Categorical trace
In category theory, a branch of mathematics, the categorical trace is a generalization of the trace of a matrix.
Definition
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 [math]\displaystyle{ \otimes }[/math]. (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 [math]\displaystyle{ X^\vee }[/math] playing the role of a dual object of X. In this situation, the trace of a morphism [math]\displaystyle{ f: X \to X }[/math] is defined as the composition of the following morphisms: [math]\displaystyle{ \mathrm{tr}(f) : 1 \ \stackrel{coev}{\longrightarrow}\ X \otimes X^\vee \ \stackrel{f \otimes \operatorname{id}}{\longrightarrow}\ X \otimes X^\vee \ \stackrel{twist}{\longrightarrow}\ X^\vee \otimes X \ \stackrel{eval}{\longrightarrow}\ 1 }[/math] 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
- If C is the category of vector spaces over a fixed field k, the dualizable objects are precisely the finite-dimensional vector spaces, and the trace in the sense above is the morphism
- [math]\displaystyle{ k \to k }[/math]
- which is the multiplication by the trace of the endomorphism f in the usual sense of linear algebra.
- If C is the ∞-category of chain complexes of modules (over a fixed commutative ring R), dualizable objects V in C are precisely the perfect complexes. The trace in this setting captures, for example, the Euler characteristic, which is the alternating sum of the ranks of its terms:
- [math]\displaystyle{ \mathrm{tr}(\operatorname{id}_V) = \sum_i (-1)^i \operatorname {rank} V_i. }[/math][2]
Further applications
(Kondyrev Prikhodko) 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
- Kondyrev, Grigory; Prikhodko, Artem (2018), "Categorical Proof of Holomorphic Atiyah–Bott Formula", J. Inst. Math. Jussieu 19 (5): 1–25, doi:10.1017/S1474748018000543
- Ponto, Kate; Shulman, Michael (2014), "Traces in symmetric monoidal categories", Expositiones Mathematicae 32 (3): 248–273, doi:10.1016/j.exmath.2013.12.003, Bibcode: 2011arXiv1107.6032P
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