35 research outputs found
Permutation 2-groups I: structure and splitness
By a 2-group we mean a groupoid equipped with a weakened group structure. It
is called split when it is equivalent to the semidirect product of a discrete
2-group and a one-object 2-group. By a permutation 2-group we mean the 2-group
of self-equivalences of a groupoid
and natural isomorphisms between them, with the product given by composition of
self-equivalences. These generalize the symmetric groups , , obtained when is a finite discrete groupoid.
After introducing the wreath 2-product of
the symmetric group with an arbitrary 2-group , it
is shown that for any (finite type) groupoid the permutation
2-group is equivalent to a product of wreath
2-products of the form $\mathsf{S}_n\wr\wr\
\mathbb{S}ym(\mathcal{B}\mathsf{G})\mathcal{B}\mathsf{G}\mathsf{G}\mathbb{S}ym(\mathcal{G})\mathbb{S}ym(\mathcal{G})\mathcal{B}\mathsf{1}\mathcal{B}\mathsf{G}\mathbb{Z}_2[1]\times\mathbb{Z}_2[0]\mathbb{Z}_2[0]\mathbb{Z}_2[1]\mathbb{Z}_2$ thought of as a discrete
and a one-object 2-group, respectively.Comment: 45 pages; v2, expository and language improvement
2-cosemisimplicial objects in a 2-category, permutohedra and deformations of pseudofunctors
In this paper we take up again the deformation theory for -linear
pseudofunctors initiated in a previous work (Adv. Math. 182 (2004) 204-277). We
start by introducing a notion of a 2-cosemisimplicial object in an arbitrary
2-category and analyzing the corresponding coherence question, where the
permutohedra make their appearence. We then describe a general method to obtain
cochain complexes of K-modules from (enhanced) 2-cosemisimplicial objects in
the 2-category of small -linear categories and prove that the
deformation complex introduced in the above mentioned work can be obtained by
this method from a 2-cosemisimplicial object that can be associated to the
pseudofunctor. Finally, using a generalization to the context of -linear
categories of the deviation calculus introduced by Markl and Stasheff for
-modules (J. Algebra 170 (1994) 122), it is shown that the obstructions to
the integrability of an -order deformation of a pseudofunctor indeed
correspond to cocycles in the third cohomology group, a question which remained
open in our previous work.Comment: 43 pages, 6 figures; this is the revised version as published in the
JPA
On the representations of 2-groups in {Baez-Crans} 2-vector spaces
We prove that the theory of representations of a finite 2-group
in Baez-Crans 2-vector spaces over a field of characteristic zero
essentially reduces to the theory of -linear representations of the group of
isomorphism classes of objects of , the remaining homotopy
invariants of playing no role. It is also argued that a similar
result is expected to hold for topological representations of compact
topological 2-groups in suitable topological Baez-Crans 2-vector spaces.Comment: 9 page
The 2-group of symmetries of a split chain complex
We explicitly compute the 2-group of self-equivalences and (homotopy classes of) chain homotopies between them for any {\it split} chain complex in an arbitrary \kb-linear abelian category (\kb any commutative ring with unit). In particular, it is shown that it is a {\it split} 2-group whose equivalence class depends only on the homology of , and that it is equivalent to the trivial 2-group when is a split exact sequence. This provides a description of the {\it general linear 2-group} of a Baez and Crans 2-vector space over an arbitrary field and of its generalization to chain complexes of vector spaces of arbitrary length.Preprin