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Categorical trace

From Wikipedia, the free encyclopedia

In category theory, a branch of mathematics, the categorical trace is a generalization of the trace of a matrix.

Definition

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The trace is defined in the context of a symmetric monoidal category C, i.e., a category equipped with a suitable notion of a product . (The notation reflects that the product is, in many cases, a kind of a tensor product.) An object X in such a category C is called dualizable if there is another object playing the role of a dual object of X. In this situation, the trace of a morphism is defined as the composition of the following morphisms: where 1 is the monoidal unit and the extremal morphisms are the coevaluation and evaluation, which are part of the definition of dualizable objects.[1]

The same definition applies, to great effect, also when C is a symmetric monoidal ∞-category.

Examples

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which is the multiplication by the trace of the endomorphism f in the usual sense of linear algebra.
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Further applications

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Kondyrev & Prikhodko (2018) have used categorical trace methods to prove an algebro-geometric version of the Atiyah–Bott fixed point formula, an extension of the Lefschetz fixed point formula.

References

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Further reading

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  • Kondyrev, Grigory; Prikhodko, Artem (2018), "Categorical Proof of Holomorphic Atiyah–Bott Formula", J. Inst. Math. Jussieu, 19 (5): 1–25, arXiv:1607.06345, doi:10.1017/S1474748018000543
  • Ponto, Kate; Shulman, Michael (2014), "Traces in symmetric monoidal categories", Expositiones Mathematicae, 32 (3): 248–273, arXiv:1107.6032, Bibcode:2011arXiv1107.6032P, doi:10.1016/j.exmath.2013.12.003, S2CID 119129371