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Crossed modules in Cat and Brown-Spencer theorem for 2-categories. (English) Zbl 0575.18006

Starting with considerations on cohomology theory and extensions of categories, this paper developes the theory of crossed modules in the category of small categories. Given a category C, a C-structure is a split extension of categories \(H\to K\to C\) in the sense of the reviewer [Rend. Mat., VI. Ser. 7, 169-192 (1974; Zbl 0361.18012)]. The category of C-structures is equivalent to the category \(Groups^ C\) and a crossed module is a particular C-structure. The obtained category is shown to be equivalent to a subcategory of 2-Cat.
The interest of these notions is indicated linking crossed modules and internal categories as in Brown-Spencer theorems [R. Brown and C. B. Spencer, Nederl. Akad. Wet., Proc., Ser. A 79, 296-302 (1976; Zbl 0333.55011); Cah. Topologie Géom. Différ. 17, 343-362 (1976; Zbl 0344.18004)].
Reviewer: G.Hoff

MSC:

18G10 Resolutions; derived functors (category-theoretic aspects)
18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
18D35 Structured objects in a category (MSC2010)
55Q99 Homotopy groups
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References:

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