Categorical notions of fibration

By Fosco Loregiàn, Emily Riehl. Expositiones mathematicae, 2018

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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 $\cal V$-$\bf Cat$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/783663e4631ae76efdee3418ca9f9f3b7757bea6