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Enhanced twisted arrow categories

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Fernando Abellán García and Walker H. Stern

Given an ∞-bicategory D with underlying ∞-category D_0, we construct a Cartesian fibration Tw(D) --> Tw(D_0) x D_0^op, which we call the enhanced twisted arrow ∞-category, classifying the restricted mapping category functor Map_D:D_0^op x D_0 --> D^op \times D --> Cat_∞. With the aid of this new construction, we provide a description of the ∞-category of natural transformations Nat(F,G) as an end for any functors F and G from an ∞-category to an ∞-bicategory. As an application of our results, we demonstrate that the definition of weighted colimits studied by Gepner-Haugseng-Nikolaus satisfies the expected 2-dimensional universal property.

Keywords:
(∞,2)-category, twisted arrow category, natural transformation, weighted colimit

2020 MSC:
18N10, 18N60, 18N65

*Theory and Applications of Categories,*
Vol. 39, 2023,
No. 4, pp 98-149.

Published 2023-01-26.

http://www.tac.mta.ca/tac/volumes/39/4/39-04.pdf

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