On the relation between continuous and combinatorial

Abstract

Axiomatic Cohesion proposes that the contrast between cohesion and non-cohesion may be expressed by means of a geometric morphism $p :\mathcal{E} \to \mathcal {S}$ (between toposes) with certain special properties that allow to effectively use the intuition that the objects of $\mathcal{E}$ are `spaces' and those of $\mathcal{S}$ are `sets'. Such geometric morphisms are called (pre-)cohesive. We may also say that $\mathcal{E}$ is pre-cohesive (over $\mathcal{S}$). In this case, the topos $\mathcal{E}$ determines an $\mathcal{S}$-enriched `homotopy' category. 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, which are analogous to Kan complexes in the topos of simplicial sets. Also, given a geometric morphism $g:\mathcal{F} \to\mathcal{E}$ between pre-cohesive toposes $\mathcal{F}$ and $\mathcal{E}$ (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 $\mathcal{F}$ and $\mathcal{E}$, and that the direct image $g_*:\mathcal{F}\to \mathcal{E}$ preserves weakly Kan objects. These and other results support the intuition that the inverse image of $g$ is `geometric realization'. In particular, since Kan complexes are weakly Kan in the pre-cohesive topos of simplicial sets, the result relating $g$ and weakly Kan objects is analogous to the fact that the singular complex of a space is a Kan complex.