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Appendix: Sites for Topoi Appendix: Sites for Topoi A Grothendieck topos was defined in Chapter III to be a category of sheaves on a small site---or a category equivalent to such a sheaf cate­ gory. For a given Grothendieck topos [; there are many sites C for which [; will be equivalent to the category of sheaves on C. There is usually no way of selecting a best or canonical such site. This Appendix will prove a theorem by Giraud, which provides conditions characterizing a Grothendieck topos, without any reference to a particular site. Gi­ raud's theorem depends on certain exactness properties which hold for colimits in any (cocomplete) topos, but not in more general co complete categories. Given these properties, the construction of a site for topos [; will use a generating set of objects and the Hom-tensor adjunction. The final section of this Appendix will apply Giraud's theorem to the comparison of different sites for a Grothendieck topos. 1. Exactness Conditions In this section we consider a category [; which has all finite limits and all small colimits. As always, such colimits may be expressed in terms of coproducts and coequalizers. We first discuss some special properties enjoyed by colimits in a topos. Coproducts. Let E = It, Ea be a coproduct of a family of objects Ea in [;, with the coproduct inclusions ia: Ea -> E. This coproduct is said to be disjoint when every ia is mono and for every 0: =I- (3 the pullback Ea XE E{3 is the initial object in [;. Thus in the category Sets the coproduct, often described as the disjoint union, is disjoint in this sense. (On the other hand, in the category of commutative rings the coproduct is the tensor product which is not generally disjoint.) The coproduct E = IL Ea is said to be stable (under pullback) when for every map E' -> E in [; the pullbacks E' x EEa -> E' of the coproduct inclusions ia: Ea -> E yield an isomorphism lla(E' XE Ea) ~ E'. Hence, if every coproduct in [; is stable, then for each map u: E' -> E the pullback functor u* : [; / E -> [; / E' preserves coproducts. Thus, in 574 1. Exactness Conditions 575 this case every square of the form a a 1 1 (1) E'----------~)E is a pullback in f; or, equivalently, the pullback operation - XE E' is distributive over coproducts, as in (2) a a Equivalence Relations. An equivalence relation on an object E of f is a subobject R <;;;; Ex E satisfying the usual axioms for a reflexive, symmetric, and transitive relation, as expressed in the appropriate dia­ grammatic way. If we write (80 , 81 ): R >---+ E x E for the monomorphism representing the sub object R, these axioms are (i) (reflexive) the diagonal D.: E -+ E x E factors through (80 ,8d: R>---+E x E; (ii) (symmetric) the map (81 ,80 ): R -+ E x E factors through (80 , 8d: R>---+ Ex E; (iii) if R * R denotes the following pullback (3) R -----:----+) E, 8 1 then (80 71"1,8171"2): R * R -+ Ex E factors through R. To explain the condition (iii), observe, in the case where f is the category of sets, that the pullback R * R is just the set of all those quadruples (x, y, z, w) with (x, y) E R, (z, w) E Rand y = z. So in this case, (iii) expresses the usual transitivity condition for an equivalence relation on a set. The quotient E / R of an equivalence relation R <;;;; E x E in f is defined, much as in Sets, to be the object E / R given by the following co equalizer diagram in f 8 R~E~E/R, (4) 8 1 provided such a coequalizer exists. The coequalizing map q, when it exists, is always an epimorphism. 576 Appendix: Sites for Topoi If u: E -> D is any morphism in [;, the kernel pair of u is a parallel pair of arrows 80, 81 : R -> E, universal with the property that u80 = u81 ; thus the object R is the pullback of u along itself, as in the diagram (5) E ----:u.,---....) D. It follows readily that (80, 8 1): R -> E x E is a monomorphism, and an equivalence relation. Such an equivalence relation, arising as the kernel pair of some arrow u, is automatically also the kernel pair of its quotient map q: E -> E / R when the latter exists. However, the converse assertion (every equivalence relation is the kernel pair of some map, or equivalently, of its quotient map) holds in Sets, but not always; we will need it as a requirement for a Grothendieck topos. Similarly, the coequalizer of any parallel pair of arrows is necessarily also the co equalizer of its kernel pair, but we must require for a topos that every epi is the coequalizer of some parallel pair (or equivalently, of its kernel pair). A diagram of the form of a "fork" (6) is said to be exact if q is the coequalizer of 80 and 81 , while these form the kernel pair of q. This diagram (6) is stably exact if it remains exact after pulling back along any map Q' -> Q in [;. This means that the diagram R xQ Q' ====t E xQ Q' ~ (Q xQ Q') = Q', (7) obtained from (6) by pullback, is again exact. [As in the case of coprod­ ucts, if every exact diagram (6) in [; is stably so, then for any arrow u: A -> B in [; the pullback functor u*: [; / B -> [; / A preserves exact forks.] A set of objects {Ci liE I} of [; is said to generate [; when, for any two parallel arrows u, v: E -> E' in [;, the identity uw = vw for all arrows w: Ci -> E from any object Ci implies u = v. Equivalently, this means that for each object E, the set of all arrows Ci -> E (for all i E I) is an epimorphic family. Again this means that whenever two parallel arrows u, v: E -> E' are different, there is an arrow w: Ci -> E from some Ci with uw =I- vw. We can now formulate the Giraud theorem for Grothendieck topoi. 2. Construction of Coequalizers 577 Theorem 1 (Giraud). A category £ with small hom-sets and all finite limits is a Grothendieck topos iff it has the following properties: (i) £ has all small coproducts, and they are disjoint and stable under pullback; (ii) every epimorphism in £ is a coequalizer; (iii) every equivalence relation R =t E in £ is a kernel pair and has a quotient; (iv) every exact fork R =t E ......, Q is stably exact; (v) there is a small set of objects of £ which generate £. Note that (ii) and (iv) together state that for each epimorphism B ......, A, the fork B XA B =t B ......, A is stably exact. Also (iii) and (iv) together state that each equivalence relation R ~ E x E has a quotient E / R for which the resulting fork R =t E ......, E / R is stably exact. The necessity of these conditions for a Grothendieck topos is eas­ ily established. Indeed, properties (i)-(iv) obviously hold for the cat- egory of sets. Hence, they are true in the functor category Setscop constructed from any given small category C because limits and col­ imits in such a functor category are computed pointwise. Finally, if J is any Grothendieck topology on such a C, the inclusion func- cop tor Sh(C, J) >-+ Sets from the category of all J-sheaves has, as in COP Chapter III, a left adjoint, the associated sheaf functor a: Sets ......, Sh(C, J). The sheaf category Sh(C, J) is closed under finite limits in COP Sets , hence has all finite limits. The colimits in Sh(C, J) are com­ puted, as in Chapter III, by first constructing the colimit in SetsCOP and then applying the associated sheaf functor a. Since this functor a is a left adjoint and is left exact, it must preserve colimits and finite lim­ its. Hence, Sh(C, J) inherits the exactness properties (i)-(iv) from the CoP presheaf category Sets • The property (v) requiring generators also holds for each Grothendieck topos £, as observed at the end of §III.6. The sufficiency of the conditions of Theorem 1 will be proved at the end of §3. 2. Construction of Coequalizers In this short section we will prove Proposition 1. Any category £ with small hom-sets and all finite limits which satisfies conditions (i)-(iv) of the Giraud theorem has all coequalizers, hence is cocomplete. The following factorization lemma is a first step. Lemma 2. Every arrow u: E ......, A in £ can be factored as an epi followed by a mono, as in E --* B >-+ A. 578 Appendix: Sites for Topoi The subobject B is then called the image of u. Proof: Take the kernel pair R = E x A E of u: E -t A, with its projections 1Tl and 1T2: E x A E -t E. Let p: E -t E / R be the coequal­ izer of 1Tl and 1T2 [this co equalizer exists by condition (iii) of Giraud's theorem]. Since U1Tl = U1T2, we can factor U through this coequalizer p as in the commutative diagram (1) E/R. Since p is a coequalizer, it is epi. To prove that the other factor v is mono, we use the fact that epimorphisms in £ are stable under pullback [a consequence of conditions (ii) and (iv) of the theorem]. So, to show that v is mono, consider any two parallel arrows f, g: T -t E / R in £ for which vf = vg. Then the pair (f,g): T -t (E/R) x (E/R) factors through the pullback (E/R) XA (E/R).
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