The categories of topoi and logoi

A morphism between topoi can be viewed in two opposite directions. The algebraic direction records a left exact colimit-preserving functor \(f^*\colon S \to T\), while the geometric direction records its right adjoint \(f_*\colon T \to S\). We write \(\Logos\) and \(\Topos\) for the resulting categories. Although a morphism carries the same adjunction in either language, the two directions illuminate different constructions: generators, relations, and quotients are most naturally expressed in \(\Logos\), while slices, base change, and geometric morphisms are most naturally expressed in \(\Topos\).

The algebraic perspective allows topoi to be constructed by presentations. Given a small category \(C\), we first form the free logos \(\An[C]\) generated by \(C\). Given a congruence \(K\) in a logos \(T\), we then form a quotient \(T/K\) in which the morphisms of \(K\) become invertible. Combining these constructions produces presentations by generators and relations, and every logos admits such a presentation. In Section 4.2, we develop this formalism and its universal properties.

Presentations provide enough control to construct all small limits and colimits in \(\Logos\) and \(\Topos\). Some of these constructions are computed in the underlying categories, while others require choosing presentations and combining their generators and relations. We carry this out in Section 4.3; among the consequences are that \(\An\) is the terminal topos and that products of topoi are given by the tensor product of cocomplete categories.

The geometric part of the chapter concerns étale morphisms, the topos-theoretic analogues of local homeomorphisms. The fundamental classification says that the étale topoi over a fixed topos \(T\) are precisely its slice topoi:

Thus the internal objects of \(T\) classify a distinguished class of geometric objects over \(T\). The class of étale morphisms is stable under base change, and, more surprisingly, the wide subcategory \(\Topos^{\et} \subseteq \Topos\) admits colimits which are preserved by its inclusion into \(\Topos\). The proof of this colimit theorem occupies Section 4.5; its main ingredients are univalent families, dependent products, and a recognition criterion for étale morphisms.