Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: theory: category theory

Topic: Idempotent splitting completion of a monoidal category

Jean-Baptiste Vienney (Jun 29 2024 at 17:40):

It seems to me that maybe the idempotent splitting completion of a monoidal category could be made into a monoidal category with tensor product on objects $(A,e_1) \otimes (B,e_2) = (A \otimes B, e_1 \otimes e_2)$, tensor product on morphisms as usual and monoidal unit $(I,\mathrm{id}_I)$.

Is there something like this in the literature? I found a paper which explains that the Cauchy completion of a monoidal category can be made into a monoidal category but I didn't find the equivalent for the idempotent splitting completion.

Todd Trimble (Jun 29 2024 at 17:49):

In the context of $\mathsf{Set}$-enriched category theory, the Cauchy completion is the idempotent-splitting completion.

Jean-Baptiste Vienney (Jun 29 2024 at 18:06):

Oh, ok.

Matteo Capucci (he/him) (Jun 30 2024 at 20:06):

IIRC some theory is developed here: https://link.springer.com/article/10.1007/s10485-005-2958-5

Paolo Perrone (Jul 01 2024 at 07:50):

For the case of Markov categories, we develop some theory here, in Section 4.5.
In Proposition 4.5.2 we give a sketch of the argument for generic monoidal categories.

Jean-Baptiste Vienney (Jul 01 2024 at 14:53):

Thanks, Proposition 4.5.2 is exactly what I wanted.