Journal
ADVANCES IN MATHEMATICS
Volume 217, Issue 5, Pages 2268-2300Publisher
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.aim.2007.10.004
Keywords
2-categories; character; 2-vector spaces
Categories
Funding
- Division Of Mathematical Sciences
- Direct For Mathematical & Physical Scien [0801198] Funding Source: National Science Foundation
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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 V 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 V. 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. (c) 2007 Elsevier Inc. All rights reserved.
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