On completeness and cocompleteness in and around small categories
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 counterexample does not prove the claim made in section 0.2 that the main theorem does not hold without some additional hypothesis such as (a) or (b)\}.
equivalence of completeness and cocompleteness
scientific article; zbMATH DE number 791274
scientific article