Classifying Topoi

The idea of “classifying” geometric or algebraic structures or spaces by maps into a given space is familiar from topology. For example, for any abelian group 7 and any n, there is a classifying space K(π, n) for cohomology: for each space X, cohomology classes α ∈ Hn(X, π) correspond to (“are classified by”) maps X→ K(π, n). After reviewing some of these topological examples in more detail, we introduce a similar notion of a classifying topos. Again, the idea is to classify structures over topoi by maps into one suitably constructed topos. For example, a topos R. is said to be a classifying topos for commutative rings when for any topos ε there is a natural equivalence between ring objects in E and geometric morphisms E → R. An application of the results on continuous filtering functors from the previous chapter will construct such a classifying topos R; it will turn out to be the topos of set-valued functors on the familiar category of finitely presented commutative rings. This will follow from the fact that this category is “freely generated” by the polynomial ring Z[X], in a suitable sense to be formulated below (see Proposition 5.1).