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Zbl 0880.18006
Garzon, Antonio R.; Miranda, Jesus G.
Homotopy theory for (braided) cat-groups. (English)
[J] Cah. Topologie Géom. Différ. Catég. 38, No.2, 99-139 (1997). ISSN 0008-0004

A cat-group is an internal category in the category of groups. It is thus essentially the same as a $\text {cat}^1$-group in the sense of {\it R. Brown} and {\it J.-L. Loday} [Topology 26, 311-335 (1987; Zbl 0622.55009)] and is equivalent to a crossed module. Another description is as a 1-truncated simplicial group [cf. {\it J.-L. Loday}, J. Pure Appl. Algebra 24, 179-202 (1982; Zbl 0491.55004)] and it represents a homotopy 2-type. Homotopy 3-types can be modelled by 2-truncated simplicial groups [cf. {\it D. Conduché}, J. Pure Appl. Algebra 34, 155-178 (1984; Zbl 0554.20014)], by braided crossed modules [cf. {\it R. Brown} and {\it N. D. Gilbert}, Proc. Lond. Math. Soc., III. Ser. 59, No. 1, 51-73 (1989; Zbl 0645.18007)], by quadratic modules [see {\it H. J. Baues}, ``Combinatorial homotopy and 4-dimensional complexes'', De Gruyter Expo. Math. 2 (1991; Zbl 0716.55001)] or as in this paper by braided cat-groups. The braiding refers to a pairing on the object group of the cat-group taking values in the arrow group and measuring the difference between the two products $xy$ and $yx $, $x,y\in$ object group. It is very like the symmetry map in a symmetric monoidal category. \par In this article to the category of braided cat-groups is given the structure of a closed model category structure in the sense of Quillen. Analogues of path spaces, cylinders, loop spaces and suspensions are constructed and the overall structure is expected to shed light on the classification of sets of homotopy classes of maps between connected topological spaces whose homotopy groups are trivial except those of dimensions $n$ and $n+1$ for some $n> 1$.
[T.Porter (Bangor)]

MSC 2000:: *18D10 Monoidal categories
55U35 Abstract homotopy theory
18G30 Simplicial objects in a category

Keywords: cat-group; internal category; crossed module; simplicial groups; quadratic modules; braiding; symmetry map; closed model category; sets of homotopy classes of maps; connected topological spaces

Citations: Zbl 0668.18013; Zbl 0622.55009; Zbl 0491.55004; Zbl 0554.20014; Zbl 0645.18007; Zbl 0716.55001

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