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Stream: theory: category theory

Topic: ✔ Axioms of Traced (non-strict) Monoidal Categories

Chad Nester (Feb 22 2024 at 15:59):

I'm looking for a reference in which the axioms of a traced monoidal category are explicitly written down, including the associators and unitors. For example one usually encounters:

$\mathsf{Tr}^{C \otimes D}_{A,B}(f) = \mathsf{Tr}^C_{A,B}(\mathsf{Tr}^D_{A\otimes C,B \otimes C}(f))$

but this is not technically correct, instead I guess we should have:

$\mathsf{Tr}^{C \otimes D}_{A,B}(f) = \mathsf{Tr}^C_{A,B}(\mathsf{Tr}^D_{A\otimes C, B \otimes C}(\alpha^{-1}_{A,C,D}f\alpha_{B,C,D}))$

While it is not terribly difficult to insert the necessary associators and unitors in the rest of the axioms, I am hoping that an explicit version exists somewhere in the literature. I can't find it!

Cole Comfort (Feb 22 2024 at 16:08):

Chad Nester said:

I'm looking for a reference in which the axioms of a traced monoidal category are explicitly written down, including the associators and unitors. For example one usually encounters:

$\mathsf{Tr}^{C \otimes D}_{A,B}(f) = \mathsf{Tr}^C_{A,B}(\mathsf{Tr}^D_{A\otimes C,B \otimes C}(f))$

but this is not technically correct, instead I guess we should have:

$\mathsf{Tr}^{C \otimes D}_{A,B}(f) = \mathsf{Tr}^C_{A,B}(\mathsf{Tr}^D_{A\otimes C, B \otimes C}(\alpha^{-1}_{A,C,D}f\alpha_{B,C,D}))$

While it is not terribly difficult to insert the necessary associators and unitors in the rest of the axioms, I am hoping that an explicit version exists somewhere in the literature. I can't find it!

I think they are contained explicitly in this paper:
https://arxiv.org/pdf/2109.00589.pdf

Jean-Baptiste Vienney (Feb 22 2024 at 16:35):

It looks like they are correct in this paper too (section 2): Traced Monads and Hopf Monads

Jean-Baptiste Vienney (Feb 22 2024 at 16:36):

Oh sorry, but there is only the definition of a traced symmetric monoidal category (definition 2.2).

Chad Nester (Feb 22 2024 at 16:50):

Cole Comfort said:

Chad Nester said:

I'm looking for a reference in which the axioms of a traced monoidal category are explicitly written down, including the associators and unitors. For example one usually encounters:

$\mathsf{Tr}^{C \otimes D}_{A,B}(f) = \mathsf{Tr}^C_{A,B}(\mathsf{Tr}^D_{A\otimes C,B \otimes C}(f))$

but this is not technically correct, instead I guess we should have:

$\mathsf{Tr}^{C \otimes D}_{A,B}(f) = \mathsf{Tr}^C_{A,B}(\mathsf{Tr}^D_{A\otimes C, B \otimes C}(\alpha^{-1}_{A,C,D}f\alpha_{B,C,D}))$

While it is not terribly difficult to insert the necessary associators and unitors in the rest of the axioms, I am hoping that an explicit version exists somewhere in the literature. I can't find it!

I think they are contained explicitly in this paper:
https://arxiv.org/pdf/2109.00589.pdf

This has the unitors, but not the associators!

Chad Nester (Feb 22 2024 at 16:51):

Jean-Baptiste Vienney said:

It looks like they are correct in this paper too (section 2): Traced Monads and Hopf Monads

This is perfect! Thanks.

Chad Nester (Feb 22 2024 at 16:51):

Jean-Baptiste Vienney said:

Oh sorry, but there is only the definition of a traced symmetric monoidal category (definition 2.2).

That's actually what I'm looking for :)

Notification Bot (Feb 22 2024 at 16:52):

Chad Nester has marked this topic as resolved.