On completeness and cocompleteness in and around small categories
type: Item
instance of: scholarly article
title: On completeness and cocompleteness in and around small categories
zbMATH Open document ID: 0829.18001
DOI: 10.1016/0168-0072(94)00035-2
author: Dusko Pavlovic
published in: Annals of Pure and Applied Logic
publication date: 1996-01-17T00:00:00Z
review text: For many important classes of categories (including toposes and accessible categories), the property of completeness is equivalent to cocompleteness. The adjoint functor theorem implies that this ``ought to be true'' in general; but the solution-set condition may fail, and so we may only draw this conclusion for small categories -- and, as \textit{P. Freyd} pointed out long ago, the result for small categories says less than it claims, since (classically) a complete small category is a preorder. However, around ten years ago \textit{E. Moggi} observed that in the non-classical context of \(\mathcal S\)-indexed categories (that is, of categories fibred over \(\mathcal S\), where \(\mathcal S\) is a suitable base category such as \textit{M. Hyland}'s effective topos), one can exhibit nontrivial examples of small categories (indeed, of small full subcategories of \(\mathcal S\) itself) which are complete and cocomplete. The present paper tackles the question: given a small \(\mathcal S\)-indexed category \(\mathbb{B}\), can we deduce the cocompleteness of \(\mathbb{B}\) from its completeness? The author sketches two possible approaches to proving such a result, based loosely on the adjoint functor theorem and on \textit{R. Paré}'s proof that complete toposes are cocomplete, and shows that they can be used to give positive answers to the question if either (a) \(\mathbb{B}\) is cartesian closed or (b) \(\mathcal S\) is locally cartesian closed. \{Reviewer's remark: the latter result does not seem to be best possible -- it should be possible to deduce the same conclusion if \(\mathcal S\) is merely cartesian closed.\} He also shows that a small full subcategory of \(\mathcal S\) which is complete must be cocomplete in all cases, and gives an example to show that the converse does not hold. \{Unfortunately for the terminology, a ``small full subcategory'' is not actually small -- or even locally small -- unless the base \(\mathcal S\) is locally cartesian closed; so this …
Mathematics Subject Classification ID: 18A35
Mathematics Subject Classification ID: 03F65
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18D30
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 791274
zbMATH Keywords: equivalence of completeness and cocompleteness
zbMATH Keywords: toposes
zbMATH Keywords: accessible categories
zbMATH Keywords: adjoint functor theorem
zbMATH Keywords: small categories
zbMATH Keywords: cartesian closed
reviewed by: Peter Johnstone
describes a project that uses: Automath
MaRDI profile type: Publication
full work available at URL: 0168-0072(94)00035-2
OpenAlex ID: W1966573209
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description: scientific article; zbMATH DE number 791274