论连续与组合的关系
On the relation between continuous and combinatorial
摘要：
公理内聚提出内聚和非内聚之间的对比可以通过具有某些特殊性质的几何态射$p :\mathcal{E} \to \mathcal {S}$（在拓扑之间）来表达，这些性质允许有效地使用$\mathcal{E}$的对象是“空间”而$\mathcal{S}$的对象是“集合”的直觉。这样的几何态射被称为（前）凝聚。我们也可以说$\mathcal{E}$是前内聚的（在$\mathcal{S}$之上）。在这种情况下，拓扑$\mathcal{E}$确定一个$\mathcal{S}$ -富集的“同伦”范畴。本文的目的是研究这个同伦理论的某些方面。我们引入弱Kan对象在一个前内聚拓扑，这是类似于Kan复合物的拓扑的单纯集。此外，给定预内聚拓扑$\mathcal{F}$和$\mathcal{E}$之间的几何态射$g:\mathcal{F} \to\mathcal{E}$（在相同的基上），我们定义了$g$保留片段的含义。我们证明，如果$g$保留件，那么它会导致由$\mathcal{F}$和$\mathcal{E}$确定的同伦类别之间的附加，并且直接图像$g_*:\mathcal{F}\to \mathcal{E}$保持弱Kan对象。这些和其他结果支持的直觉，逆像的$g$是“几何实现”。特别地，由于Kan复形在单纯集的前凝聚拓扑中是弱Kan，因此将$g$和弱Kan对象联系起来的结果类似于空间的奇异复形是Kan复形的事实。
英文摘要：
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.
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