Dany Majard

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On Double Groups and the Poincaré group

arXiv: Mathematical Physics, Dec 29, 2011

In [22], Crane and Sheppard considered the structure of the Poincare group as a 2-Group, and deri... more In [22], Crane and Sheppard considered the structure of the Poincare group as a 2-Group, and derived important information about its representations in a 2-Category suited for representations of non-compact 2groups, following a lead of [23]. In this paper, starting from the position that the most natural structure to describe cobordisms with corners, as in the recently published work of A. Voronov [27], is the cubical approach to higher category theory of Ehreshman, we explore some possibilities given by double groups to build TQFTs. Our main theorem is an extension of the work of [4], where we prove a theorem on the structure of maximally exclusive double groups. This result gives a presentation of the Poincare group where the distinction between boosts, rotations and translations is part of the structure, from which a TQFT could be build with space and spacetime transformations kept separate. This article drafts a program that will hopefully yield new state sum models of physical interest.

$Actually the definition should have a map between the chosen limits for any n but the relations would impose that they are the unique map given by the pullback. It will be understood from now on that the induced unique arrow is written Fy := 7 F," 7)F\, or more generally F,, := pr F\'"..."a)F n times, and we will give a functor by a pair of maps only F' := (Fo, F:). Note that (¢dq, id,) is always a functor and that functors have an obvious compo- sition. Composing F' := (Fo, Fi) with G := (Go, G1) gives F := (Fo Go, Fi Gi) and since it is reduced to the composition in the underlying category it is there- fore associative and with unit. Then, given a category C, one can build IntCat(C) whose objects are internal categories and arrows internal functors. In the case where C is Set, this cate- gory is simply Cat, the category of all small categories. As the real appeal for categories came with natural transformations, internal categories should have little to say were we to stop here. The construction of natural transformations on categories and functor allows us to generalize and define the notion internally.$

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N-tuple groups and matched n-tuples of groups

arXiv: Category Theory, 2011

This paper proves that the category of vacant n-tuple groupoids is equivalent to the category of ... more This paper proves that the category of vacant n-tuple groupoids is equivalent to the category of factorizations of groupoids by n subgroupoids. Moreover it extends this equivalence to the category of maximally exclusive n-tuple groupoids, that we define, and (n+1)-factorizations of groupoids with a normal abelian subgroupoid. Finally it shows that in the smooth case, such a factorization gives a presentation of the Poincar\'e group as a triple groupoid.

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N-Tuple Groupoids and Optimally Coupled Factorizations

In this paper, we prove that the category of vacant n tuple groupoids is equivalent to the catego... more In this paper, we prove that the category of vacant n tuple groupoids is equivalent to the category of factorizations of groupoids by n factors that satisfy some Yang-Baxter type equation. Moreover we extend this equivalence to the category of maximally exclusive n-tuple groupoids, which we define, by dropping one assumption. The paper concludes by a note on how these results could tell us more about some Lie groups of interest.

format_quoteA vacant n-tuple groupoid has all information reduced to boundary arrows targeted at its sink, guaranteeing its existence and isomorphism.format_quote

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Cubical categories, TQFTs and possible new representations for the Poincare group

On Double Groups and the Poincar 'e group

N-tuple groups and matched n-tuples of groups

Eprint Arxiv 1201 0059, Dec 29, 2011

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