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Isbell conjugacy and the reflexive completion
<!-- Modified 2021-06-01 by Geoff Cruttwell --> <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"> <html> <head> <title> Isbell conjugacy and the reflexive completion </title> <link REL="stylesheet" TYPE="text/css" href="../../../tac.css"> <meta name="citation_title" content="Isbell conjugacy and the reflexive completion"> <meta name="citation_author" content="Tom Avery"> <meta name="citation_author" content="Tom Leinster"> <meta name="citation_publication_date" content="2021/06/01"> <meta name="citation_journal_title" content="Theory and Applications of Categories"> <meta name="citation_volume" content="36"> <meta name="citation_issue" content="12"> <meta name="citation_firstpage" content="306"> <meta name="citation_lastpage" content="347"> <meta name="citation_pdf_url" content="http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf"> </head> <body> <h1> Isbell conjugacy and the reflexive completion </h1> <h2> Tom Avery and Tom Leinster </h2> <p> 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. </p><p> Keywords: Isbell conjugacy, reflexive completion, small functor, Cauchy completion, Isbell envelope </p> <p> 2020 MSC: 18A99 (primary), 18A35, 18A40 </p> <p><i>Theory and Applications of Categories,</i> <font face="times new roman"> Vol. 36, 2021, No. 12, pp 306-347.</font> </p><p> Published 2021-06-01. </p><p> <a href="http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf"> http://www.tac.mta.ca/tac/volumes/36/12/36-12.pdf</a><br> </p> <a href ="../../../index.html"> TAC Home </a> </body></html>