Schreier theory for singular extensions of categorical groups and homotopy classification ∗ ∗

Applied Categorical Structures

Garzón

2004

Abstract. For any categorical group H, we introduce the categorical group Out(H) and then the well-known group exact sequence 1 → Z(H ) → H → Aut(H ) → Out(H ) → 1 is raised to a categorical group level by using a suitable notion of exactness. Breen's Schreier theory for extensions of categorical groups is codified in terms of homomorphism to Out(H) and then we develop a sort of Eilenberg-Mac Lane obstruction theory that solves the general problem of the classification of all categorical group extensions of a group G by a categorical group H, in terms of ordinary group cohomology.

“…If (G, c) and (H, c) are braided categorical groups, compatibility with c is also required [16]. If T : G → H is a homomorphism, there exists an isomorphism [16,6], µ 0 : …”

Section: Preliminariesmentioning

confidence: 99%

“…Note that there is an action [6] of Out(H) on the braided categorical group Z(H) given by the homomorphism Out(H) → Eq(Z(H)), T → T .…”

Section: And the Composition Of Two Arrows (A ϕ A ) : (T µ) → (T mentioning

confidence: 99%

Section: (G Z(h ))mentioning

confidence: 99%

See 1 more Smart Citation

Obstruction Theory for Extensions of Categorical Groups

Applied Categorical Structures

Garzón

2004

“…Nevertheless, in view of the potential interest of non-necessarily strict categorical groups on one hand, and the recent development of several cohomology sets with coefficients in categorical groups (see [4,7,39,2,34]) on the other hand, we define a cohomology theory of simplicial sets with coefficients in symmetric categorical groups, whose study this paper is mainly dedicated to. There are several advantages of considering general symmetric categorical groups instead of the strict ones.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, the study of categorical groups enjoys a lively interest in the literature. Thus, extensions of categorical groups are studied in [1,2,5,9,7,34]; the notions of kernel, cokernel and factorization systems of symmetric categorical groups are considered in [27,40]; categorical torsors are introduced in [12] and graded categorical groups, originally introduced by Frölich and Wall [21], and (co)fibred categorical groups are studied in [10,11,13]. All of them, together with other more "classical" ones [28,35,37,38] show that the study of categorical groups, as algebraic objects in their own right, is a subject of interest.…”

Section: Introductionmentioning

confidence: 99%

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

Applied Categorical Structures

Martínez-Moreno

2004

Self Cite

Abstract. In this paper we introduce and study a cohomology theory {H n (−, A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A, n)} n≥0 , which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n ≥ 3, the functor K(−, n) is right adjoint to the functor ℘ n , where ℘ n (X • ) is defined as the fundamental groupoid of the n-loop complex n (X • ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with π i (Y ) = 0 for all i = n, n + 1 and n ≥ 3; and also we obtain a classification theorem for those spaces:Mathematics Subject Classifications (2000): 18D10, 18G50, 18G60.

The classifying space of a categorical crossed module

Mathematische Nachrichten

Cegarra

Garzón

2010

Self Cite

Any pointed CW-complex X has associated a categorical crossed module WX whose homotopy groups coincide with those of the space up to dimension 3. Here we associate WX more closely with the homotopy 3-type of X. We introduce the nerve of a categorical crossed module L and define its classifying space BL as the geometrical realization of the nerve. Then we prove that there is a map X −→ BWX inducing isomorphism of the homotopy groups πi for i ≤ 3. Finally, comparison with other algebraic models of 3-types is achieved.