Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the relation between continuous and combinatorial   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.

29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
5
The categories of this paper are: gr-qc
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
11
The categories of this paper are: hep-lat
9
The categories of this paper are: nlin
3
The categories of this paper are: physics
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
23
The categories of this paper are: solv-int
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The categories of this paper are: atom-ph
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The categories of this paper are: plasm-ph
30
The categories of this paper are: q-alg