Homotopy classification of braided graded categorical groups

Abstract

For any group G, a certain cohomology theory of G-modules is developed. This cohomology arises from the homotopy theory of G-spaces and it is called the "abelian cohomology of G-modules". Then, as the main results of this paper, natural one-toone correspondences between elements of the 3 rd cohomology groups of G-modules, G-equivariant pointed simply-connected homotopy 3-types and equivalence classes of braided G-graded categorical groups are established. The relationship among all these objects with equivariant quadratic functions between G-modules is also discussed.

A Cohomology Theory for Commutative Monoids

“…Then, making a dimensional shift, we state the following definition. [18,19] and by H n 1 (G, A) in [25]). Although these cohomology groups arise from algebraic topology, they come with algebraic interest.…”

“…The coherence condition (18), (20) and (21) follow from the three-cocycle condition ∂ 3 (h, µ) = (0, 0, 0), while the coherence condition (19) holds due to the normalization condition h(a, 1, b) = 0.…”

“…Since M is strictly unitary, the equations in (19) and (22) give the normalization conditions h(a, 1, b) = 0 = h(1, a, b) = h(a, b, 1) for h, while the equations in (23) imply the normalization conditions µ(a, 1) = 0 = µ(1, a) for µ. Thus, (h, µ) ∈ C 3 c (M, A) is a commutative three-cochain, which is actually a three-cocycle, since the coherence conditions (18), (20) and (21) Since an easy comparison (see Example 4.1) shows that M = A h,µ M , the proof of this part is complete, under the hypothesis of being M totally disconnected and strictly unitary.…”

“…Conversely, suppose that F = (F, ϕ, ϕ 0 ) : A h,µ M → A h ,µ M is any braided equivalence. By [18], there is no loss of generality in assuming that F is strictly unitary in the sense that ϕ 0 = 0 1 : 1 → 1 = F 1. As the underlying functor establishes an equivalence between the underlying groupoids,…”

“…G is an Eilenberg-Mac Lane minimal complex K(G, 2), for any abelian group A (regarded as a constant coefficient system on G), the commutative cohomology groups H n c (G, A) are precisely the Eilenberg-Mac Lane cohomology groups of the abelian group G with coefficients in A [12][13][14][15] (also denoted by H n ab (G, A) in [18,19]). In this paper, we are mainly interested in the lower cohomology groups H n c (M, A), for n ≤ 3.…”