model category, model $\infty$-category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of $(\infty,1)$-categories
Model structures
for $\infty$-groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant $\infty$-groupoids
for rational $\infty$-groupoids
for rational equivariant $\infty$-groupoids
for $n$-groupoids
for $\infty$-groups
for $\infty$-algebras
general $\infty$-algebras
specific $\infty$-algebras
for stable/spectrum objects
for $(\infty,1)$-categories
for stable $(\infty,1)$-categories
for $(\infty,1)$-operads
for $(n,r)$-categories
for $(\infty,1)$-sheaves / $\infty$-stacks
A model category structure on the category of crossed complexes, due to Brown & Golasiński 1989.
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 $\mathbf I$ extended to be a crossed complex.
Notice that by other work of Brown and 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 can be deduced from that on strict cubical omega-groupoids.
The original article:
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).
Dimitri Ara and François Métayer, The Brown-Golasiński model structure on strict ∞-groupoids revisited. Homology Homotopy Appl. 13 (2011), no. 1, 121–142.
Georges Maltsiniotis La catégorie cubique avec connexions est une catégorie test stricte. Homology, Homotopy Appl. (11) (2) (2009) 309 – 326.
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