On the relation between continuous and combinatorial

TL;DR

This paper explores the relationship between continuous and combinatorial structures through the lens of axiomatic cohesion, introducing weakly Kan objects in pre-cohesive toposes and analyzing how geometric morphisms preserve homotopical properties.

Contribution

It introduces the concept of weakly Kan objects in pre-cohesive toposes and studies how geometric morphisms preserve homotopy-theoretic structures, linking geometric realization to inverse images.

Findings

Weakly Kan objects are analogous to Kan complexes in simplicial sets.

Geometric morphisms preserving pieces induce adjunctions between homotopy categories.

Inverse images of such morphisms preserve weakly Kan objects.

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.