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Isbell conjugacy and the reflexive completion

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Tom Avery and Tom Leinster

The reflexive completion of a category consists of the Set-valued
functors on it that are canonically isomorphic to their double
conjugate. After reviewing both this construction and Isbell conjugacy
itself, we give new examples and revisit Isbell's main results from 1960
in a modern categorical context. We establish the sense in which reflexive
completion is functorial, and find conditions under which two categories
have equivalent reflexive completions. We describe the relationship between
the reflexive and Cauchy completions, determine exactly which limits and
colimits exist in an arbitrary reflexive completion, and make precise the
sense in which the reflexive completion of a category is the intersection
of the categories of covariant and contravariant functors on it.

Keywords:
Isbell conjugacy, reflexive completion, small functor,
Cauchy completion, Isbell envelope

2020 MSC:
18A99 (primary), 18A35, 18A40

*Theory and Applications of Categories,*
Vol. 36, 2021,
No. 12, pp 306-347.

Published 2021-06-01.

http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf

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