Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
173 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Duality in Monoidal Categories (2301.03545v2)

Published 9 Jan 2023 in math.CT and math.QA

Abstract: We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal-hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the latter, the internal-hom functor is implemented by tensoring with the respective duals. This raises the question: can one decide whether a closed monoidal category is rigid, simply by verifying that the internal-hom is tensor-representable? We provide a counterexample in terms of finitely-generated projective objects in an abelian k-linear category. A byproduct of our work is that we obtain characterisations of the Grothendieck--Verdier duality, also called *-autonomy, and rigidity of functor categories endowed with Day convolution as their tensor product. Applied to Mackey functors, this yields a proof of a sketched argument by Bouc linking rigidity of an object to it being finitely-generated projective.

Citations (2)

Summary

We haven't generated a summary for this paper yet.