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A 2-Categorical Approach To Change Of Base And Geometric Morphisms II

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

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* $G$.

Keywords: equipment, adjunction, span.

1991 MSC: 18A25.

*Theory and Applications of Categories*, Vol. 4, 1998, No. 5, pp 82-136.

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