Geodesic
Isbell conjugacy and the reflexive completion
Theory and applications of categories, The Rosebrugh Festschrift, Tome 36 (2021), pp. 306-347

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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.

Publié le :
Classification : 18A99 (primary), 18A35, 18A40
Keywords: Isbell conjugacy, reflexive completion, small functor, Cauchy completion, Isbell envelope 
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     title = {Isbell conjugacy and the reflexive completion},
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Tom Avery; Tom Leinster. Isbell conjugacy and the reflexive completion. Theory and applications of categories, The Rosebrugh Festschrift, Tome 36 (2021), pp. 306-347. http://geodesic.mathdoc.fr/item/TAC_2021_36_a11/