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Contravariance through enrichment

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Michael Shulman

We define strict and weak duality involutions on 2-categories, and prove a
coherence theorem that every bicategory with a weak duality involution is
biequivalent to a 2-category with a strict duality involution. For this
purpose we introduce "2-categories with contravariance", a sort of
enhanced 2-category with a basic notion of "contravariant morphism",
which can be regarded either as generalized multicategories or as enriched
categories. This enables a universal characterization of duality
involutions using absolute weighted colimits, leading to a conceptual
proof of the coherence theorem.

Keywords:
opposite category, contravariant functor, generalized
multicategory, enriched category, coherence theorem

2010 MSC:
18D20, 18D05

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 5, pp 95-130.

Published 2018-01-22.

http://www.tac.mta.ca/tac/volumes/33/5/33-05.pdf

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