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arXiv 1806.06129math.CT
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纤维化的范畴概念

Categorical notions of fibration

Fosco Loregian, Emily Riehl

备注：
These notes were initially written by the second-named author to accompany a talk given in the Algebraic Topology and Category Theory Proseminar in the fall of 2010 at the University of Chicago. A few years later, the now first-named author joined to expand and improve in minor ways the exposition. To appear on "Expositiones Mathematicae"
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摘要：

范畴$B$上的纤维化由Grothendieck引入范畴论，编码伪函子$B^{op} \rightsquigarrow {\bf Cat}$，而离散纤维化的特殊情况编码预层$B^{op} \to {\bf Set}$。双侧离散变异编码函子$B^{op} \times A \to {\bf Set}$，也称为从$A$到$B$的前函子。通过Street的工作，所有这些纤维化概念都可以在内部定义为任意的2-范畴或双范畴。虽然双侧离散纤维化在${\bf Cat}$内部对原函子进行建模，但出乎意料的是，双重双侧共离散纤维化对于在$\cal V$ - $\bf Cat$内部对$\cal V$ -原函子进行建模是必要的。

英文摘要：

Fibrations over a category $B$, introduced to category theory by Grothendieck, encode pseudo-functors $B^{op} \rightsquigarrow {\bf Cat}$, while the special case of discrete fibrations encode presheaves $B^{op} \to {\bf Set}$. A two-sided discrete variation encodes functors $B^{op} \times A \to {\bf Set}$, which are also known as profunctors from $A$ to $B$. By work of Street, all of these fibration notions can be defined internally to an arbitrary 2-category or bicategory. While the two-sided discrete fibrations model profunctors internally to ${\bf Cat}$, unexpectedly, the dual two-sided codiscrete cofibrations are necessary to model $\cal V$-profunctors internally to $\cal V$-$\bf Cat$.

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