Classifying Toposes

Algebraic theories can be interpreted ‘over’ a topological space. For example, an interpretation of the theory of groups over a topological space $X$ assigns a group $G(U)$ to each open set $U$ of $X$, in a suitably compatible way: for instance, if $V \subseteq U$, then there is a canonical homomorphism $G(U) \to G(V)$. Such a collection of groups, called a sheaf, can be thought of as encapsulating different states of knowledge about a single object $G$, which we can analyse in terms of the sentences that it satisfies. Any given sentence in the language of groups is classically either true or false for each $G(U)$. Analogously to Heyting semantics, the truth value of a sentence $\varphi$ interpreted in $G$ will be an open set of $X$, namely, the union of the open sets $U$ for which $\varphi$ holds in $G(U)$. For example, the group axioms will take the maximal truth value $X$, so in this sense $G$ ‘is’ a group. We will show that this method of reasoning is logically sound, provided that we only use a constructive form of logic, and that the sentences in question are suitably geometric. In this essay, we will study the idea of interpreting theories over spaces in detail.