Flexible limits for 2-categories

G. J. Bird*, G. M. Kelly, A. J. Power, R. H. Street

*Corresponding author for this work

    Research output: Contribution to journalArticle

    33 Citations (Scopus)

    Abstract

    Many important 2-categories - such as Lex, Fib/B, elementary toposes and logical morphisms, the dual of Grothendieck toposes and geometric morphisms, locally-presentable categories and left adjoints, the dual of this last, and the Makkai-Paré 2-category of accessible categories and accessible functors - fail to be complete, lacking even equalizers. These examples do in fact admit all bilimits - those weakenings of the limit notion that represent not by an isomorphism but only by an equivalence - but much more is true: they admit important classes of honest limits, including products, cotensor products, comma objects, Eilenberg-Moore objects, descent objects, inserters, equifiers, inverters, lax limits, pseudo limits, and idempotent-splitting. We introduce the class of flexible limits, which includes all of the above and is, in the technical sense, a closed class. Note that such honest limits, when they exist, have many advantages over bilimits: they are unique to within isomorphism, and their universal properties are both stronger and more convenient to use, a whole level of coherent families of invertible 2-cells being avoided.

    Original languageEnglish
    Pages (from-to)1-27
    Number of pages27
    JournalJournal of Pure and Applied Algebra
    Volume61
    Issue number1
    DOIs
    Publication statusPublished - 5 Nov 1989

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