Representation theory of 2-groups on Kapranov and Voevodsky's 2-vector spaces

By J. Elgueta. 2004

62 citations recorded by the source corpus.

Abstract In this paper the representation theory of 2-groups in 2-categories is considered, focusing the attention on the 2-category Rep 2 Mat K ( G ) of representations of a 2-group G in (a version of) Kapranov and Voevodsky's 2-category of 2-vector spaces over a field K. The set of equivalence classes of such representations is computed in terms of the invariants π 0 ( G ) , π 1 ( G ) and [ α ] ∈ H 3 ( π 0 ( G ) , π 1 ( G ) ) classifying G, and the categories of intertwiners are described in terms of categories of vector bundles endowed with a projective action. In particular, it is shown that the monoidal category of finite dimensional linear representations (more generally, the category of [ z ] -projective representations, for any given cohomology class [ z ] ∈ H 2 ( π 0 ( G ) , K ∗ ) ) of the first homotopy group π 0 ( G ) as well as its category of representations on finite sets both live in Rep 2 Mat K ( G ) , the first as the monoidal category of endomorphisms of the trivial representation (more generally, as the category of intertwiners between suitable 1-dimensional representations) and the second as a non-full subcategory of the homotopy category of Rep 2 Mat K ( G ) .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3099cb769bbf408be085c5cdce80fca80a3447e5

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In Algebra and Number Theory

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Citations in the corpus

Representation and character theory in 2-categories (2006, background): “Proposition 7.3 (compare [Ost01, Ex.3.4.,Th.2] and [ Elgb , 6.5]).”
Universal Non‐Invertible Symmetries (2022, background): “The structure of this 2-category is discussed in [35], where the classification of the objects also appears.”