The classifying space of a categorical crossed module

Carrasco, P.; Cegarra, Antonio M.; Garzón, Antonio R.

doi:10.1002/mana.200610827

Cited by 4 publications

(5 citation statements)

References 25 publications

(51 reference statements)

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“…Proof Consider the 2-exact sequence (3) associated to the homotopy fibration (6) ... → ℘ 3 BGLR + → ℘ 2 F R ℘2dR −→ ℘ 2 BGLR → ℘ 2 BGLR + → · · · Deduced from the universal properties of the homotopy kernel Ker℘ 2 d R and the homotopy cokernel Coker℘ 2 d R (see [7]) there are morphisms of categorical groups K : ℘ 3 BGLR + = K 2 R −→ Ker℘ 2 d R and C : Coker℘ 2 d R −→ ℘ 2 BGLR + = K 1 R such that the following diagram of 2-exact sequences is commutative…”

Section: Theoremmentioning

confidence: 99%

“…According to Definition 1, associated to the homotopy fibration (6) there is also the categorical crossed module…”

Section: K-theory Categorical Groupsmentioning

confidence: 99%

“…Categorical crossed modules are algebraic models for homotopy 3-types (see [5,6]) and the homotopy groups of the classifying space of any categorical crossed module H, G, T, ν, χ are given by π 0 CokerT, π 0 KerT ∼ = π 1 CokerT and π 1 KerT, that is, they are the so-called homotopy groups of the categorical crossed module. The following corollary reveals that the 3-type associated to ℘ 2 d R is controlled by the K-groups K i R, i = 1, 2, 3, or, in other words, that the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by the categorical crossed module ℘ 2 d R .…”

Section: Theoremmentioning

confidence: 99%

See 2 more Smart Citations

On Algebraic K-theory Categorical Groups

Garzón

Río

2009

Appl Categor Struct

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Homotopy categorical groups of any pointed space are defined via the fundamental groupoid of iterated loop spaces. This notion allows, paralleling the group case, to introduce the notion of K-categorical groups K i R of any ring R. We also show the existence of a fundamental categorical crossed module associated to any fibre homotopy sequence and then, K 1 R and K 2 R are characterized, respectively, as the homotopy cokernel and kernel of the fundamental categorical crossed module associated to the fibre homotopy sequence F R dR G G BGLR qR G G BGLR + .As consequence, the 3th level of the Postnikov tower of the K-theory spectrum of R is classified by this categorical crossed module.

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Section: Theoremmentioning

confidence: 99%

“…According to Definition 1, associated to the homotopy fibration (6) there is also the categorical crossed module…”

Section: K-theory Categorical Groupsmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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On Algebraic K-theory Categorical Groups

Garzón

Río

2009

Appl Categor Struct

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“…Categorical precrossed modules and categorical crossed modules have been introduced in [10], and used in [7] as algebraic models for connected homotopy 3-types. Next we recall the construction of the 2-category of categorical crossed modules and some results on the quotient categorical group associated with a categorical crossed module that we will use below (c.f.…”

Section: Let Nowmentioning

confidence: 99%

Singular extensions and the second cohomology categorical group

Garzón

2016

J. Homotopy Relat. Struct.

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We give a classification of the 2-groupoid of singular extensions of categorical groups by means of a second cohomology categorical group, whose definition uses the notion of nerve of a categorical group. This second cohomology is, then, shown to be involved in a five term 2-exact sequenceà la Hochschild-Serre which is associated with any essentially surjective homomorphism of categorical groups.This work has been partially supported by MCI of Spain (Project: MTM2011-22554); Consejería de Innovación de J. de Andalucía (FQM-168).

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“…This construction appeared in [33] , with different conventions, and also in a slightly different language in [34] [35].…”

Section: And −(Dϕmentioning

confidence: 99%

On 3-gauge transformations, 3-curvatures, and Gray-categories

Wang

2014

Journal of Mathematical Physics

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In the 3-gauge theory, a 3-connection is given by a 1-form A valued in the Lie algebra g, a 2-form B valued in the Lie algebra h and a 3-form C valued in the Lie algebra l, where (g, h, l) constitutes a differential 2-crossed module. We give the 3-gauge transformations from a 3-connection to another, and show the transformation formulae of the 1-curvature 2form, the 2-curvature 3-form and the 3-curvature 4-form. The gauge configurations can be interpreted as smooth Gray-functors between two Gray 3-groupoids: the path 3-groupoid P3(X) and the 3-gauge group G L associated to the 2-crossed module L , whose differential is (g, h, l). The derivatives of Gray-functors are 3-connections, and the derivatives of lax-natural transformations between two such Gray-functors are 3-gauge transformations. We give the 3-dimensional holonomy, the lattice version of the 3-curvature, whose derivative gives the 3curvature 4-form. The covariance of 3-curvatures easily follows from this construction. This Gray-categorical construction explains why 3-gauge transformations and 3-curvatures have the given forms. The interchanging 3-arrows are responsible for the appearance of terms concerning the Peiffer commutator {, }.

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The classifying space of a categorical crossed module

Cited by 4 publications

References 25 publications

On Algebraic K-theory Categorical Groups

On Algebraic K-theory Categorical Groups

Singular extensions and the second cohomology categorical group

On 3-gauge transformations, 3-curvatures, and Gray-categories

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