Representation and character theory in 2-categories

By Nora Ganter, M. Kapranov. 2006

88 citations recorded by the source corpus.

We develop a (2-)categorical generalization of the theory of group representations and characters. We categorify the concept of the trace of a linear transformation, associating to any endofunctor of any small category a set called its categorical trace. In a linear situation, the categorical trace is a vector space and we associate to any two commuting self-equivalences a number called their joint trace. For a group acting on a linear category we define an analog of the character which is the function on commuting pairs of group elements given by the joint traces of the corresponding functors. We call this function the 2-character of . Such functions of commuting pairs (and more generally, n-tuples) appear in the work of Hopkins, Kuhn and Ravenel [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553�594 (electronic)] on equivariant Morava E-theories. We define the concept of induced categorical representation and show that the corresponding 2-character is given by the same formula as was obtained in [Michael J. Hopkins, Nicholas J. Kuhn, Douglas C. Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (3) (2000) 553�594 (electronic)] for the transfer map in the second equivariant Morava E-theory

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

View Paper: https://www.semanticscholar.org/paper/81438cc929d4e3b428fd260f7d3d915d39c679d9

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Citations in the corpus

Notes on generalized global symmetries in QFT (2015, background): “(For current work on representations of 2-groups, see for example [24, 80, 81].”
A categorification of quantum sl(n) (2008, unknown): “This 2-category has the structure of an additive k-linear 2-category, see [12] and [21, Section 5].”
Bimodules in bordered Heegaard Floer homology (2010, background): “8 Group actions on algebraic categories have seen considerable interest recently; see for instance Ganter and Kapranov [9], as well as the references in Khovanov and Thomas [16].”
Higher traces, noncommutative motives, and the categorified Chern character (2015, methodology): “When X is the classifying stack of an algebraic group, their construction is an enhancement of Ganter and Kapranov’s 2-character theory [16].”
Coherent presentations of Artin monoids (2012, background): “It is also studied by Ganter and Kapranov in [16] in the special case of groups.”
Chern characters and Hirzebruch-Riemann-Roch formula for matrix factorizations (2010, background): “the functor TrC sends (Vij) to the vector space L i∈I Vii. Note however that the functor TrC is diﬀerent from the categorical trace introduced in the context of 2-categories by Ganter and Kapranov in [15]. Instead of the functor of tensor product with ∆ they use the functor HomC(∆,?) (which leads to”
Generalized trace and modified dimension functions on ribbon categories (2010, background): “Of particular relevance to the work here, Ganter and Kapranov categorified the notion of the categorical trace and used this to study representations and character theory in 2-categories [26].”