nLab model structure on crossed complexes

Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

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 I\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.

References

The original article:

See also

Last revised on July 6, 2022 at 10:05:38. See the history of this page for a list of all contributions to it.