:sig]{Signature},
867\hyperref[def:Sigmacomm]{$\Sigma$-commutation} and related concepts
868\item
869\hyperref[def:LCS]{Local Coordinate Space (LCS)} and related concepts
870\item
871\hyperref[def:lin]{Linear space and related concepts}
872\item
873\hyperref[def:trivck]{Trivial $\Ck$ linear model space} and related
874concepts
875\item \hyperref[def:BunAtl]{$G$-$\rho$ bundle atlas}
876\footnote{Similar to coordinate bundles}
877and related concepts
878\end{enumerate}
879
880\part {Conventions}
881\label{part;conv}
882An arrow with an Equal-Tilde ($A \toiso_\phi B$) represents an
883isomorphism. One with a hook ($A \underset{i}{\hookrightarrow} B$)
884represents an inclusion map. One with a tail ($A \into_i B$) represents
885a monomorphism. One with a double head ($A \onto_\pi B$) represents a
886surjection.
887
888When a superscript ends in $-1$, e.g., $\funcname{f}^{i-1}$, it is to be
889taken as function inverse rather than subtraction of 1.
890
891All diagrams shown are commutative; none are exact.
892
893A definition of a base term several more restrictive terms may
894specify the modifiers in parenthese in the base definition and then
895give the restrictions for each modifier, e.g., if
896"$\funcseqname{f}$ is a (semi-strict, strict) prestructure morphism
897of $\seqname{P}^1$ to $\seqname{P}^2$ iff" is followed by the
898definition of prestructure morphism, then the restrictions for
899strict and semi-strict prestructure morphisms.
900
901When a definition defines a base propositional function and variant
902propositional functions with a qualifier given as a superscript,
903then the form $\mathrm{base}^{(qualifiers)}$ will refer to
904either $\mathrm{base}^{\mathrm{qualifer}}$
905or $\mathrm{base}$, e.g.,
906$\strict[semi-]{\isAtl[(classic,near)]_\Ar}$ refers to either
907$\strict[semi-]{\isAtl[classic]_\Ar}$,
908$\strict[semi-]{\isAtl[near]_\Ar}$
909or $\strict[semi-]{\isAtl_\Ar}$.
910
911Alternatively, a definition may specify a numbered list of alternatives,
912and subsequently specify additional numbered lists with items
913corresponding to those in the first list, e.g.,
914$\funcseqname{f}$ is also a
915\begin{enumerate}
916\item full
917\item semi-maximal
918\item maximal
919\item full semi-maximal
920\item full maximal
921\end{enumerate}
922$E^1$-$E^2$ $\Ck$ near morphism of
923$\seqname{A}^1$ to $\seqname{A}^2$ in the coordinate spaces $C^1$,
924$C^2$, abbreviated as
925\begin{enumerate}
926\item abbreviation for full
927\item abbreviation for semi-maximal
928\item abbreviation for maximal
929\item abbreviation for full semi-maximal
930\item abbreviation for full maximal
931\end{enumerate}
932iff
933\begin{enumerate}
934\item definition for full
935\item definition for semi-maximal
936\item definition for maximal
937\item definition for full semi-maximal
938\item definition for full maximal
939\end{enumerate}
940
941A Corollary, Lemma or Theorem that applies to related terms defined
942with the above convention will specify the modifiers in parentheses
943to indicate that it applies to the base term and to the more
944restrictive terms, e.g., "a (semi-strict, strict)
945$\seqname{E}^i$-$\seqname{E}^{i+1}$ m-atlas morphism" If it applies
946only to more restrictive terms them it will specify the first
947relevant modifier followed by the remaining relevant modifiers in
948parentheses, e.g., "If $\catname{S}^i \SUBCAT[full-]
949\catname{S}'^i$ and $\funcseqname{f}^i$ is a semi-strict (strict)
950prestructure morphism".
951
952Blackboard bold upper case will denote specific sets, e.g., the
953Naturals.
954
955Bold lower case italic letters will refer to sets, sequences and tuples
956of functions, e.g.,
957$\funcseqname{f} \defeq (\funcname{f}_1, \funcname{f}_2)$.
958
959Bold lower case Latin letters will refer to sequence valued functions of
960sequences and tuple valued functions of tuples, e.g., $\Range$ yields
961the sequence of ranges of a sequence of functions.
962
963Bold upper case italic letters will refer to sequences or tuples, e.g.,
964$\seqname{A}=(x,y,z)$, to sets of them, to sets of topological spaces or
965to sets of open sets.
966
967Bold upper case script letters will refer to
968sequences of categories, e.g.,
969$\catseqname{A} \defeq (\catname{A}_\alpha, \alpha \in \Alpha)$.
970
971Fraktur will refer to topologies and to topology-valued functions, e.g.,
972$\Topology$.
973
974Functions have a range, domain and relation, not just a relation. Unless
975otherwise stated, they are assumed to be continuous.
976
977Groups are assumed to be topological groups. The ambiguous notation
978$x^{-1}$ will be used when it is obvious from context what the group
979operation $\star$ and the group identity $\mathbf{1}_G$ are.
980
981Lower case Greek letters other than $\pi$, $\rho$, $\sigma$, $\phi$ and
982$\psi$ will refer to ordinals, possibly transfinite, and to formal
983labels.  A letter with a Greek superscript and a letter with a Latin or
984numeric superscript always refer to distinct variables.
985
986Lower case $\pi$ will refer to a projection operator
987
988Lower case $\rho$ will refer to a continuous effective group action,
989i.e., a continuous representation of a group in a homeomorphism group.
990
991Lower case $\sigma$ will refer to a sequence of ordinals, referred to as
992a signature.
993
994Lower case $\phi$ will refer to a coordinate function.
995
996Lower case italic and Latin letters will refer to
997\begin{enumerate}
998\item elements of a set or sequence
999\item functions
1000\item morphisms of a category
1001\item natural numbers
1002\item objects of a category
1003\item ordinal numbers
1004\end{enumerate}
1005
1006Upper case Greek letters other than $\Sigma$ may refer to
1007\begin{enumerate}
1008\item ordinal used as the limit of a sequence of consecutive ordinals,
1009e.g., $x_\alpha, \alpha \preceq \Alpha$
1010\item ordinal used as the order type of a sequence of consecutive
1011ordinals, e.g., $x_\alpha, \alpha \prec \Alpha$
1012\end{enumerate}
1013
1014Upper case $\Sigma$ will refer to a sequence of signatures
1015
1016Upper case Latin letters will refer to
1017\begin{enumerate}
1018\item Natural numbers
1019\item Topological spaces
1020\item Open sets
1021\item Elements of a sequence or tuple of functions, e.g.,
1022$\funcname{f}_E$ might be $\funcname{f}_0 \maps E_1 \to E_2$.
1023\end{enumerate}
1024
1025Upper case Script Latin letters will refer to categories and functors.
1026
1027Upright Latin letters will be used for long names.
1028
1029The term $\Ck$ includes $\mathrm{C}^\infty$ (smooth) and
1030$\mathrm{C}^\omega$ (analytic).
1031
1032This paper uses the term morphism in preference to arrow, but uses
1033the conventional $\Ar$.
1034
1035The term sequence without an explicit reference to $\mathbb{N}$
1036will refer to a general ordinal sequence, possibly transfinite.
1037
1038Sequence numbering, unlike tuple numbering, starts at 0 and the
1039exposition assumes a von Neumann definition of ordinals, so that
1040$\alpha \in \beta \equiv \alpha \prec \beta$.
1041
1042Except where explicitly stated otherwise, all categories mentioned are
1043small categories with underlying sets, but the morphisms will often not
1044be set functions between the objects and there will not always be a
1045forgetful function to $\Set$ or $\Cat{Top}$. By abuse of language no
1046distinction will be made between a category $\catname{A}$ of topological
1047spaces and the concrete category $(\catname{A},\catname{U})$ over
1048$\Cat{Top}$.  Similarly, no distinction will be made among the object $U
1049\in \Ob(\catname{A})$, the topological space $\catname{U}(U)$ and the
1050underlying set.
1051
1052The notation $G^V$ will refer only to the set of continuous functions
1053from $V$ to $G$, never to the set of all functions from $V$ to $G$.
1054
1055When defining a category, the Ordered pair $(\seqname{O}, \seqname{M})$
1056refers to the smallest concrete category over $\Set$ or $\Cat{Top}$
1057whose objects are the elements in $\seqname{O}$, whose morphisms include
1058all of the elements of $\seqname{M}$ and whose morphisms from
1059$o^1 \in \seqname{O}$ to $o^2 \in \seqname{O}$ are functions
1060$\funcname{f} \maps o^1 \to o^2$ and whose composition is function
1061composition.
1062
1063When defining a category, the Ordered triple
1064$(\seqname{O}, \seqname{M}, C)$ refers to the small category whose
1065objects are in $\seqname{O}$, whose morphisms are in $\seqname{M}$,
1066whose $\Hom$ is
1067
1068\begin{equation}
1069\Hom_{(\seqname{O}, \seqname{M}, C)}
1070  (o_1 \in \seqname{O}, o_2 \in \seqname{O})
1071\defeq
1072\set
1073  {
1074    (
1075      \funcseqname{f},
1076      o_1,
1077      o_2
1078    )
1079    \in \seqname{M}
1080  }
1081\end{equation}
1082and whose composition is C.
1083
1084By abuse of language I may write
1085``$\catname{S}$'' for $\Ob(\catname{S})$,
1086``$A \in \catname{A}$'' for  $A \in\ \Ob(\catname{A})$,
1087``$A \subset \catname{A}$'' for $A \subset \Ob(\catname{A})$,
1088``$A \in \catname{A} \subset B \in \catname{B}$'' for
1089``the underlying set of $A$ is contained in the underlying set of $B$
1090and the inclusion $\funcname{i} \maps x \in A \hookrightarrow x \in B$
1091is a morphism'' and
1092``$\funcname{f} \maps A \to B$'' for
1093$\funcname{f} \in \Hom_{\catname{C}}(A,B)$, where $\catname{C}$ is
1094understood by context.
1095
1096By abuse of language I shall use the same nomenclature for sequences and
1097tuples, and will use parentheses around a single expression both for
1098grouping and for a tuple with a single element; the intent should be
1099clear from context.
1100
1101By abuse of language I will omit parenthese around the operands of
1102Functors when they can be assumed by context.
1103
1104By abuse of language I shall use the $\times$ and $\bigtimes$ symbols
1105for both Cartesian products of sets and Cartesian products of
1106functions on those sets.
1107
1108By abuse of language, and assuming AOC,  I shall refer to some sets as
1109ordinal sequences, e.g., ``$(C_\alpha, \alpha \in \Alpha)$'' for
1110``$\{C_\alpha \mid \alpha \in \Alpha\}$'', in cases where the order is
1111irrelevant.
1112
1113By abuse of language, I may omit universal quantifiers in cases where
1114the intent is clear.
1115
1116In some cases I define a notion similar to a conventional notion and
1117also need to refer to the conventional notion. In those cases I prefix
1118a letter or phrase to the term, e.g., m-paracompact versus paracompact.
1119
1120\part {General notions}
1121\label{part;notions}
1122This section describes nomenclature used throughout the paper. In
1123some cases this reflects new nomenclature or notions, in others it
1124simply makes a choice from among the various conventions in the
1125literature.
1126
1127\begin{definition}[Operations on categories]
1128\label{def:catprop}
1129If $\catname{C}$ is a category then $x \objin \catname{C}$ iff $x$ is an
1130object of $\catname{C}$ and $y \arin \catname{C}$ iff $y$ is a morphism
1131of $\catname{C}$.
1132
1133If $\catname{S}$ and $\catname{T}$ are categories then
1134$\catname{S} \subcat \catname{T}$ iff $S$ is a subcategory of
1135$\catname{T}$ and $\catname{S} \subcat[full-] \catname{T}$ iff $S$ is a
1136full subcategory of $\catname{T}$.
1137
1138If $\catname{S}$ and $\catname{T}$ are categories then the category
1139union of $\catname{S}$ and $\catname{T}$, abbreviated
1140$\catname{S} \unioncat \catname{T}$, is the category whose objects are
1141in $\catname{S}$ or in $\catname{T}$ and whose morphisms are in
1142$\catname{S}$ or in $\catname{T}$.
1143\end{definition}
1144
1145\begin{definition}[Identity]
1146$\Id_S$ is the identity function on the space $S$,
1147
1148$\Id_o$ is the
1149identity morphism for the object $o$\footnote{
1150The object is often expressed as a tuple, e.g.,
1151$\Id_{(\seqname{A}, \seqname{B})}$ is the identity morphism for the
1152object $(\seqname{A}, \seqname{B})$
1153},
1154
1155$\Id_{U,V}$, for $U \subseteq V$, is the inclusion map\footnote{
1156$U$ and $V$ need not have the same topology.
1157}.
1158
1159$\Id_\catname{C}$ is the identity functor on the category $\catname{C}$.
1160
1161$\ID_{\seqname{S}^i}$, $i=1,2$, is the sequence of identity functions
1162for the elements of the sequence
1163$\seqname{S}^i \defeq (\seqname{S}^1_\alpha,\ \alpha \prec \Alpha)$. Let
1164$\seqname{S}^1 \SUBSETEQ \seqname{S}^2$. Then
1165$\ID_{\seqname{S}^1,\seqname{S}^2}$ is the sequence of inclusion maps
1166$(\Id_{\seqname{S}^1_\alpha,\seqname{S}^2_\alpha}),\ \alpha \prec \Alpha$
1167for the elements of the sequences $\seqname{S}^i$.
1168
1169The subscript may be omitted
1170when it is clear from context.
1171\end{definition}
1172
1173\begin{definition}[Images]
1174$\funcname{f} [U] \defeq \set { {\funcname{f}(x)} }[x \in U]$ is the
1175image of $U$ under $\funcname{f}$ and
1176$\funcname{f}^{-1} [V] \defeq \set {x}[\funcname{f}(x) \in V]$ is the
1177inverse image of $V$ under $\funcname{f}$.
1178
1179\begin{remark}
1180  This notation, adopted from \cite{GenTop}, avoids the ambiguity in
1181  the traditional $\funcname{f}(U)$ and $\funcname{f}^{-1}(V)$.
1182\end{remark}
1183\end{definition}
1184
1185\begin{definition}[Projections]
1186\label{projections}
1187$\pi_\alpha$ is the projection function that maps a sequence into
1188element $\alpha$ of the sequence.  $\pi_i$ is also the projection
1189function that maps a tuple into element $i$ of the tuple.
1190\end{definition}
1191
1192\begin{definition}[Topological category]
1193\label{def:topcat}
1194A topological category is a small subcategory of $\Cat{Top}$ or its
1195concrete category over $\Set$.
1196
1197$\catname{T}$ is a full topological category iff it is a topological
1198category and whenever $U^i,V^i \objin \catname{T}$, $i=1,2$,
1199$V^i \subseteq U^i$,
1200$\funcname{f} \maps U^1 \to U^2 \arin \catname{T}$ and
1201$\funcname{f}[V^1] \subseteq V^2$ then \\
1202$\funcname{f} \maps V^1 \to V^2 \arin \catname{T}$.
1203\end{definition}
1204
1205\begin{lemma}[Inclusions in topological categories are morphisms]
1206\label{lem:topInc}
1207Let $\catname{T}$ be a full topological category,
1208$S^i \objin \catname{T}$, $i=1,2$, and $S^1 \subseteq S^2$. Then
1209$\Id_{S^1,S^2}$ is a morphism of $\catname{T}$
1210
1211\begin{proof}
1212$\Id_{S^2} \arin \catname{T}$, $S^1 \subseteq S^2$ by hypothesis and
1213$S^1 \subseteq S^2$, so $\Id_{S^1,S^2} \arin \catname{T}$ by
1214\cref{def:topcat}.
1215\end{proof}
1216\end{lemma}
1217
1218\begin{definition}[Local morphisms]
1219\label{def:topLocal}
1220Let $\catname{S}^i$, $i=1,2$, be a full topological category and
1221$S^i \objin \catname{S}^i$, A continuous function
1222$\funcname{f} \maps S^1 \to S^2$ is locally a
1223$\catname{S}^1$-$\catname{S}^2$ morphism of $S^1$ to $S^2$ iff
1224$\catname{S}^1 \subcat[full-] \catname{S}^2$ and for every
1225$u \in S^1$ there is an open neighborhood $U_u$ for $u$ and an open
1226neighborhood $V_u$ for $v \defeq \funcname{f}(u)$ such that
1227$\funcname{f}[U_u] \subseteq V_u$ and $\funcname{f} \maps U_u \to V_u$
1228is a morphism of $\catname{S}^2$.
1229
1230\begin{remark}
1231The condition $\funcname{f} \maps U_u \to V_u \arin \catname{S}^2$
1232ensures that $U_u \objin \catname{S}^1$ and $V_u \objin \catname{S}^2$
1233\end{remark}
1234
1235Let $\catname{T}$ be a full topological category and
1236$S^i \objin \catname{T}$, $i=1,2$. A continuous function
1237$\funcname{f} \maps S^1 \to S^2$ is locally a $\catname{T}$ morphism of
1238$S^1$ to $S^2$ iff it is locally a $\catname{T}$-$\catname{T}$ morphism
1239of $S^1$ to $S^2$.
1240\end{definition}
1241
1242\begin{lemma}[Local morphisms]
1243\label{lem:topLocal}
1244Let $\catname{T}^i$, $i \in [1,3]$, be a full topological category,
1245$S^i \objin \catname{T}^i$ and
1246$\catname{T}^i \subcat[full-] \catname{T}^{i+1}$.
1247
1248If $\funcname{f}^i \maps S^i \to S^{i+1} \arin \catname{T}^{i+1}$ then
1249$\funcname{f}^i$ is locally a $\catname{T}^i$-$\catname{T}^{i+1}$
1250morphism of $S^i$ to $S^{i+1}$.
1251
1252\begin{proof}
1253Let $u \in S^i$ and $v \defeq \funcname{f}^i(u) \in S^{i+1}$. $S^i$ is
1254an open for $u$, $S^{i+1}$ is an open neighborhood for $v$ and
1255$\funcname{f}^i \maps S^i \to S^{i+1} \arin \catname{T}^{i+1}$ by
1256hypothesis.
1257\end{proof}
1258
1259If each $\funcname{f}^i \maps S^i \to S^{i+1}$, is locally a
1260$\catname{T}^i$-$\catname{T}^{i+1}$ morphism of $S^i$ to
1261$S^{i+1}$ then
1262$\funcname{f}^2 \compose \funcname{f}^1 \maps S^1 \to S^3$ is locally a
1263$\catname{T}^1$-$\catname{T}^3$ morphism of $S^1$ to $S^3$.
1264
1265\begin{proof}
1266Since $\catname{T}^1 \subcat[full-] \catname{T}^2$ and
1267$\catname{T}^2 \subcat[full-] \catname{T}^3$,
1268$\catname{T}^1 \subcat[full-] \catname{T}^3$.
1269Let $u \in S^1$, $v \defeq \funcname{f}^1(u)$ and
1270$w \defeq \funcname{f}^2(v)$. There exist an open neighborhood $U_u$
1271for $u$, open neighborhoods $V_u$, $V'_v$ for $v$ and an open
1272neighborhood $W_v$ of $w$ such that
1273$\funcname{f}^1[U_u] \subseteq V_u$,
1274$\funcname{f}^1 \maps U_u \to V_u$ is a morphism of $\catname{T}^2$,
1275$\funcname{f}^2[V'_v] \subseteq W_v$ and
1276$\funcname{f}^2 \maps V'_v \to W_v$ is a morphism of $\catname{T}^3$.
1277Then $\hat{V_u} \defeq V_u \cap V'_v \neq \emptyset$, $\hat{V_u}$ is an
1278open neighborhood of $v$ and
1279$\hat{U_u} \defeq \funcname{f}^{i-1}_1[\hat{V_u}]$ is an open
1280neighborhood for $u$$\funcname{f}^1 \maps \hat{U_u} \to \hat{V_u}$
1281and $\funcname{f}^2 \maps \hat{V_u} \to W_v$ are morphisms of
1282$\catname{T}^3$ by \pagecref{def:topcat} and thus
1283$\funcname{f}^2 \compose \funcname{f}^1 \maps \hat{U_u} \to W_v$ is a
1284morphism of $\catname{T}^3$.
1285\end{proof}
1286\end{lemma}
1287
1288\begin{corollary}[Local morphisms]
1289\label{cor:topLocal}
1290Let $\catname{T}^i$, $i=1,2$, be a full topological category,
1291$\catname{T}^i \subcat[full-] \catname{T}^{i+1}$,
1292$S^i \objin \catname{T}^i$ and $S^1 \subseteq S^2$. Then $\Id_{S^1,S^2}$
1293is locally a $\catname{T}^1$-$\catname{T}^2$ morphism of $S^1$ to $S^2$
1294and $\Id_{S^i}$ is locally a $\catname{T}$ morphism
1295of $S^i$ to $S^i$.
1296
1297\begin{proof}
1298$S^1 \objin \catname{T}^2$ because $S^1 \objin \catname{T}^1$ and
1299$\catname{T}^1 \subcat \catname{T}^2$,
1300$S^2 \objin \catname{T}^2$ by hypothesis and
1301$S^1 \subseteq S^2$ by hypothesis, so
1302$\Id_{S^1,S^2} \arin \catname{T}^2$ by \cref{lem:topInc}.
1303
1304$\Id_{S^i} \defeq \Id_{S^i,S^i}$.
1305\end{proof}
1306\end{corollary}
1307
1308\begin{definition}[Sequence functions]
1309Let $\seqname{S} \defeq (s_\alpha, \alpha \prec \Alpha)$ be a sequence
1310of functions. Then
1311\begin{equation}
1312\Domain(\seqname{S}) \defeq \bigl ( \domain(s_\alpha), \alpha \prec \Alpha \bigr )
1313\end{equation}
1314\begin{equation}
1315\Range(\seqname{S}) \defeq \bigl ( \range(s_\alpha), \alpha \prec \Alpha \bigr )
1316\end{equation}
1317
1318Let $\seqname{T} \defeq (t_\alpha, \alpha \prec \Alpha)$ be a sequence
1319of functions with $\Range(\seqname{S}) = \Domain(\seqname{T})$. Then
1320their composition is the sequence
1321$
1322  \seqname{T} \compose[()] \seqname{S} \defeq
1323  (t_\alpha \compose s_\alpha, \alpha \prec \Alpha)
1324$,
1325
1326Let $\seqname{S} \defeq (s_\gamma, \gamma \preceq \Gamma)$, then these
1327functions extract information about the sequence:
1328\begin{equation}
1329\head(\seqname{S},\Omega) \defeq (s_\gamma, \gamma \prec \Omega)
1330\end{equation}
1331\begin{equation}
1332\head(\seqname{S}) \defeq \head(\seqname{S},\Gamma)
1333\end{equation}
1334\begin{equation}
1335\lengtho(\seqname{S}) \defeq \Gamma
1336\end{equation}
1337\begin{equation}
1338\tail(\seqname{S}) \defeq S_\Gamma
1339\end{equation}
1340
1341Let $\seqname{S} \defeq (s_\gamma, \gamma \prec \Gamma)$, then
1342\begin{equation}
1343\length(\seqname{S}) \defeq \Gamma
1344\end{equation}
1345
1346\begin{remark}
1347If $\lengtho(\seqname{S})$ is defined then
1348$\length(\seqname{S}) = \lengtho(\seqname{S}) + 1$.
1349$\lengtho(\seqname{S})$ is the ordinal type of
1350$\head(\seqname{S})$, not the ordinal type of $\seqname{S}$.
1351\end{remark}
1352
1353Let $\catseqname{S} \defeq (\catname{S}_\alpha, \alpha \prec \Alpha)$
1354and $\catseqname{T} \defeq (\catname{T}_\alpha, \alpha \prec \Alpha)$ be
1355sequences of categories.  Then $\catseqname{S}$ is a subcategory
1356sequence of $\catseqname{T}$, abbreviated
1357$\catseqname{S} \SUBCAT \catseqname{T}$, iff every category in
1358$\catseqname{S}$ is a subcategory of the corresponding category in
1359$\catseqname{T}$, i.e.,
1360$
1361  \uquant%
1362    {\alpha \prec \Alpha}
1363    {\catname{S}_\alpha \subcat \catname{T}_\alpha}
1364$,
1365and $\catseqname{S}$ is a full subcategory sequence of $\catseqname{T}$,
1366abbreviated $\catseqname{S} \SUBCAT[full-] \catseqname{T}$, iff every
1367category in $\catseqname{S}$ is a full subcategory of the corresponding
1368category in $\catseqname{T}$, i.e.,
1369$
1370  \uquant%
1371    {\alpha \prec \Alpha}
1372    {\catname{S}_\alpha \subcat[full-] \catname{T}_\alpha}
1373$.
1374
1375The category sequence union of $\catseqname{S}$ and $\catseqname{T}$,
1376abbreviated $\catseqname{S} \UNIONCAT \catseqname{T}$, is the sequence
1377of category unions of corresponding categories in $\catseqname{S}$ and
1378$\catseqname{T}$, i.e.,
1379$(\catseqname{S}_\alpha \unioncat \catseqname{T}_\alpha)$.
1380
1381\end{definition}
1382
1383\begin{lemma}[Sequence functions]
1384\label{lem:seqfunc}
1385Let $\funcseqname{f}^i \defeq (\funcname{f}^i_\alpha, \alpha \prec \Alpha)$,
1386$i \in [1,3]$, be sequences of functions with
1387$\Domain(\funcseqname{f}^2) = \Range(\funcseqname{f}^1)$ and
1388$\Domain(\funcseqname{f}^3) = \Range(\funcseqname{f}^2)$. Then
1389$
1390(\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1 =
1391\funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1)
1392$.
1393\begin{proof}
1394\begin{equation*}
1395\begin{split}
1396  (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1
1397  & =
1398  \bigl ( (\funcname{f}^3_\alpha \compose \funcname{f}^2_\alpha) \compose \funcname{f}^1_\alpha, \alpha \prec A \bigr )
1399\\*
1400  & =
1401  \bigl ( \funcname{f}^3_\alpha \compose (\funcname{f}^2_\alpha \compose \funcname{f}^1_\alpha), \alpha \prec A \bigr )
1402\\*
1403  & =
1404  \funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1)
1405\end{split}
1406\end{equation*}
1407\end{proof}
1408
1409Let $\funcseqname{f} \defeq (\funcname{f}_\alpha, \alpha \prec \Alpha)$
1410be a sequence of functions, $\seqname{D} = \Domain(\funcseqname{f})$ and
1411$\seqname{R} = \Range(\funcseqname{f})$. Then $ID_\seqname{R}$ is a left
1412$\compose[()]$ identity for $\funcseqname{f}$ and $ID_\seqname{D}$ is a
1413right $\compose[()]$ identity for $\funcseqname{f}$.
1414
1415\begin{proof}
1416\begin{equation*}
1417\begin{split}
1418  \ID_\seqname{R} \compose[()] \funcseqname{f}
1419  & =
1420  (\Id_{\range(\funcname{f}_\alpha)} \compose \funcname{f}_\alpha, \alpha \prec \Alpha)
1421\\*
1422  & =
1423  (\funcname{f}_\alpha, \alpha \prec \Alpha)
1424\\*
1425  & =
1426  \funcseqname{f}
1427\end{split}
1428\end{equation*}
1429\begin{equation*}
1430\begin{split}
1431  \funcseqname{f} \compose[()] \ID_\seqname{D}
1432  & =
1433  (\funcname{f}_\alpha \compose \Id_{\domain(\funcname{f}_\alpha)}, \alpha \prec \Alpha)
1434\\*
1435  & =
1436  (\funcname{f}_\alpha, \alpha \prec \Alpha)
1437\\*
1438  & =
1439  \funcseqname{f}
1440\end{split}
1441\end{equation*}
1442\end{proof}
1443\end{lemma}
1444
1445\begin{definition}[Tuple functions]
1446Let $\seqname{S} \defeq (s_n, n \in [1,N])$ be a tuple of functions. Then
1447\begin{equation}
1448\Domain(\seqname{S}) \defeq \bigl ( \domain(s_n), n \in [1,N] \bigr )
1449\end{equation}
1450\begin{equation}
1451\Range(\seqname{S}) \defeq \bigl ( \range(s_n), n \in [1,N] \bigr )
1452\end{equation}
1453
1454Let $\seqname{T} \defeq (t_n, n \in [1,N])$ be a tuple of functions with
1455$\Range(\seqname{S})=\Domain(\seqname{T})$,
1456Then their composition is the tuple
1457$
1458  \seqname{T} \compose[()] \seqname{S} \defeq
1459  (t_n \compose s_n, n \in [1,N])
1460$
1461
1462Let $\seqname{S} \defeq (s_m, m \in [1,M])$ and
1463$\seqname{T} \defeq (t_n, n \in [1,N])$ be tuples.
1464Then the following are tuple functioms
1465\begin{equation}
1466\head(S,I) \defeq (s_m, m \in [1,I])
1467\end{equation}
1468\begin{equation}
1469\head(S) \defeq \head(S,M-1)
1470\end{equation}
1471\begin{equation}
1472\tail(T,I) \defeq (t_n, n \in [I,N])
1473\end{equation}
1474\begin{equation}
1475\tail(T) \defeq t_N
1476\end{equation}
1477\begin{equation}
1478\join(S,T) \defeq (s_1, \dots, s_M, t_1, \dots, t_N)
1479\end{equation}
1480\end{definition}
1481
1482\begin{lemma}[Tuple functions]
1483\label{lem:tupfunc}
1484Let $\funcseqname{f}^i \defeq (\funcname{f}^i_n, n \in [1,N])$,
1485$i \in [1,3]$, be tuples of functions with
1486$\Domain(\funcseqname{f}^2) = \Range(\funcseqname{f}^1)$ and
1487$\Domain(\funcseqname{f}^3) = \Range(\funcseqname{f}^2)$. Then
1488$
1489(\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1 =
1490\funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1)
1491$.
1492
1493\begin{proof}
1494\begin{equation*}
1495\begin{split}
1496  (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2) \compose[()] \funcseqname{f}^1
1497  & =
1498  \bigl ( (\funcname{f}^3_n \compose \funcname{f}^2_n) \compose \funcname{f}^1_n, n \in [1,N] \bigr )
1499\\*
1500  & =
1501  \bigl ( \funcname{f}^3_n \compose (\funcname{f}^2_n \compose \funcname{f}^1_n), n \in [1,N] \bigr )
1502\\*
1503  & =
1504  \funcseqname{f}^3 \compose[()] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1)
1505\end{split}
1506\end{equation*}
1507\end{proof}
1508
1509Let $\funcseqname{f} \defeq (\funcname{f}_\alpha, \alpha \prec \Alpha)$
1510be a sequence of functions,
1511$\seqname{D} \defeq \Domain(\funcseqname{f})$ and \\*
1512$\seqname{R} \defeq \Range(\funcseqname{f})$. Then $\ID_\seqname{R}$ is a
1513left $\compose[()]$ identity for $\funcseqname{f}$ and $\ID_\seqname{D}$
1514is a right $\compose[()]$ identity for $\funcseqname{f}$.
1515
1516\begin{proof}
1517\begin{equation*}
1518\begin{split}
1519  \ID_\seqname{R} \compose[()] \funcseqname{f}
1520  & =
1521  (\Id_{\range(\funcname{f}_\alpha)} \compose \funcname{f}_\alpha, \alpha \prec \Alpha)
1522\\*
1523  & =
1524  (\funcname{f}_\alpha, \alpha \prec \Alpha)
1525\\*
1526  & =
1527  \funcseqname{f}
1528\end{split}
1529\end{equation*}
1530\begin{equation*}
1531\begin{split}
1532  \funcseqname{f} \compose[()] \ID_\seqname{D}
1533  & =
1534  (\funcname{f}_\alpha \compose \Id_{\domain(\funcname{f}_\alpha)}, \alpha \prec \Alpha)
1535\\*
1536  & =
1537  (\funcname{f}_\alpha, \alpha \prec \Alpha)
1538\\*
1539  & =
1540  \funcseqname{f}
1541\end{split}
1542\end{equation*}
1543\end{proof}
1544\end{lemma}
1545
1546\begin{definition}[Tuple composition for labeled morphisms]
1547\label{def:labcomp}
1548Let $\seqname{M}^i \defeq (\funcseqname{f}^i, o^i_1, o^i_2)$, $i=1,2$,
1549be tuples such that each $\funcseqname{f}^i$ is a sequence of functions
1550or each $\funcseqname{f}^i$ is a tuple of functions,
1551$\Range(\funcseqname{f}^1) = \Domain(\funcseqname{f}^2)$ and
1552$0^1_2=o^2_1$. Then
1553
1554\begin{equation}
1555\seqname{M}^2 \compose[A] \seqname{M}^1 \defeq
1556\bigl (
1557  \funcseqname{f}^2 \compose[()] \funcseqname{f}^1,
1558  o^1_1,
1559  o^2_2
1560\bigr )
1561\end{equation}
1562\end{definition}
1563
1564\begin{lemma}[Tuple composition for labeled morphisms]
1565\label{lem:labcomp}
1566Let $\seqname{M}^i \defeq (\funcseqname{f}^i, o^i_1, o^i_2)$,
1567$i \in [1,3]$,
1568be a tuple such that $\funcseqname{f}^i$ is a sequence or tuple of
1569functions,
1570$\Range(\funcseqname{f}^i) = \Domain(\funcseqname{f}^{i+1})$
1571and $o^i_2 = o^{i+1}_1$, $i=1,2$.
1572Then
1573\begin{equation}
1574\seqname{M}^3 \compose[A] \bigl ( \seqname{M}^2 \compose[A] \seqname{M}^1 \bigr )
1575=
1576\bigl ( \seqname{M}^3 \compose[A] \seqname{M}^2 \bigr ) \compose[A] \seqname{M}^1
1577\end{equation}
1578
1579\begin{proof}
1580From \fullcref{def:labcomp}\negmedspace, \\
1581{
1582  \showlabelsinline
1583  \pagecref{lem:seqfunc}
1584}
1585and \fullcref{lem:tupfunc}\negmedspace, we have
1586\begin{equation*}
1587\begin{split}
1588  \seqname{M}^3 \compose[A] (\seqname{M}^2 \compose[A] \seqname{M}^1)
1589  & =
1590  \seqname{M}^3 \compose[A] (\funcseqname{f}^2 \compose[()] \funcseqname{f}^1, o^1_1, o^2_2)
1591\\*
1592  & =
1593  (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2 \compose[()] \funcseqname{f}^1, o^1_1, o^3_2)
1594\\*
1595  & =
1596  (\funcseqname{f}^3 \compose[()] \funcseqname{f}^2, o^2_1, o^3_2) \compose[A] \seqname{M}^1
1597\\*
1598  & =
1599  (\seqname{M}^3 \compose[A] \seqname{M}^2) \compose[A] \seqname{M}^1
1600\end{split}
1601\end{equation*}
1602\end{proof}
1603
1604Let $\seqname{D}^i \defeq \Domain(\funcseqname{f}^i)$ and
1605$\seqname{R}^i \defeq \Range(\funcseqname{f}^i)$. Then
1606\begin{equation}
1607(\ID_{\seqname{R}^i}, o^i_2, o^i_2) \compose[A] \seqname{M}^i =
1608\seqname{M}^i
1609\end{equation}
1610\begin{equation}
1611\seqname{M}^i \compose[A] (\ID_{\seqname{D}^i}, o^i_1, o^i_1) =
1612\seqname{M}^i
1613\end{equation}
1614
1615\begin{proof}
1616\begin{equation*}
1617\begin{split}
1618  (\ID_{\seqname{R}^i}, o^i_2, o^i_2) \compose[A] \seqname{M}^i
1619  & =
1620  (\ID_{\seqname{R}^i} \compose[()] \funcseqname{f}^i, o^i_1, o^i_2)
1621\\*
1622  & =
1623  (\funcseqname{f}^i, o^i_1, o^i_2)
1624\\*
1625  & =
1626 \seqname{M}^i
1627\end{split}
1628\end{equation*}
1629\begin{equation*}
1630\begin{split}
1631  \seqname{M}^i \compose[A] (\ID_{\seqname{D}^i}, o^i_1, o^i_1)
1632  & =
1633  (\funcseqname{f}^i \compose[()] \ID_{\seqname{D}^i}, o^i_1, o^i_2)
1634\\*
1635  & =
1636  (\funcseqname{f}^i, o^i_1, o^i_2)
1637\\*
1638  & =
1639 \seqname{M}^i
1640\end{split}
1641\end{equation*}
1642\end{proof}
1643\end{lemma}
1644
1645\begin{definition}[Cartesian product of sequence]
1646\label{def:Cart}
1647Let $\seqname{S}^i \defeq (S^i_\alpha, \alpha \prec \Alpha)$, $i=1,2$,
1648be a sequence and
1649$\funcseqname{f} \defeq (\funcname{f}_\alpha \maps S^1_\alpha \to
1650S^2_\alpha, \alpha \prec \Alpha)$
1651be a sequence of functions, then
1652$
1653  \bigtimes \seqname{S}^i \defeq
1654  \bigtimes_{\alpha \prec \Alpha} S^i_\alpha
1655$
1656is the generalized Cartesian product of the sequence $\seqname{S}^i$ and
1657$
1658  \bigtimes \funcseqname{f} \maps \seqname{S}^1 \to \seqname{S}^2
1659  \defeq
1660  \bigtimes_{\alpha \prec \Alpha}\funcname{f}_\alpha
1661$
1662is the generalized Cartesian product of the function sequence
1663$\funcseqname{f}$.
1664\end{definition}
1665
1666\begin{definition}[underline]
1667Let $\seqname{S}^1 \defeq (S^1_\alpha, \alpha \preceq \Alpha)$,
1668$\seqname{S}^2 \defeq (S^2_\alpha, \alpha \preceq \Alpha)$ be
1669sequences and
1670$
1671  \funcseqname{f} \defeq
1672  (
1673    \funcname{f}_\alpha \maps S^1_\alpha \to S^2_\alpha,
1674    \alpha \preceq \Alpha
1675  )
1676$
1677be a sequence of functions, then
1678\begin{equation}
1679\underline{\funcname{f}} \maps head(S_1) \to head(S_2) \defeq
1680\bigtimes \head(\funcseqname{f}) =
1681\bigtimes_{\alpha \prec \Alpha} \funcname{f}_\alpha
1682\end{equation}
1683is the function mapping
1684$(s_\alpha \in S^1_\alpha, \alpha \prec \Alpha)$
1685into $(\funcname{f}_\alpha(s_\alpha), \alpha \prec \Alpha)$.
1686\end{definition}
1687
1688\begin{definition}[Head and tail compositions]
1689Let \\
1690$\funcname{f}^1 \maps (S^1_\alpha, \alpha \prec \Alpha) \to S^1_\Alpha$,
1691$\funcname{f}^2 \maps (S^2_\alpha, \alpha \prec \Alpha) \to S^2_\Alpha$
1692and
1693$
1694  \funcseqname{g} \defeq
1695  (
1696    \funcname{g}_\alpha \maps S^1_\alpha \to S^2_\alpha,
1697    \alpha \preceq \Alpha
1698  )
1699$.
1700Then (see \cref{fig:fg,fig:gf})
1701
1702\begin{subequations}
1703\begin{equation}
1704\funcname{f}^2 \composeh \funcseqname{g} \defeq
1705\funcname{f}^2 \compose \underline{\funcseqname{g}}
1706\end{equation}
1707\begin{equation}
1708\funcseqname{g} \composet \funcname{f}^1 \defeq
1709\tail(\funcseqname{g}) \compose \funcname{f}^1
1710\end{equation}
1711\end{subequations}
1712\end{definition}
1713
1714\begin{figure}
1715\[ \bfig
1716\node s1(0,0)[{\head(S^1)}]
1717\node s2(0,-1000)[{\head(S^2)}]
1718\node s2a(1000,-1000)[{S^2_\Alpha}]
1719\arrow |l|[s1`s2;{\underline{\funcseqname{g}}}]
1720\arrow |r|[s1`s2a;{\funcseqname{f}^2 \composeh \funcseqname{g}}]
1721\arrow |b|[s2`s2a;{\funcseqname{f^2}}]
1722\efig \]
1723\caption{$\funcname{f}^2 \composeh \funcseqname{g}$}
1724\label{fig:fg}
1725\end{figure}
1726
1727\begin{figure}
1728\[ \bfig
1729\node s1(0,0)[{\head(S^1)}]
1730\node s1a(0,-1000)[{S^1_\Alpha}]
1731\node s2a(1000,-1000)[{S^2_\Alpha}]
1732\arrow |l|[s1`s1a;{\funcname{f}^1}]
1733\arrow |r|[s1`s2a;{\funcseqname{g} \composet \funcseqname{f}^1}]
1734\arrow |b|[s1a`s2a;{\tail(\funcseqname{g}) = \funcseqname{g}_\Alpha}]
1735\efig \]
1736\caption{$\funcseqname{g} \composet \funcname{f}^1$}
1737\label{fig:gf}
1738\end{figure}
1739
1740\begin{definition}[Topology functions]
1741Let $S$ be a topological space and $Y$ a subset. Then \\
1742\begin{enumerate}
1743\item $\Topology(S)$ is the topology of $S$.
1744\item
1745$\Topology(Y,S) \defeq \set{U \cap Y}[{U \in \Topology(S)}]$
1746is the relative topology of $Y$.
1747\item
1748$\Top(Y,S) \defeq \bigl ( Y, \Topology(Y,S) \bigr )$ is $Y$ with the
1749relative topology.
1750\item
1751$
1752  \op{S} \defeq
1753  \set
1754    {(U, \Topology(U,S))}%
1755    [{{ U \in \Topology(S) \setminus \{ \emptyset \} }}]
1756$
1757is the set of all non-null open subspaces of $S$.
1758\end{enumerate}
1759
1760Let $\seqname{S}$ be a set of topological spaces. Then
1761$\op{\seqname{S}} \defeq \union [{S \in \seqname{S}}] { {\op{S}}}$ is the set
1762of open subspaces in $\seqname{S}$.
1763
1764Let $S$ and $T'$ be spaces, $T \subseteq T'$ be a subspace and
1765$\funcname{f} \maps S \to T$ a function. Then
1766$\funcname{f} \maps S \to T' \defeq \Id_{T,T'} \compose \funcname{f}$
1767is $\funcname{f}$ considered as a function from $S$ to $T'$.
1768
1769Let $S'$ and $T'$ be spaces, $S \subseteq S'$, $T \subseteq T'$ be
1770subspaces and $\funcname{f}' \maps S' \to T'$ a function such that
1771$\funcname{f}'[S] \subseteq T$. Then $\funcname{f}' \maps S \to T$, also
1772written $\funcname{f}' \restrictto_{S,T}$, is
1773$\funcname{f}' \restrictto_S$ considered as a function from $S$ to
1774$T$.
1775
1776Let $\seqname{S}^i \defeq (S^i_\alpha, \alpha \preceq \Alpha)$, $i-1,2$,
1777be a sequence of spaces, $\seqname{S}^1 \SUBSETEQ \seqname{S}^2$ and \linebreak
1778$\funcname{f}^2 \maps \head(\seqname{S}^2) \to \tail(\seqname{S}^2)$ a
1779function.
1780$
1781  \funcname{f}^2 \restrictto_{\head(\seqname{S}^1)} \defeq
1782  \funcname{f}^2 \restrictto_{\bigtimes \head(\seqname{S}^1)}
1783$.
1784If \linebreak
1785$
1786  \funcname{f}^2 \restrictto_{\head(\seqname{S}^1)}
1787    [\bigtimes \head(\seqname{S}^1)]
1788  \subseteq \tail(\seqname{S}^1)
1789$
1790then
1791$ \funcname{f}^2 \maps \head(\seqname{S}^1) \to \tail(\seqname{S}^1)$,
1792also written
1793$
1794  \funcname{f}^2 \restrictto_%
1795    {\head(\seqname{S}^1),\tail(\seqname{S}^1)}
1796$,
1797is $\funcname{f}^2 \restrictto_{\head(\seqname{S}^1)}$ considered as a
1798function from $\bigtimes \head(\seqname{S}^1)$ to $\tail(\seqname{S}^1)$.
1799\end{definition}
1800
1801\begin{definition}[Truth space]
1802$\false \defeq \emptyset$, $\true \defeq \set{\emptyset}$,
1803the truth set is \linebreak
1804$\truthset \defeq \set{\false, \true}$,
1805$\truthtop \defeq \set {\emptyset,\truthset}$ and the truth space
1806$\truthspace \defeq (\truthset, \truthtop)$ is $\truthset$
1807with the indiscrete topology.
1808\end{definition}
1809
1810\begin{definition}[Truth category]
1811The truth category is
1812$
1813   \truthcat \defeq \\
1814   \bigl (
1815     \truthspace,
1816     \set{\truthspace \to \truthspace}
1817   \bigr )
1818$.
1819The truth model space is \\*
1820$\seqname{Truthspace} \defeq (\truthspace, \truthcat)$.
1821\end{definition}
1822
1823\begin{definition}[Constraint functions]
1824A constraint function is a continuous function with range
1825$\truthspace$ or a model
1826function with range $\seqname{Truthspace}$.
1827\end{definition}
1828
1829\begin{definition}[Sequence inclusion]
1830\label{def:seqin}
1831Let $\seqname{S} \defeq (S_\alpha, \alpha \prec \Alpha)$ and
1832$\seqname{T} \defeq (T_\alpha, \alpha \prec \Alpha)$ be sequences.
1833$\seqname{S} \seqin \seqname{T}$ iff
1834$\uquant{\alpha \prec \Alpha} {S_\alpha \in T_\alpha}$ or
1835$\uquant{\alpha \prec \Alpha} {S_\alpha \objin T_\alpha}$.
1836\end{definition}
1837
1838\begin{lemma}[Sequence inclusion]
1839\label{lem:seqin}
1840Let $\seqname{S} \defeq (S_\alpha, \alpha \prec \Alpha)$
1841be a sequence,
1842$
1843  \catseqname{T}^i \defeq \\
1844  (\catname{T}^i_\alpha, \alpha \prec \Alpha)
1845$,
1846$i=1,2$, a sequence of
1847categories, $\catseqname{T}^1 \SUBCAT \catseqname{T}^2$ and
1848$\seqname{S} \seqin \catseqname{T}^1$ Then
1849$\seqname{S} \seqin \catseqname{T}^2$.
1850
1851\begin{proof}
1852If
1853$\uquant{\alpha \prec \Alpha} {S_\alpha \objin \catname{T}^1_\alpha}$
1854then
1855$\uquant{\alpha \prec \Alpha} {S_\alpha \objin \catname{T}^2_\alpha}$.
1856\end{proof}
1857\end{lemma}
1858
1859\begin{thebibliography}{9}
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1861\bibitem[Ad\'amek, Herrlich, Strecker, 1990] {JoyCat} Ji\v{r}\'i Ad\'amek,
1862Horst Herrlich,
1863George E. Strecker.
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1865John Wiley and Sons, Inc., 1990.
1866
1867\bibitem[Kelley, 1955] {GenTop} John L. Kelley,
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1869D. Van Nostrand Company (first edition), 1955.
1870
1871\bibitem[Kobayashi, 1996] {FoundDiffGeo1} Shoshichi Kobayashi,
1872Katsumi Nomizu,
1873\textit{Foundations of Differential Geometry, Volume I},
1874ISBN 0-471-15733-3, John Wiley and Sons, Inc., 1996.
1875
1876%\bibitem[Lee, 2000] {DiffPhys} Jeffrey Marc lee,
1877%\textit{Differential Geometry},
1878%Analysis and Physics, 2000.
1879
1880\bibitem[Mac Lane, 1998] {CftWM} Saunders Mac Lane,
1881\textit{Categories for the Working Mathemation, 2nd edition},
1882ISBN 0-387-98403-8, Springer-Verlag, 1998.
1883
1884\bibitem[Steenrod, 1999] {TopFib} Norman Steenrod,
1885\textit{The Topology Of Fibre Bundles}, ISBN 0-691-00548-6,
1886Prineton University Press (seventh printing), 1999.
1887
1888\end{thebibliography}
1889
1890\end{document}