A 2-categorical approach to change of base and geometric morphisms II

A. Carboni, G. M. Kelly, D. Verity, R. J. Wood

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as $\rel\K$, $\spn\K$, $\par\K$, and $\pro\K$ for a suitable category $ \K$, along with related constructs such as the $\V$-$\pro$ arising from a suitable monoidal category $\V $. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors $F:\eL -> \K$ induce equipment arrows $\rel F:\rel\eL ->\rel\K$, $\spn F:\spn\eL -> \spn\K$, and so on, and similarly for arbitrary monoidal functors $\V -> \W$. The article I with the title above dealt with those equipments $\M$ having each $\M(A,B)$ only an ordered set, and contained a detailed analysis of the case $\M =\rel\K$; in the present article we allow the $\M(A,B)$ to be general categories, and illustrate our results by a detailed study of the case $\M=\spn\K$. We show in particular that $\spn$ is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those $\spn G$ which arise from a geometric morphism $G$.

    LanguageEnglish
    Pages82-136
    Number of pages55
    JournalTheory and Applications of Categories
    Volume4
    Publication statusPublished - 1998

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    Morphisms
    Categorical
    Functor
    Adjunction
    Monoidal Category
    Ordered Set
    Arbitrary
    Morphism
    Faithful
    Generalise

    Keywords

    • Adjunction
    • Equipment
    • Span

    Cite this

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    title = "A 2-categorical approach to change of base and geometric morphisms II",
    abstract = "We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as $\rel\K$, $\spn\K$, $\par\K$, and $\pro\K$ for a suitable category $ \K$, along with related constructs such as the $\V$-$\pro$ arising from a suitable monoidal category $\V $. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors $F:\eL -> \K$ induce equipment arrows $\rel F:\rel\eL ->\rel\K$, $\spn F:\spn\eL -> \spn\K$, and so on, and similarly for arbitrary monoidal functors $\V -> \W$. The article I with the title above dealt with those equipments $\M$ having each $\M(A,B)$ only an ordered set, and contained a detailed analysis of the case $\M =\rel\K$; in the present article we allow the $\M(A,B)$ to be general categories, and illustrate our results by a detailed study of the case $\M=\spn\K$. We show in particular that $\spn$ is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those $\spn G$ which arise from a geometric morphism $G$.",
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    A 2-categorical approach to change of base and geometric morphisms II. / Carboni, A.; Kelly, G. M.; Verity, D.; Wood, R. J.

    In: Theory and Applications of Categories, Vol. 4, 1998, p. 82-136.

    Research output: Contribution to journalArticleResearchpeer-review

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    T1 - A 2-categorical approach to change of base and geometric morphisms II

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    AU - Kelly, G. M.

    AU - Verity, D.

    AU - Wood, R. J.

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