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Zbl 0795.18009
Breen, Lawrence
Higher Schreier theory. (Théorie de Schreier supérieure.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 25, No. 5, 465-514 (1992). ISSN 0012-9593

A number of authors over the last 30 years have given definitions and interpretations of cohomology with non-abelian coefficients. The primary emphasis of this paper is on the definition of pointed sets which play the role of an $H\sp 3$. A $gr$-category is defined to be a monoidal category with an associativity isomorphism satisfying the coherence condition, and such that objects have inverses for the monoidal structure and morphisms have inverses for composition. Automorphisms (resp. self- equivalences) of a $gr$-category ${\germ G}$ form a $gr$-category $\Aut{\germ G}$ (resp. $\text{Eq}({\germ G}))$ which acts on ${\germ G}$; the map ${\germ G}\to\Aut{\germ G}$ is called a crossed square (a crossed square of groups extends the notion of crossed module).\par An extension of the $gr$-category ${\germ K}$ by ${\germ G}$ is a further $gr$-category ${\germ H}$, together with an essentially surjective additive functor $p:{\germ H}\to{\germ K}$ satisfying the property of lifing morphisms, and with kernel equivalent to ${\germ G}$.\par In particular, take ${\germ K}$ to be the category $\underline K$ with a single object whose morphisms form the group $K$. Then the set of equivalence classes of extensions of $\underline K$ by ${\germ G}$ corresponds bijectively to a set $H\sp 1(B\underline K;{\germ G}\to\text{Eq}({\germ G}))$. This example in fact motivates the definition of such $H\sp 1$ groups -- detailed formulations in terms of cocyles and equivalence are given.\par The author then discusses variants and special cases, e.g., when the associativity isomorphism is the identity. A more substantial case is the study of $gr$-fields ${\germ G}$ over a topos ${\cal T}$. A field $C$ over ${\cal T}$ is a ${\germ G}$-bitorsor if composition laws ${\germ G}\times C\to C$, $C\times{\germ G}\to C$ are given, which satisfy appropriate identities. These are classified by a group $H\sp 0(e,{\germ G}\to\text{Eq}({\germ G}))$. If $K$ is a group over ${\cal T}$, extensions of $\underline K$ by ${\germ G}$ are classified: formally this is as above.\par The paper concludes with a discussion of possible extensions to higher degrees: it seems likely that such will involve crossed cubes etc. and they should relate to homotopy classification of topological fibrations with base a $K(\pi,1)$.
[C.T.C.Wall (Liverpool)]

MSC 2000:: *18G50 Nonabelian homological algebra
18G55 Nonabelian homotopical algebra
57T99 Homology and homotopy of topological groups
20J99 Connections of group theory with homological algebra
18F99 Categories and geometry
18D10 Monoidal categories

Keywords: $H\sp 3$; $H\sp 1$; cohomology with non-abelian coefficients; monoidal category; coherence condition; crossed square; crossed module; homotopy classification of topological fibrations

Cited in: Zbl 1047.18009 Zbl 0887.18005 Zbl 0856.18006 Zbl 0856.18005

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