model structure on crossed complexes

Brown and Golasinski gave a model category structure on the category of crossed complexes in

  • Ronald Brown and Marek Golasinski, A model structure for the homotopy theory of crossed complexes, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 30 no. 1 (1989), p. 61-82 (pdf)

The key facts used here are that the category of crossed complexes is monoidal closed, and that there is a good interval object, essentially the groupoid I\mathbf I extended to be a crossed complex.

Notice that by other work of Brown-Higgins, crossed complexes are equivalent to strict globular omega-groupoids, and also to strict cubical omega-groupoids with connections. In fact the monoidal closed structure on crossed complexes is deduced from that on strict cubical omega-groupoids.

See also

  • Ronald Brown, Philip J. Higgins, and Rafael Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics Vol 15, (2011).

  • Ara, Dimitri; Métayer, François, The Brown-Golasiński model structure on strict ∞-groupoids revisited. Homology Homotopy Appl. 13 (2011), no. 1, 121–142.

  • Maltsiniotis, G. La catégorie cubique avec connexions est une catégorie test stricte. Homology, Homotopy Appl. (11) (2) (2009) 309 – 326.

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