Every elementary topos is a finitary extensive category. In fact, more is true: if an elementary topos has coproducts indexed by any arity class , then it is -extensive. In particular, a cocomplete elementary topos is infinitary extensive.
There are several possible proofs. In all cases, stability of coproducts under pullback is automatic since a topos is locally cartesian closed, so it suffices to consider disjointness.
As shown at extensive category, if we assume classical logic then , so that in a finitary-extensive category, any existing coproducts are stable and disjoint. Thus, in this case it suffices to deal with finite coproducts. However, in topos theory we are often interested in toposes “over an arbitrary base”, in which case the ambient logic is constructive logic and so this argument does not apply.
Nevertheless, we mention a couple of proofs that apparently apply only to the finitary case in addition to the fully general version.
A proof applying to finite coproducts is given in Johnstone, A2.4.4: since and are monic, it suffices to show that the pushout of a mono is a mono and also a pullback.
To see this, let be monic and any morphism, and let be the name of , i.e. is the pullback along . In other words, factors through iff factors through (i.e. for all in the internal logic). This is in particular the case if (i.e. for all in the internal logic), which implies that the square
is a pullback.
Now given a pushout
with monic, with as above we have an induced by the universal property of the pushout, with and . Since is monic, must also be monic, and since the previous square is a pullback, so must this one be.
A different proof applying to finite coproducts can be obtained by combining A1.5.14 and A1.6.2 of Johnstone. Define and to make the following squares pullbacks:
Specifically, is the kernel pair of . In particular, there is an induced diagonal such that . On the other hand, since pullback preserves colimits, is a coproduct diagram. Thus, the pair of morphisms and factor (uniquely) through some , so that in particular . Thus, has both a left and a right inverse, so it is an isomorphism, and hence is monic. Dually, is monic.
Now we claim that any two morphisms with domain are equal. For the pair of morphisms and factor uniquely through (which, recall, is the coproduct ). But since is an isomorphism, this factorization must be the left coprojection . Therefore, the composite is equal to the composite , and similarly so is the composite . Since is monic by the previous paragraph, .
Finally, since is a strict initial object, is monic. And of course is also monic, and these two monomorphisms are classified by two maps into the subobject classifier. By the previous paragraph, , hence and is initial.
Finally, we give a proof that applies constructively to arbitrary coproducts. This proof is due to Jibladze and can be found (in fibered-topos language) in Streicher, Appendix A. (Moens himself always just assumed that internal sums are stable and disjoint whereas Jibladze proved that for every fibered topos with internal sums these are actually stable and disjoint.)
Suppose a topos with -indexed coproducts for all belonging to some arity class . Then is a logical functor with a left adjoint , so (by A2.3.8 of Johnstone, see also logical functor) to show that induces equivalences (which is one characterization of -extensivity) it suffices for to be faithful. As with any adjunction, to show the left adjoint to be faithful it suffices to show that the unit is monic, which means in this case that the injections are all monic.
Now since is an arity class, also belongs to it, and hence so do the fibers of the diagonal . Thus, has coproducts indexed by the subsingletons determined by the equalities for any . Moreover, for any we can write , and under this identification the injection is obtained by applying the functor to the evident family of maps . But since is the left adjoint of a logical functor, it preserves monomorphisms (see A2.4.8 of Johnstone, also logical functor). Thus, it suffices to prove that each is monic.
Now, the functor has a logical right adjoint sending to the subsingleton-indexed family , and the units of its adjunction are isomorphisms (to define an inverse, it suffices to consider the object under the assumption , in which case this object is isomorphic to ). Thus, is fully faithful, and in particular faithful, and since its right adjoint is also logical, it induces an equivalence .
However, the unit of a pullback adjunction is , and if this is an isomorphism then in particular and thus is subterminal. Moreover, in this case the counits of the adjunction are the projections , which are pullbacks of the mono and therefore also monic. Finally, in our case where , the counit associated to is precisely the map that we wanted to prove to be monic.
Thomas Streicher, Fibered categories a la Jean Benabou, citeseer
Last revised on January 27, 2018 at 12:21:35. See the history of this page for a list of all contributions to it.