Crossed complexes and chain complexes with operators

Abstract

Chain complexes with a group of operators are a well known tool in algebraic topology, where they arise naturally as the chain complex of cellular chains of the universal cover of a reduced CW-complex X. The group of operators here is the fundamental group of X.

Free Crossed Resolutions of Groups and Presentations of Modules of Identities among Relations

“…Our method yields a resolution dependent functorially on the presentation. However a count of the numbers of generators in various dimensions shows that the module resolution obtained from our crossed resolution by the process of [14] is not the same as the Gruenberg resolution [22]. We are grateful to Justin Smith for pointing out this reference.…”

Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems

“…All these results are generalised in [26] to the non free case and to the non reduced case, which requires a groupoid of operators, thus giving functors…”

A new higher homotopy groupoid: the fundamental globular $ω$-groupoid of a filtered space

“…However the proofs for the cubical cases, particularly the properties of thin elements and T -complexes, involve also the use of crossed complexes and the equivalence of categories (a), (b) of diagram (5). Crossed complexes also have a well-developed homotopy theory [BG89], and they have a clear relation with chain complexes with operators [BH90].…”