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% (find-angg "LATEX/istanbulquotes.tex") % (find-angg "LATEX/istanbulquotes.lua") % (defun c () (interactive) (find-LATEXsh "lualatex istanbulquotes.tex")) % (defun d () (interactive) (find-xpdfpage "~/LATEX/istanbulquotes.pdf")) % (defun e () (interactive) (find-LATEX "istanbulquotes.tex")) % (find-xpdfpage "~/LATEX/istanbulquotes.pdf") % % (find-istfile "glyphs.el") % (load "~/LATEX/istanbulglyphs.el") % (eev-uc-set-composes) % % (find-books "__cats/__cats.el" "johnstone-elephant") % Â«.charsÂ»(to "chars") % % Â«.luaÂ»(to "lua") % Â«.fourmanÂ»(to "fourman") % Â«.mmoeÂ»(to "mmoe") % Â«.elephant-A4Â»(to "elephant-A4") % Â«.elephant-A4.1.1Â»(to "elephant-A4.1.1") % Â«.elephant-A4.1.4Â»(to "elephant-A4.1.4") % Â«.elephant-A4.1.5Â»(to "elephant-A4.1.5") % Â«.elephant-A4.1.8Â»(to "elephant-A4.1.8") % Â«.elephant-A4.1.10Â»(to "elephant-A4.1.10") % Â«.elephant-A4.2.6Â»(to "elephant-A4.2.6") % Â«.elephant-A4.2.7Â»(to "elephant-A4.2.7") % Â«.elephant-A4.2.9Â»(to "elephant-A4.2.9") % Â«.elephant-A4.2.10Â»(to "elephant-A4.2.10") % Â«.elephant-A4.2.12Â»(to "elephant-A4.2.12") % Â«.elephant-A4.3.6Â»(to "elephant-A4.3.6") % Â«.elephant-A4.3.9Â»(to "elephant-A4.3.9") % Â«.elephant-A4.5.2Â»(to "elephant-A4.5.2") % Â«.elephant-A4.5.8Â»(to "elephant-A4.5.8") % Â«.elephant-A4.5.9Â»(to "elephant-A4.5.9") % Â«.elephant-A4.5.10Â»(to "elephant-A4.5.10") % Â«.elephant-A4.5.20Â»(to "elephant-A4.5.20") % Â«.elephant-A4.6.2Â»(to "elephant-A4.6.2") % Â«.elephant-A4.6.5Â»(to "elephant-A4.6.5") % Â«.elephant-A4.6.6Â»(to "elephant-A4.6.6") % Â«.elephant-A4.6.10Â»(to "elephant-A4.6.10") \documentclass[oneside]{book} \usepackage[utf8]{inputenc} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{tikz} \usepackage{luacode} \usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref") \usepackage{edrx15} % (find-angg "LATEX/edrx15.sty") % \usepackage{proof} % For derivation trees ("%:" lines) \input diagxy % For 2D diagrams ("%D" lines) %\xyoption{curve} % For the ".curve=" feature in 2D diagrams % \begin{document} % \directlua{dofile "\jobname.lua"} \widemtos % Â«charsÂ»(to ".chars") \input istanbuldefs % (find-istfile "defs.tex") \catcode`Â°=13 \defÂ°{^\degree} \catcode`ÂΉ=13 \defÂΉ{^{-1}} \catcode`Â²=13 \defÂ²{^2} \catcode`Â³=13 \defÂ³{^3} %\catcode`ÂΌ=13 \defÂΌ{} %\catcode`Â½=13 \defÂ½{} %\catcode`ÂΎ=13 \defÂΎ{} \catcode`Â±=13 \defÂ±{\pm} \catcode`Ã·=13 \defÃ·{\div} \catcode`Â·=13 \defÂ·{\cdot} \catcode`Ã=13 \defÃ{\times} \catcode`Â¬=13 \defÂ¬{\neg} \catcode`Â§=13 \defÂ§{\S} \catcode`Ξ=13 \defΞ{\Delta} \catcode`Ξ=13 \defΞ{\Theta} \catcode`Ξ©=13 \defΞ©{\Omega} \catcode`Ξ±=13 \defΞ±{\alpha} \catcode`Ξ²=13 \defΞ²{\beta} \catcode`Ξ³=13 \defΞ³{\gamma} \catcode`Ξ∧=13 \defΞ∧{\delta} \catcode`Ξ΅=13 \defΞ΅{\epsilon} \catcode`Ξ·=13 \defΞ·{\eta} \catcode`ΞΈ=13 \defΞΈ{\theta} \catcode`Ξ»=13 \defΞ»{\lambda} \catcode`Ï=13 \defÏ{\pi} \catcode`Ï=13 \defÏ{\omega} \catcode`â=13 \defâ{\ot} \catcode`â=13 \defâ{\upto} \catcode`â=13 \defâ{\to} \catcode`â=13 \defâ{\dnto} \catcode`â=13 \defâ{\bij} \catcode`â=13 \defâ{\updownarrow} \catcode`â=13 \defâ{\nwarrow} \catcode`â=13 \defâ{\nearrow} \catcode`â=13 \defâ{\searrow} \catcode`â=13 \defâ{\swarrow} \catcode`â£=13 \defâ£{\epito} \catcode`â£=13 \defâ£{\twoheadrightarrow} \catcode`â¦=13 \defâ¦{\mapsto} \catcode`â=13 \defâ{\funto} \catcode`â=13 \defâ{\forall} \catcode`â=13 \defâ{\exists} \catcode`â=13 \defâ{\in} \catcode`â=13 \defâ{\circ} \catcode`â=13 \defâ{\infty} \catcode`â§=13 \defâ§{\land} \catcode`â¨=13 \defâ¨{\lor} \catcode`â©=13 \defâ©{\cap} \catcode`âª=13 \defâª{\cup} \catcode`â=13 \defâ{\simeq} \catcode`â =13 \defâ {\cong} \catcode`â€=13 \defâ€{\le} \catcode`â₯=13 \defâ₯{\ge} \catcode`â=13 \defâ{\subset} \catcode`â=13 \defâ{\supset} \catcode`â=13 \defâ{\subseteq} \catcode`â=13 \defâ{\supseteq} \catcode`â’=13 \defâ’{\vdash} \catcode`â£=13 \defâ£{\dashv} \catcode`â€=13 \defâ€{\top} \catcode`â₯=13 \defâ₯{\bot} \catcode`â=13 \defâ{\flat} \catcode`â―=13 \defâ―{\sharp} %\catcode`*=13 \def*{\ensuremath{\bullet}} %\catcode`=13 \def{\mathscr} %\catcode`=13 \def{\mathbf} %\catcode`=13 \def{\par\noindent} %\catcode`=13 \def{\par} %\catcode`=13 \def{\mathcal} \def\bfy{\mathbf{y}} \def\bfyU{\mathbf{y}U} \def\BF#1{\noindent{\bf#1}\quad} \def\liml{\underleftarrow {\lim}{}} \def\limr{\underrightarrow{\lim}{}} \def\frakCat{\mathfrak{Cat}} \def\frakTop{\mathfrak{Cat}} \def\elephantpage#1{((page #1))} \def\ob{{\operatorname{ob}}} \def\sh{\mathbf{sh}} \def\Sh{\mathbf{Sh}} \def\Sp{\mathbf{Sp}} \def\Lop{\mathbf{Lop}} \def\Hom{\mathrm{Hom}} \def\sdd{\ssk{\scriptsize (...)}\ssk} \def\mdd{\msk{\scriptsize (...)}\msk} % Â«luaÂ»(to ".lua") % (find-LATEX "2015logicandcats.lua") % \directlua{ % dednat6dir = "/home/edrx/dednat5/" % dofile(dednat6dir.."dednat6.lua") % dofile(os.getenv("LUA_INIT"):sub(2)) % } % % % \catcode`\^^J=10 % (find-es "luatex" "spurious-omega") % \directlua{dofile "istanbulall.lua"} % (find-ist "all.lua") % \directlua{output = mytexprint} % (find-ist "all.lua" "output") % \directlua{output = printboth} % (find-ist "all.lua" "output") % \directlua{tf = TexFile.read(tex.jobname..".tex")} % \def\Diag#1{\directlua{tf:processuntil()}\diag{#1}} % \def\Ded #1{\directlua{tf:processuntil()}\ded{#1}} % \def\Expr#1{\directlua{tf:processuntil() output(#1)}} % \def\Expr#1{\directlua{tf:processuntil() output(tostring(#1))}} % %L output(preamble1) -- (find-dn5 "preamble6.lua") % \directlua{tf:processuntil()} \catcode`\^^J=10 \directlua{dednat6dir = "dednat6/"} \directlua{dofile(dednat6dir.."dednat6.lua")} \directlua{texfile(tex.jobname)} \directlua{verbose()} \directlua{output(preamble1)} \def\expr#1{\directlua{output(tostring(#1))}} \def\eval#1{\directlua{#1}} \def\pu{\directlua{pu()}} % ------------------------------------------- % _____ % | ___|__ _ _ _ __ _ __ ___ __ _ _ __ % | |_ / _ \| | | | '__| '_ ` _ \ / _` | '_ \ % | _| (_) | |_| | | | | | | | | (_| | | | | % |_| \___/ \__,_|_| |_| |_| |_|\__,_|_| |_| % % Â«fourmanÂ»(to ".fourman") % (find-books "__cats/__cats.el" "fourman") % (find-slnm0753page (+ 14 329) "2.18 Elementary J-operators") This is an excerpt -- pages 329-331 -- from M.P. Fourman and D.S. Scott's ``Sheaves and Logic'', that was published in SLNM0753 (``Applications of Sheaves: Proceedings of the Research Symposium on Applications of Sheaf Theory to Logic, Algebra and Analysis - Durham, july 9-21, 1977''). \bsk \BF{2.18. ELEMENTARY J-OPERATORS.} In these examples $Ξ©$ is a given cHa. (i) {\sl The closed quotient.} The operator is defined by % $$J_a p = a â¨ p.$$ % This is obviously a J-operator, and the congruence relation is: % $$aâ¨p = aâ¨q.$$ % The set of fixed points (quotient lattice) is: % $$\setofst{p â Ξ©}{aâ€p}.$$ Classically speaking in the spatial case where $Ξ©=\Opens(X)$, the quotient corresponds to the topology on the {\sl closed} subspace complementary to the open set $a$. This quotient makes the element $a$ ``false'' and is the least such. \msk (ii) {\sl The open quotient.} The operator is defined by: % % (find-slnm0753page (+ 14 330)) % $$J^a p = aâp.$$ % The congruence relation is: % $$aâ§p = aâ§q \qquad \text{(equivalently, $a â€ pâq$)}.$$ % The set of fixed points is thus isomorphic to % $$Ξ©_a = \setofst{pâΞ©}{pâ€a}.$$ Intuitionistically speaking in the spatial case, this quotient corresponds to the topology on the open subspace $a$. This quotient makes $a$ ``true'' and is the leat such. \msk (iii) {\sl The Boolean quotient}. The operator is defined by: % $$B_a p = (pâa)âa.$$ % The congruence relation is: % $$pâa = qâ a.$$ % The set of fixed points is % $$\setofst{pâΞ©}{(pâa)âaâ€p}.$$ In case $a=â₯$, this is the well-known $Â¬Â¬$-quotient giving the (complete) Boolean algebra of ``stable'' elements. For arbitrary $a$, we could first form $Ξ©/J_a$ and follow this by the $Â¬Â¬$-quotient to obtain $Ξ©/B_a$. (In general, if $Jâ€K$, then $Ξ©/K$ is a quotient of $Ξ©/J$.) We remark that in general $Ξ©/J$ is a cBa iff $J=B_{J_â₯}$. Further, is $Ξ©$ is already Boolean, then {\sl every} J-operator on $Ξ©$ is of the form $B_a = J_a$. \msk (iv) {\sl The forcing quotient}. The operator is a combination of previous ones: % $$(J_aâ§J^b)p = (aâ¨p)â§(bâp).$$ % The congruence relation is a conjunction: % $$aâ¨b=aâ¨q \quad \text{and} \quad bâ§p=bâ§q.$$ % The set of fixed points is: % $$\setofst{pâΞ©}{bâpâ€ aâp}.$$ The point of the quotient is that it provides the {\sl least} J-operator such that $Jaâ€Jb$; that is, we take the least quotient that ``forces'' $aâb$ to be true. If we want to force a sequence of statements $a_iâb_i$, for $i<n$, the operator needed is $\bigvee_{i<n} (J_{a_i}â§J{b_i})$. It is important to note that in general sup's of J-operators cannot be calculated pointwise. We shall see below, however, that it is possible to find a finite expression for this particular sup. (We owe this remark to John Cartmell). \msk % (find-slnm0753page (+ 14 331)) (vi) {\sl A mixed quotient.} The interest of this example lies in the fact that it has a neat finite definition: % $$(B_aâ§J^a)p = (pâa)âp.$$ % The congruence relation is: % $$(pâa)âp = (qâa)âq,$$ % which is equivalent to the conjunction: % $$aâ§p = aâ§q \quad\text{and}\quad pâa = qâa.$$ % The set of fixed points is: % $$\setofst{pâΞ©}{(pâa)âp â€ p}.$$ It is difficult to make this set vivid except to say that it is the set of elements p satisfying Pierce's law (for a fixed a). \msk If we take a polynomial in $â$, $â§$, $â¨$, $â₯$, say $f(p,a,b,\ldots)$, it is a decidable question whether for all $a$, $b$, $\ldots$ it defines a J-operator. This does not, however, help us very much in cataloguing such operators (nor in seeing what good they are!) Some techniques can be developed from the following formulae, which were pointed out to us by Cartmell, see also [33] and [47]. \msk \BF{2.19. PROPOSITION.} In the following, $K$ is an arbitrary J-operator, $â€$ is the constant function (the greatest J-operator with the most trivial quotient), and $â₯$ is the least J-operator (namely, the identity function on $Ξ©$): $$ \def\li#1 #2 #3 #4 #5 #6 #7 #8 {\text{#1}& &\text{#5}& \\} \begin{array}{rlclcrlclc} \li (i) J_aâ¨J_b = J_{aâ¨b} (ii) J^aâ¨J^b = J^{aâ§b} \li (iii) J_aâ§J_b = J_{aâ§b} (iv) J^aâ§J^b = J^{aâ¨b} \li (v) J_aâ§J^a = â₯ (vi) J_aâ¨J^a = â€ \li (vii) J_aâ¨K = KâJ_a (viii) J^aâ¨K = J^aâK \li (ix) J_aâ¨B_a = B_a (x) J^aâ¨B_b = B_{aâb} \end{array} $$ Proof. Equations (i)-(iv) are easy calculations; while (v) comes down to showing $p=(aâ¨p)â§(aâp)$. Formula (vi) is a direct consequence of (vii) (equally, of (viii)). \newpage % ------------------------------------------------------- % __ __ _ ____ __ _ _ _ _ % | \/ | | / / \/ | ___ ___ _ __ __| (_)(_) | __ % | |\/| | | / /| |\/| |/ _ \ / _ \ '__/ _` | || | |/ / % | | | | |___ / / | | | | (_) | __/ | | (_| | || | < % |_| |_|_____/_/ |_| |_|\___/ \___|_| \__,_|_|/ |_|\_\ % |__/ % Â«mmoeÂ»(to ".mmoe") % (find-books "__cats/__cats.el" "moerdijk") % (find-maclanemoerdijkpage (+ -29 69) "2. Sieves and Sheaves") % (find-maclanemoerdijktext (+ -29 69)) This is an excerpt -- pages 69-71 -- from MacLane and Moerdijk's ``Sheaves in Geometry and Logic - A First Introduction to Topos Theory'' (1992). \bsk \BF{2. Sieves and Sheaves} On any space $X$, each open set $U$ determines a presheaf $\Hom(-, U)$ defined, for each open set $V$, by % % (find-maclanemoerdijkpage (+ -29 70)) % $$ \Hom(V, U) = \begin{cases} 1 & \text{if} \quad $V â U$, \\ \empty & \text{otherwise,} \\ \end{cases} % \qquad\qquad(1) $$ % where 1 is the one-point set. This presheaf is clearly a sheaf; it is the representable presheaf $\bfy(U) = \Hom(â, U)$ on the category $\Opens(X)$. Recall from Â§1.4 that a {\sl sieve} $S$ on $U$ in this category is defined to be a subfunctor of $\Hom(â, U)$. Replacing the sieve $S$ by the set (call it $S$ again) of all those $V â U$ with $SV = 1$, we may also describe a sieve on $U$ as a subset $S â \Opens(U)$ of objects such that $V_0 â V â S$ implies $V_0 â S$. Each indexed family $\setofst{V_i â U}{i â I}$ of subsets of U generates ($=$ ``spans'') a sieve $S$ on $U$; namely, the set $S$ consisting of all those open $V$ with $V â V_i$ for some $i$; in particular, each $V_0 â U$ determines a {\sl principal sieve} $(V_0)$ on $U$, consisting of all $V$ with $V â V_0$. It is not difficult to see that a sieve $S$ on $U$ is principal iff the subfunctor $S$ of $\bfy(U)$ is a subsheaf (Exercise 1). A sieve $S$ on $U$ is said to be a {\sl covering} sieve for $U$ when $U$ is the union of all the open sets $V$ in $S$. In the definition of a sheaf, we may replace {\sl open coverings} by {\sl covering sieves}, as follows: \msk \BF{Proposition 1.} A presheaf P on X is a sheaf if and only if, for every open set $U$ of $X$ and every covering sieve $S$ on $U$, the inclusion $i_s: S â \bfyU$ of functors induces an isomorphism, % $$\Hom(\bfyU, P) â \Hom(S, Π ). \qquad\qquad (2)$$ % (Here each $\Hom$ is the set of natural transformations.) \msk \BF{Proof:} For any presheaf $P$ on the space $X$ and any covering of an open set $U$ by $U_i$, we can construct the equalizer $E$ in the diagram % % (find-es "xypic" "two-and-three") % $$ (B â f) \diagxyto/->/^U \calC \diagxyto/->/^F \Set $$ % $$ E \diagxyto/->/^d \prod_i PU_i \two/->`->/ \prod_{i,j} P(U_i,U_j)$$ % Specifically, $E$ consists of those families of elements $x_i â PU_i$ with $x_i|_{U_iâ©U_j} = x_j|_{U_iâ©U_j}$ for all pairs of indices $(i,j)$. Now replace the covering $U_i$ by the corresponding sieve $S$, consisting of all open sets $V$ with $V â U_i$ for some $i$, and for each $V$ define $x_V$ to be $x_i|_V$. By the assumption that the $x_i$ match on intersections $U_iâ©U_j$, the $x_V$ so defined are independent of the choice of the index $i$ with $V â U_i$. Therefore, the equalizer $E$ can be described as the set of those families of elements $x_V â PV$ for $V â S$ with $x_V|_{V'} = x_{V'}$ whenever $V' â V$. Now regard $S$ as a functor $\Opens(X)^\op â \Sets$ with $SV = 1$ for those $V â S$ and $SV = \empty$ otherwise. Each element $x_V â PV$ is then a map $SV â PV$, so the equalizer $E$ is now described as the set of natural transformations % % (find-maclanemoerdijkpage (+ -30 71)) % $ΞΈ: S â P$ [where $ΞΈ_V(1)$ is $x_V$]. Next use the inclusion $i_S: S â \bfyU$ to form the diagram % %D diagram mmoe-p71 %D 2Dx 100 +60 +60 %D 2D 100 A B B' %D 2D %D 2D +40 C D %D 2D %D (( A .tex= \Hom(S,P) B .tex= \prod_{i}PU_i B' .tex= \prod_{i,j}P(U_iâ©U_j) %D C .tex= \Hom(\bfyU,P) D .tex= PU, %D A B -> .plabel= a d %D B B' -> sl^ .plabel= a p %D B B' -> sl_ .plabel= b q %D C A -> .plabel= l (i_S)^* %D D B -> .plabel= r e %D C D -> .plabel= a \bfy %D C D -> .plabel= b â %D )) %D enddiagram %D $$\Diag{mmoe-p71} \qquad\qquad (3)$$ % where $\bfy$ is the isomorphism given by the Yoneda lemma, while the maps $e$, $p$, and $q$ are described as before in (1.2), and the equalizer $d$ is the function which sends each natural transformation $ΞΈ: S â P$ to the family $ΞΈ_{U_i}(1) â PU_i$ of its values, for $i â I$. For this diagram, one verifies that the square in the middle always commutes, so that $e$ does, in fact, always factor through the equalizer $d$ of $p$ and $q$. Therefore, $P$ is a sheaf (i.e., $e$ is the equalizer) if and only if, for every covering $U_i$, the left-hand vertical map $(i_S)*$ is an isomorphism, where $S$ is the corresponding covering sieve. This proposition has the theoretical advantage of describing sheaves wholly in terms of objects (presheaves and sieves) of the category of presheaves. It also slightly simplifies some proofs of facts about sheaves. Moreover, it will be used as a definition of sheaves in terms of a more general notion of covering (Chapter III). As said before, the category $\Sh(X)$ for the space $X$ is a full subcategory of the functor category (the category of presheaves) $\Sets^{\Opens(X)^\op}$: % $$ \Sh(X) â£ \Opens(X) = \Sets^{\Opens(X)^\op} \qquad\qquad (4) $$ We will soon see (Â§5) that this inclusion functor has a left adjoint. This will imply the second part of \BF{Proposition 2.} For any space X the category Sh(X) has all small limits, and the inclusion of sheaves in presheaves preserves all these limits. % \end{document} % (find-854 "" "yoneda-lemma") % (find-854 "" "yoneda-lemma-2") % (find-854page 32 "yoneda-lemma") % (find-854page 33 "yoneda-lemma-2") \newpage % ----------------------------------------- % _____ _ _ _ % | ____| | ___ _ __ | |__ __ _ _ __ | |_ % | _| | |/ _ \ '_ \| '_ \ / _` | '_ \| __| % | |___| | __/ |_) | | | | (_| | | | | |_ % |_____|_|\___| .__/|_| |_|\__,_|_| |_|\__| % |_| All the extracts below are from Peter Johstone's ``Sketches of an Elephant'', vol.1, section A4: ``Geometric Morphisms - Basic Theory''. % (setq last-kbd-macro (kbd "M-A 2*<up> C-a C-SPC <down> C-w M-o C-y M-o <down>")) % Â«elephant-A4Â»(to ".elephant-A4") % Â«elephant-A4.1.1Â»(to ".elephant-A4.1.1") % (find-elephanttext (+ 17 161)) % (find-elephantpage (+ 17 161) "A4 Geometric Morphisms - Basic Theory") % (find-elephantpage (+ 17 161) "Definition 4.1.1") \msk \BF{Definition 4.1.1} (a) Let $\calE$ and $\calF$ be toposes. A geometric morphism $f: \calF â \calE$ consists of a pair of functors $f_*: \calF â \calE$ (the direct image of f) and $f^*: \calE â \calF$ (the inverse image of $f$) together with an adjunction ($f^* â£ f_*$), such that $f^*$ is cartesian (i.e. preserves finite limits). \ssk (b) Let $f$ and $g: \calF â \calE$ be geometric morphisms. A geometric transformation $Ξ±: f â g$ is defined to be a natural transformation $Ξ±: f^* â g^*$. \bsk % Â«elephant-A4.1.4Â»(to ".elephant-A4.1.4") % (find-elephanttext (+ 17 163)) % (find-elephantpage (+ 17 163) "Example 4.1.4") \BF{Example 4.1.4} Let $f: \calC â \calD$ be a functor between small categories. Then composition with $f$ defines a functor $f^*: [\calD, \Set] â [\calC, \Set]$, which has adjoints on both sides, the left and right {\sl Kan extensions} along $f$: for example, the right Kan extension $\liml_f$ sends a functor $F: \calC â \Set$ to the functor whose value at an object $B$ of $\calD$ is the limit of the diagram % % (find-es "xypic" "two-and-three") $$ (B â f) \diagxyto/->/^U \calC \diagxyto/->/^F \Set $$ % (here $(B â f)$ is the comma category whose objects are pairs $(A,\phi)$ with $\phi: B â fA$ in $\calD$, and $U$ is the forgetful functor from this category to $\calC$). Thus $f^*$ is the inverse image of a geometric morphism $[\calC, \Set] â [\calD, \Set]$, whose direct image is $\liml_f$. Moreover, any natural transformation $Ξ±: f â g$ between functors $\calC â \calD$ induces a natural transformation $f^* â g^*$ (whose value at $F$ is the natural transformation $FΞ±: Ff â Fg$), i.e. a geometric transformation \elephantpage{164} $(\liml_f,f^*) â (\liml_g,g^*)$. Thus the assignment $\calC \mto [\calC,\Set]$ can be made into a functor (that is, a 2-functor) from the 2-category $\frakCat$ of small categories, functors and natural transformations into $\frakTop$ (in fact into $\frakTop/\Set$). \sdd We note that the geometric morphisms which arise as in 4.1.4, though not as special as those of 4.1.2, still have the property that their inverse image functors have left adjoints as well as right adjoints. We call a geometric morphism $f$ {\it essential} if it has this property; we normally write $f_!$ for the left adjoint of $f^*$. With the aid of this notion, we can prove a partial converse to 4.1.4: \bsk % Â«elephant-A4.1.5Â»(to ".elephant-A4.1.5") % (find-elephanttext (+ 17 164)) % (find-elephantpage (+ 17 164) "Lemma 4.1.5") \BF{Lemma 4.1.5} Let $\calC$ and $\calD$ be small categories such that $\calD$ is Cauchy-complete (cf.\ 1.1.10). Then every essential geometric morphism $f: [\calC,\Set] â [\calD, \Set]$ is induced as in 4.1.4 by a functor $\calC â \calD$. \bsk % Â«elephant-A4.1.8Â»(to ".elephant-A4.1.8") % (find-elephanttext (+ 17 165)) % (find-elephantpage (+ 17 165) "Example 4.1.8") \BF{Example 4.1.8} Let $(\calC,T)$ be a small site, as defined in 2.1.9. The inclusion functor $\Sh(\calC,T) â [\calC^\op,\Set]$ has a cartesian left adjoint (the {\it associated sheaf functor} --- this is a special case of a result which we shall prove in 4.4.8 below), so it is the direct image of a geometric morphism. \bsk % Â«elephant-A4.1.10Â»(to ".elephant-A4.1.10") % (find-elephanttext (+ 17 165)) % (find-elephantpage (+ 17 165) "Example 4.1.10") \BF{Example 4.1.10} Let $\calC$ and $\calD$ be small cartesian categories, and $f:\calCâ\calD$ a cartesian functor. We shall show that in this case the left Kan extension functor $\limr_f [\calC^\op,\Set], [\calD^\op,\Set]$, whose {\it direct} image is $f^*$ (compare 4.1.4). To verify this, note that for any $B â \ob \calD$, the functor $\limr_f(-)(B) [\calC^\op,\Set] â \Set$ may be described as the composite % $$ [\calC^\op,\Set] \diagxyto/->/^{U^*} [(B â f)^\op, \Set] \diagxyto/->/^{\limr} \Set $$ % \elephantpage{166} where $U:(B â f) â \calC$ is the forgetful functor, as before. \bsk % Â«elephant-A4.2.6Â»(to ".elephant-A4.2.6") % (find-elephanttext (+ 17 180)) % (find-elephantpage (+ 17 180) "Lemma 4.2.6 (iii) and (iv)") \BF{Lemma 4.2.6} Let $f: \calF â \calE$ be a geometric morphism. The following conditions are equivalent: \sdd (iii) $f^*$ is faithful. (iv) The unit $Ξ·$ of the adjunction $(f^* â£ f_*)$ is monic. \sdd A geometric morphism satisfying the equivalent conditions of Lemma 4.2.6 is called a {\it surjection}. We next list some typical examples. % Â«elephant-A4.2.7Â»(to ".elephant-A4.2.7") % (find-elephanttext (+ 17 181)) % (find-elephantpage (+ 17 181) "Examples 4.2.7 (b) and (c)") % (find-elephantpage (+ 17 182) "inclusion") \ssk \BF{Examples 4.2.7} (...) (b) Let $f: \calC â \calD$ be a functor between small categories. If $f$ is surjective on objects, then it is easily verified that the functor $f^*: [\calD, \Set] â [\calC, \Set]$ is conservative; for a natural transformation a between functors $\calD â \Set$ is an isomorphism iff $Ξ±_B$ is bijective for every object $B$ of $\calD$. So the geometric morphism $[\calC, \Set] â [\calD, \Set]$ induced by $f$ as in 4.1.4 is surjective. \sdd (c) Let $f: X â Y$ be a continuous map of topological spaces. If $f$ is surjective, then the geometric morphism $\Sh(X) â \Sh(Y)$ induced by $f$ as in 4.1.11 is a surjection. \bsk % Â«elephant-A4.2.9Â»(to ".elephant-A4.2.9") % (find-elephanttext (+ 17 182)) % (find-elephantpage (+ 17 182) "Lemma 4.2.9") \BF{Lemma 4.2.9} A geometric morphism is an inclusion iff its direct image is a cartesian closed functor (i.e. preserves exponentials). \bsk % Â«elephant-A4.2.10Â»(to ".elephant-A4.2.10") % (find-elephanttext (+ 17 183)) % (find-elephantpage (+ 17 183) "Theorem 4.2.10") \BF{Theorem 4.2.10} Every geometric morphism can be factored, uniquely up to canonical equivalence, as a surjection followed by an inclusion. \bsk % Â«elephant-A4.2.12Â»(to ".elephant-A4.2.12") % (find-elephanttext (+ 17 184)) % (find-elephantpage (+ 17 184) "Examples 4.2.12 (b) and (c)") % (find-elephantpage (+ 17 189) "Grothendieck coverages") \BF{Examples 4.2.12} (...) (b) Let $f: \calC â \calD$ be a functor between small categories. If $f$ is full and faithful, then the induced geometric morphism $[\calC, \Set] â [\calD, \Set]$ is an inclusion; (...) (c) Let $f:XâY$ be a continuous map of topological spaces. Then it is straightforward to verify that $f_*: \Sh(X) â \Sh(F)$ is faithful iff it is full and faithful, iff $fÂΉ: \Opens(Y) â \Opens(X)$ is surjective. If $X$ is a subspace of $Y$ and $f$ is the inclusion, then the latter condition is satisfied; the converse holds (up to homeomorphism) provided $Y$ satisfies the $T_0$ separation axiom, in which case the surjectivity of $fÂΉ$ forces $f$ to be injective. Combining this with 4.2.7(c), we see that if we apply the factorization of 4.2.10 to the morphism $\Sh(X) â \Sh(F)$ induced by an arbitrary continuous $f:XâY$, we obtain $\Sh(I)$, where $I$ is the image of $f$ topologized as a subspace of $Y$ (that is, we obtain the coimage factorization in $\Sp$, rather than the image factorization). \bsk % Â«elephant-A4.3.6Â»(to ".elephant-A4.3.6") % (find-elephanttext (+ 17 189)) % (find-elephantpage (+ 17 189) "Lemma 4.3.6") % (find-elephantpage (+ 17 190) "(b) (i) and (ii)") \BF{Lemma 4.3.6} Let $L$ be a cartesian reflector on a cartesian category $\calE$, corresponding to a reflective subcategory $\calL$, and let $c_L$ denote the universal closure derived from $L$ as in 4.3.2. Let A be an object of â¬. Then \ssk (a) The following are equivalent: (i) A is $c_L$-separated. (ii) The unit map $Ξ·_A: A â LA$ is monic. (iii) $A$ is a subobject of an object of $C$. (iv) The diagonal map $A \mto A Ã A$ is $c_L$ -closed. \ssk (b) The following are equivalent: (i) A is a $c_L$-sheaf. (ii) The unit $Ξ·_a : A â LA$ is an isomorphism. (iii) $A$ is an object of $C$. \bsk % Â«elephant-A4.3.9Â»(to ".elephant-A4.3.9") % (find-elephanttext (+ 17 192)) % (find-elephantpage (+ 17 192) "Theorem 4.3.9") \BF{Theorem 4.3.9} Let $\calE$ be a topos, and $L$ a cartesian reflector on $\calE$, corresponding to a reflective subcategory $\calL$. Then $\calL$ is a topos, and the inclusion $\calL â \calE$ is the direct image of a geometric morphism, whose inverse image is (the factorization through $\calL$ of) L. % Â«elephant-A4.5.2Â»(to ".elephant-A4.5.2") % (find-elephanttext (+ 17 205)) % (find-elephantpage (+ 17 205) "Example 4.5.2") \bsk \BF{Example 4.5.2} Let $\calC$ be a small category, and $\calD$ a full subcategory of $\calC$. Then the geometric morphism $[\calD, \Set] â [\calC, \Set]$ induced by the inclusion $\calD â \calD$ is an inclusion by 4.2.12(b); so it corresponds to a local operator on $[\calC, Set]$. \bsk % Â«elephant-A4.5.8Â»(to ".elephant-A4.5.8") % (find-elephanttext (+ 17 209)) % (find-elephantpage (+ 17 209) "Proposition 4.5.8 (i)") \BF{Proposition 4.5.8} Let $j$ be a local operator on a topos $\calE$. The following conditions are equivalent: (i) The associated sheaf functor $L: \calE â \sh_j(\calE)$ preserves the subobject classifier. (ii) The canonical monomorphism $Ξ©_j â Ξ©$ is $j$-dense. (iii) For any $\phi: A â Ξ©$, the equalizer of $\phi$ and $j\phi$ is a $j$-dense subobject of $A$. (iv) Every monomorphism in $\calE$ may be factored (not necessarily uniquely) as a $j$-closed monomorphism followed by a $j$-dense one. (v) $j$ commutes with implication, i.e. the diagram (...) commutes. % (vi) The diagram (...) commutes. \bsk % Â«elephant-A4.5.9Â»(to ".elephant-A4.5.9") % (find-elephanttext (+ 17 211)) % (find-elephantpage (+ 17 211) "Example 4.5.9") % (find-elephantpage (+ 17 211) "We write Lop(E)") Example 4.5.9 Let $Â¬: Ξ© â Ξ©$ be the Heyting negation map, i.e. the classifying map of $â₯: 1 \monicto Ξ©$. It is straightforward to verify that the composite $Â¬Â¬$ is a local operator, i.e. that it satisfies the conditions of 4.4.1. Moreover, it satisfies the conditions of 4.5.8: to see this, observe that for any element $x$ of a Heyting algebra if, we have $x â€ (Â¬Â¬x â x)$ and $Â¬x â€ (Â¬Â¬x â x)$ (the latter since $(Â¬x â§ Â¬Â¬x) = â₯ â€ x$), and so $(Â¬Â¬x â x) â₯ (x â¨ Â¬x)$; hence $Â¬Â¬(Â¬Â¬x â x) â₯ Â¬Â¬(xâ¨Â¬x) = â€$. But this is just the statement that the diagram in (vi) of 4.5-8 commutes. Alternatively, we could use condition (iv): given a subobject $A' \monicto A$, if we set $A'' = A' âª Â¬A'$, then $A' \monicto A''$ is $Â¬Â¬$-closed (since it is complemented) and $A'' \monicto A$ is $Â¬Â¬$-dense (cf. the proof of 1.4.14). We note that the subtopos $\sh_{Â¬Â¬}(\calE)$ is Boolean; for if $A$ is any $Â¬Â¬$-sheaf, its subobjects in $\sh_{Â¬Â¬}(\calE)$ are its $Â¬Â¬$-closed subobjects in $\calE$, and these form a Boolean algebra. It is easy to see that it is not an open subtopos in general; for example, if $X$ is a $T_0$-space (such as $\R$) in which no nonempty open subspace is discrete, then $\sh_{Â¬Â¬}(\Sh(X))$ cannot be open. We shall have more to say about Boolean subtoposes in 4.5.21 below. \msk We write $\Lop(\calE)$ for the class of all local operators on a topos $\calE$ (note that it is a set if $\calE$ is locally small). $\Lop(\calE)$ carries a natural partial order, defined by $j_1 â€ j_2$ iff $â§(j_1, j_2) = j_1$; this is equivalent to saving that $J_1 < J_2$ in $\Sub(Ξ©)$, or that $Ξ©_{j_2} â€ Ξ©_{j_1}$, or that $\sh_{j_2}(\calE) â \sh_{j_1}(\calE)$ as subcategories of $\calE$ (the more dense monomorphisms we have, the more conditions an object has to satisfy to be a sheaf). We shall see eventually that $\Lop(\calE)$ is a Heyting algebra; for the moment, we note \msk % Â«elephant-A4.5.10Â»(to ".elephant-A4.5.10") % (find-elephanttext (+ 17 211)) % (find-elephantpage (+ 17 211) "Lemma 4.5.10") % (find-elephantpage (+ 17 215) "(e)") \BF{Lemma 4.5.10} The partial ordering $\Lop(\calE)$ has greatest and least elements, and binary meets. \bsk % Â«elephant-A4.5.20Â»(to ".elephant-A4.5.20") % (find-elephanttext (+ 17 219)) % (find-elephantpage (+ 17 219) "Corollary 4.5.20") % (find-elephantpage (+ 17 219) "the local operator Â¬Â¬ of 4.5.9 is dense") \BF{Corollary 4.5.20} Any geometric inclusion $\calE' â \calE$ has a unique factorization $\calE' â \calE'' â \calE$, where $\calE' â \calE''$ is dense and $\calE'' â \calE$ is closed. \bsk % Â«elephant-A4.6.2Â»(to ".elephant-A4.6.2") % (find-elephanttext (+ 17 224)) % (find-elephantpage (+ 17 224) "Examples 4.6.2 (a), (c), (f)") \BF{Examples 4.6.2} (a) Every inclusion is localic, for if $f$ is an inclusion then every object of its domain is isomorphic to one of the form $f^* A$. More generally, if $f_*$ is merely faithful, then the counit $f^*f_*B â B$ is epic for all $B$, and so $f$ is localic. \sdd (c) Let $f: \calC â \calD$ be a functor between small categories. If $f$ is faithful, then the induced geometric morphism $[\calC, \Set] â [\calD, \Set]$ of 4.1.4 is localic. For every functor $\calC â Set$ is a quotient of a coproduct of representable functors; if $f$ is faithful then the representable functor $\calC (A, â)$ is a subfunctor of $f^*(\calD(f(A),â))$; and $f^*$ preserves coproducts. The converse is also true: if $\calC (A, â)$ appears as a subquotient of some $f^*(F)$, then (being projective) it actually occurs as a subobject of $f^*(F)$, and this can only happen if there exists $x â F(f(A))$ such that $F(fΞ±)(x) \neq F(fΞ²)(x)$ whenever $Ξ±, Ξ²: A \rightrightarrows B$ are distinct morphisms of $\calC -$ which in particular forces $fΞ± \neq fΞ²$. (d) In particular, if $\calC$ is a preorder (so that the unique functor from $\calC$ to the terminal category $\mathbf{1}$ is faithful), then the unique geometric morphism $[\calC, \Set] â \Set$ of 4.1.9 is localic. (e) It is easy to verify that a composite of localic morphisms is localic, since the subquotient relation is transitive and inverse image functors preserve monomorphisms and epimorphisms. So, combining (a) and (d), we see that if $(\calC, T)$ is a small site whose underlying category is a preorder, then the unique geometric morphism $\Sh(\calC, T) â \Set$ is localic. (We shall prove a converse to this result in B3.3.5.) In particular, for any topological space $X$, $\Sh(X) â \Set$ is localic. Similarly, combining (a) and (b), we note that the surjection with Boolean domain constructed in the proof of 4.5.23 is localic. (f) It is even easier to verify that, if % $$ \calG \diagxyto/->/^{g} \calF \diagxyto/->/^{f} \calE $$ % is a composable pair of geometric morphisms and the composite $fg$ is localic, then $g$ is localic. Hence if $\calF$ and $\calG$ both admit localic morphisms to $\Set$, then any geometric morphism between them is localic. For example, the geometric morphism $\Sh(X) â \Sh(Y)$ induced by a continuous map of spaces $X â Y$, as in 4.1.11, is always localic. \bsk % Â«elephant-A4.6.5Â»(to ".elephant-A4.6.5") % (find-elephanttext (+ 17 226)) % (find-elephantpage (+ 17 226) "Theorem 4.6.5") \BF{Theorem 4.6.5} Any geometric morphism can be factored, uniquely up to equivalence, as a hyperconnected morphism followed by a localic one. % Â«elephant-A4.6.6Â»(to ".elephant-A4.6.6") % (find-elephanttext (+ 17 226)) % (find-elephantpage (+ 17 226) "Proposition 4.6.6 (i) and (iv)") \bsk \BF{Proposition 4.6.6} Let $f:\calFâ\calE$ be a geometric morphism. The following are equivalent: (i) $f$ is hyperconnected. (ii) $f^*$ is full and faithful, and its image is closed under subobjects in $\calF$. (iii) $f^*$ is full and faithful, and its image is closed under quotients in $\calF$. (iv) The unit and counit of $(f^* â£ f_*)$ are both monic. (v) $f_*$ preserves $Ξ©$, i.e. the comparison map $\tau: f_*(Ξ©_{\calF}) â Ξ©_{\calE}$ (the classifying map of $f_*(â€_{\calF})$) is an isomorphism. (vi) For each object $A$ of $\calE$, $f^*$ induces an equivalence $\Sub_{\calE}(A) â Sub_{\calF}(f^*A)$. % Â«elephant-A4.6.10Â»(to ".elephant-A4.6.10") % (find-elephanttext (+ 17 231)) % (find-elephantpage (+ 17 231) "Proposition 4.6.10 (i) and (iii)") \end{document} \end{document} % Local Variables: % coding: utf-8-unix % ee-anchor-format: "Â«%sÂ»" % End: