Also ψ′ is a crossed module.
is a crossed module and Open image in new window.
J = Ker ∂3 is commutative, since C3 → ∂3C2 is a crossed module, but more is true.
The definition of a free 2-crossed module is similar in some ways to the corresponding definition of a free crossed module.
A crossed module is an algebra morphism ∂ : C → R with an action of R on C satisfying ∂(r·c) = r ∂ c and ∂(c)·c′ = c c′ for all c,c′ ∈ C, r ∈ R.
Given a crossed module, for example, an arbitrary algebra R together with an ideal I in R, what should one mean by presentation of (R,I) or more generally of a crossed on R? The 'yoga' of crossed algebraic methods suggest several possible replies.
Crossed complexes themselves, (cf. Brown and Higgins, [17]) correspond to a class of hypercrossed complexes in which nearly all of the extra structure is trivial, so the only non-abelian algebras occur in dimensions 0 and 1 and are linked by a crossed module structure.
Any simplicial algebra yields a crossed module derived from the Moore complex [2] and any finitely generated free crossed complex C → R of commutative algebras was shown in [3] to have C ≅ R n / d ( ∧ 2 R n ) Open image in new window, i.e. the 2nd Koszul complex term modulo the 2-boundaries.
The describe of this category 2CM/PCM may seem a bit artificial, but given an algebra present as a quotient of say, a polynomial algebra by a ideal, the free crossed module of that presentation has been found to contain valuable information on the algebra.
is a crossed module in which C1 acts on C2 via C0, (we require thus that for all x ∈ C2, y ∈ C1 and z ∈ C0 that (x y z = x y z)), further, there is a C0-bilinear function giving : C 1 ⊗ C 0 C 1 → C 2, Open image in new window.
