Foundations

The definition of a topos isolates a strong exactness property of colimits. Given a diagram \(X_{\bullet}\colon I \to T\) in a category with pullbacks, there is a natural functor

The target describes compatible objects over the pieces \(X_i\), while the source describes objects over the object obtained by gluing these pieces together. The category \(T\) satisfies descent if this functor is an equivalence for every small diagram. A topos is, by definition, a presentable category in which all colimits satisfy descent.

We begin in Section 2.1 by unpacking this condition. Descent separates into two complementary assertions: colimits are universal, meaning that they are preserved by base change, and effective, meaning that compatible diagrams over the pieces glue to diagrams over the colimit. This formulation makes descent amenable to categorical arguments and shows, in particular, that left exact localizations of topoi are again topoi.

The basic covers in a topos are the effective epimorphisms. Their targets are recovered as colimits of their Čech nerves, which are groupoid objects encoding all iterated overlaps of the cover. In Section 2.2, we use the effectivity and universality of groupoid colimits to study these maps, prove the epi–mono factorization system, and establish the delooping principle for group objects. This is the first indication that the exactness axioms of a topos support an internal homotopy theory.

Finally, Section 2.3 compares the descent definition with several intrinsic and extrinsic characterizations. For a presentable category, descent is equivalent to the Giraud axioms, to the combination of universal colimits with an object classifier, and to being a left exact localization of a presheaf category. The last description is especially useful: it reduces many statements about arbitrary topoi to statements about presheaf topoi, and ultimately to the topos \(\An\) of animae. We conclude by recording general exactness properties of topoi and constructing the subobject classifier as a special case of a classifying object for a local class of morphisms.