Topics in topos theory

This chapter develops several further parts of topos theory which build on the common foundations of descent, truncation, and geometric morphisms, but which do not form a single linear theory. They fall into four broad groups: concrete forms of descent, convergence and homotopical invariants, finiteness and completeness, and global constructions in the category of topoi. Many of the sections can therefore be read independently.

The first group gives concrete models for the abstract descent condition. In Section 6.1, we construct the topos \(\Shv_{\tau}(C)\) associated with a Grothendieck site and prove left exactness of sheafification using the theory of generated congruences. In Section 6.2, we study actions and principal bundles for a groupoid object \(\Gg\) and their classification by the object \(\bB\Gg\) obtained as the colimit of \(\Gg\). Hypercovers, treated in Section 6.3, refine Čech nerves at every simplicial degree. They provide a bridge from descent to completion theory: their effectiveness characterizes hypercomplete topoi.

The second group concerns convergence and homotopical invariants. In Section 6.4, we compare localic, hypercomplete, Postnikov-complete, and bounded topoi. These notions need not agree in general. Dimension provides useful sufficient conditions: a topos which is locally of finite dimension is hypercomplete, while a uniform local dimension bound implies Postnikov-completeness. Finally, Section 6.6 studies a different kind of homotopical information. After passage to pro-categories, the inverse image functor of every geometric morphism admits a further left adjoint; its value on the terminal object defines the relative shape, and in particular the shape of a topos as a pro-anima.

The third group introduces finiteness conditions. Coherent objects provide a higher-categorical analogue of quasi-compact and quasi-separated objects and satisfy strong compactness properties after truncation. The completeness theorem shows that every locally coherent topos admits a surjection \(\An^I \to T\) for some set \(I\). In Section 6.8, we relate coherent topoi to pretopoi, their finitary counterparts, by means of sheafification for the effective epimorphism topology. Makkai completeness then reconstructs an appropriate coherent topos from its category of points together with its ultraproduct structure.

The final group studies constructions involving topoi as objects of \(\Topos\). A topos is exponentiable precisely when its underlying presentable category is compactly assembled; this is proved in Section 6.10. In Section 6.11, we ask when the category \(\int_T C\) of \(T\)-parametrized objects in a category \(C\) is itself a topos. The answer is expressed by the notion of a locus: an accessible category with pullbacks and van Kampen weakly contractible colimits. The parametrized-object material is not used later and may be skipped on a first reading.