On the relation between continuous and combinatorial

Abstract

Let E and S be toposes. A geometric morphism p: E→ S is called pre-cohesive if it is local, essential, hyperconnected and the leftmost adjoint preserves finite products. More explicitly, it is a string of adjoints p!⊣ p∗⊣ p∗⊣ p! such that p∗: S→ E is fully faithful, its image is closed under subobjects, and p!: E→ S preserves finite products. We may also say that E is pre-cohesive (over S). For example, the canonical geometric morphism Δ ^ → Set from the topos of simplicial sets is pre-cohesive. In general, a pre-cohesive geometric morphism p: E→ S allows us to effectively use the intuition that the objects of E are ‘spaces’ and those of S are ‘sets’, that p∗A is the discrete space with A as underlying set of points and that p!X is the set of pieces of the space X. For instance, such a p determines an associated S-enriched ‘homotopy’ category HE whose objects are those of E and, for each X, Y in HE, (HE) (X, Y) = p!(YX). In other words, every pre-cohesive topos has an associated ‘homotopy theory’. The purpose of the present paper is to study certain aspects of this homotopy theory. We introduce weakly Kan objects in a pre-cohesive topos. Also, given a geometric morphism g: F→ E between pre-cohesive toposes F and E (over the same base), we define what it means for g to preserve pieces. We prove that if g preserves pieces then it induces an adjunction between the homotopy categories determined by F and E, and that the direct image g∗: F→ E preserves weakly Kan objects. These and other results support the intuition that the inverse image of g is ‘geometric realization’. Also, the result relating g and weakly Kan objects is analogous to the fact that the singular complex of a space is a Kan complex.