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Zbl 0723.18006
Rosebrugh, R.; Wood, R.J.
Gamuts and cofibrations. (English)
[J] Cah. Topologie Géom. Différ. Catég. 31, No.3, 197-211 (1990). ISSN 1245-530X

For categories A, B a fibration from A to B is a span $p: A\leftarrow E\to B :q$ equipped with compatible actions $A\downarrow p\to E$, $q\downarrow B\to E$. (Here $f\downarrow g$ denotes the comma category.) This amounts to a functor $A\sp{op}\times B\to Cat$. The fibrations which amount to functors $A\sp{op}\times B\to Set$ are called discrete. Fibrations (as spans with actions) can be defined in any bicategory and the discrete ones can be distinguished. In particular, fibrations in $Cat\sp{op}$ are called cofibrations. There is an equivalence between discrete fibrations and codiscrete cofibrations in Cat; so, internal to Cat, we have two ways of viewing functors $A\sp{op}\times B\to Set.$ \par The reviewer showed [ibid. 21, 111-159 (1980; Zbl 0436.18005)] that, for the bicategory V-Cat of categories with homs enriched in a closed category V, it is the codiscrete cofibrations from A to B (and not the discrete fibrations, in general) which amount to V-functors $A\sp{op}\otimes B\to V$; that is, to left A-, right B-modules. Moreover, cofibrations in V-Cat were shown to amount to certain diagrams of modules called ``gamuts'' by the reviewer. \par The present paper nicely abstracts the last result: V-Cat is replaced by a bicategory K and the bicategory of V-enriched modules by a bicategory M such that M equips K with proarrows in the sense of the second author [ibid. 23, 279-290 (1982; Zbl 0497.18012) and ibid. 26, 135-168 (1985; Zbl 0583.18003)]. Subject to two reasonable axioms on $K\to M$, the authors prove that cofibrations in K are gamut diagrams in M. Several other examples are described.
[R.H.Street (North Ryde)]

MSC 2000:: *18D05 2-categories and generalizations
18B25 Topoi
18A25 Comma categories

Keywords: collage; comma category; fibrations; spans with actions; bicategory; cofibrations; diagrams of modules; gamuts; proarrows

Citations: Zbl 0436.18005; Zbl 0497.18012; Zbl 0583.18003

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