The enriched Grothendieck construction

We define and study opfibrations of V-enriched categories when V is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary...

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Bibliographic Details
Published in:Advances in mathematics (New York. 1965) 2019-02, Vol.344, p.234-261
Main Authors: Beardsley, Jonathan, Wong, Liang Ze
Format: Article
Language:English
Subjects:
2-category
Enriched category
Fibration
Grothendieck construction
Pseudofunctor
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Summary:We define and study opfibrations of V-enriched categories when V is an extensive monoidal category whose unit is terminal and connected. This includes sets, simplicial sets, categories, or any locally cartesian closed category with disjoint coproducts and connected unit. We show that for an ordinary category B, there is an equivalence of 2-categories between V-enriched opfibrations over the free V-category on B, and pseudofunctors from B to the 2-category of V-categories. This generalizes the classical (Set-enriched) Grothendieck correspondence.
ISSN:0001-8708
1090-2082
DOI:10.1016/j.aim.2018.12.009