Sentence examples for crossed module from inspiring English sources

Also ψ′ is a crossed module.

is a crossed module and Open image in new window.

The definition of a free 2-crossed module is similar in some ways to the corresponding definition of a free crossed module.

J = Ker ∂3 is commutative, since C3 → ∂3C2 is a crossed module, but more is true.

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.

The 3-truncated crossed complex yields a quasi 3-crossed module by taking {c ⊗ c′} = 0 ∈ C3 for all c,c′ ∈ C2 and in which the actions of C2 on C3 are both trivial.

is a 3-truncated crossed complex, then (C2,C1,C0,∂2,∂1) is a 2-crossed module, C2 is a module over C1 on which ∂2C2 acts trivially, and ∂1∂2∂3 = 0.

is a pre 2-crossed module.

Thus to analysis an ideal pair homological, one natural method to use is to compare it via a free 2-crossed module, with a free pre-crossed module.

The free pre 2-crossed module {C2,C1,C0,∂2,∂1} is totally free if ∂1 : C1 → C0 is a totally free pre-crossed module.