On the relation between continuous and combinatorial

Let \(\mathcal {E}\) and \(\mathcal {S}\) be toposes. A geometric morphism \({p:\mathcal {E}\rightarrow \mathcal {S}}\) 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 \({p_! \dashv p^* \dashv p_* \dashv p^!}\) such that \({p^*:\mathcal {S}\rightarrow \mathcal {E}}\) is fully faithful, its image is closed under subobjects, and \({p_!:\mathcal {E}\rightarrow \mathcal {S}}\) preserves finite products. We may also say that \(\mathcal {E}\) is pre-cohesive (over \(\mathcal {S}\)). For example, the canonical geometric morphism \({\widehat{\Delta } \rightarrow \mathbf {Set}}\) from the topos of simplicial sets is pre-cohesive. In general, a pre-cohesive geometric morphism \({p:\mathcal {E}\rightarrow \mathcal {S}}\) allows us to effectively use the intuition that the objects of \(\mathcal {E}\) are ‘spaces’ and those of \(\mathcal {S}\) 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 \(\mathcal {S}\)-enriched ‘homotopy’ category \({\mathbf {H}\mathcal {E}}\) whose objects are those of \(\mathcal {E}\) and, for each X, Y in \(\mathbf {H}\mathcal {E}\), \({(\mathbf {H}\mathcal {E})(X, Y) = p_!(Y^X)}\). 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 \({g:\mathcal {F}\rightarrow \mathcal {E}}\) 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 \({g_*:\mathcal {F}\rightarrow \mathcal {E}}\) 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.