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On the trace of an endofunctor of a small category
Faro, Emilio
Centre de Recerca Matemàtica
The trace of a square matrix can be defined by a universal property which, appropriately generalized yields the concept of "trace of an endofunctor of a small category". We review the basic definitions of this general concept and give a new construction, the "pretrace category", which allows us to obtain the trace of an endofunctor of a small category as the set of connected components of its pretrace. We show that this pretrace construction determines a finite-product preserving endofunctor of the category of small categories, and we deduce from this that the trace inherits any finite-product algebraic structure that the original category may have. We apply our results to several examples from Representation Theory obtaining a new (indirect) proof of the fact that two finite dimensional linear representations of a finite group are isomorphic if and only if they have the same character.
512 - Àlgebra
Categories (Matemàtica)
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Centre de Recerca Matemàtica
Prepublicacions del Centre de Recerca Matemàtica;833

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