One of the best books on a relatively new branch of mathematics, this text is the work of a leading authority in the field of topos theory. Suitable for advanced undergraduates and graduate students of mathematics, the treatment focuses on how topos theory integrates geometric and logical ideas into the foundations of mathematics and theoretical computer science.

After a brief overview, the approach begins with elementary toposes and advances to internal category theory, topologies and sheaves, geometric morphisms, and logical aspects of topos theory. Additional topics include natural number objects, theorems of Deligne and Barr, cohomology, and set theory. Each chapter concludes with a series of exercises, and an appendix and indexes supplement the text.

implies that as subobjects of Z. So , which is equivalent to saying that ϕ factors through τ. 3.2. SHEAVES 3.21 DEFINITION. Let j be a topology in a topos , F an object of . (i) We say F is (j-) separated if, given any j-dense and any pair such that fσ = gσ, we have f = g. (ii) We say F is a (j-)sheaf if, given any j-dense and any , there exists a unique such that gσ = f. We write shj() for the full subcategory of whose objects are sheaves. 3.22 EXAMPLE. Let (C, J) be a site, and j

a canonical isomorphism in lcn(Γ, α; ) (in fact an equality on inverse image functors). So the assignment defines an inverse (up to natural isomorphism) for the functor of 4.22(iii). In certain cases, the glueing construction can be used to construct actual colimits (in the sense of 2-categories) in . An important example is the following: 4.26 PROPOSITION. Pushouts of pairs of inclusions exist in . Proof. Let be a diagram of inclusions in . Apply the glueing construction to the lax diagram

morphism on {ϕ(x)} is It is straightforward to check that these definitions make into a category. Moreover, has finite limits; the object {true} is terminal (since for any ϕ, [ϕ] is easily seen to be the unique morphism from {ϕ} to {true}), and the pullback of is the object {∃z(θ(x, z) ∧ η(y, z))}. (In fact is a regular category with universal finite unions of subobjects; but we shall not need this extra information.) We now define a family of morphisms to be provably epimorphic if the

cohomology of X with coefficients in F coincide. [Use 8.28; a quasi-coherent module is one whose restriction to each affine open subset of X is a module of the form .] 9. Let X be a paracompact (Hausdorff) topological space, and A a presheaf of abelian groups on X whose stalk at every point is 0. Show that for all q. [Method: given a cochain , where is a locally finite open cover of X, let be a shrinking ([GT], p. 104) of , and define a further refinement of by choosing for each x ∈ X a

first introduced by C. Kuratowski [174]; its application to topos theory is largely the work of A. Kock, P. Lecouturier and C. J. Mikkelsen [65]. Let be a finitely-presented algebraic theory. If X is a -model in a topos and is a subobject of X, we may (even without 6.43) construct the sub--model of X generated by Y, by applying the internal intersection operator (5.34) to “the object of sub--models of X which contain Y”. In particular, this is true for the theory slat of semilattices ([LT], p.