Homotopy types for “gros” toposes

Introduction In this thesis we are investigating possibility of assigning homotopical invariants to toposes in an alternative way. The 2-category of (Grothendieck) toposes, geometric morphisms and their natural transformations (which we will denote by Top) has been used by many authors to model homotopy types particularly efficiently. Thus for example a topological space X can be represented in Top via its category of set-valued sheaves Shv(X). Moreover a small category C can be represented, according to convenience, either via the category Shv(BC) of sheaves on the geometric realization of its nerve, or just by the category Set C op of set-valued presheaves on C. There are well known ways to read off homotopical and homological invariants, such as the fundamental group or (co)homology with various coefficients, from these representations. However approximately one half of Top stays apparently useless from this point of view. It is well known, for example, that as soon as a small category is filtered or cofiltered, its classifying space is homotopically trivial (see e. g. [29]). This seemingly rules out all toposes of presheaves on, say, categories with finite limits. In fact, there is a problem of interpretation here. Usually one approximates the notion of the set of homotopy classes of maps [X, Y] between toposes by the set of connected components of the category Top(X, Y). But how to interpret the situations when this category is not small? And in Top this is by no means an exception. There is in fact a distinctive dichotomy between s. c. " petit " and " gros " toposes, pointed out from the very beginning of topos theory. Unfortunately there still does not exist a precise definition of these classes, but there are lots of definite examples of both kinds, and it is generally agreed that Top(X, Y) cannot be expected small for " gros " Y. A leading example for us will be this: any finitary algebraic theory T has a classifying topos [T] which represents the 2-functor T-mod() : Top op → Cat to (large) categories which sends a topos X to the category of T-models in X. Thus Top(X, [T]) is not a small category for most X and T. In fact, [T] can be taken to be the category of set-valued functors on the category of finitely presented T-models (in sets), so in any case it must be considered " contractible " from the …