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: "extensive" monoidal categories

Jonas Frey (Jul 14 2023 at 02:14):

Call a monoidal category $\mathcal M$ extensive, if

1. $\mathcal M/I\simeq 1$
2. for all $A,B\in\mathcal M$, the functor $\mathcal M/A\times\mathcal M/ B\to\mathcal M / (A\otimes B)$ sending $(f,g)$ to $f\otimes g$ is an equivalence.

Examples are:

1. extensive categories in the ordinary sense, wrt the coproduct monoidal structure
2. the wide subcategory of any extensive category on coproduct inclusions
3. monoids viewed as discrete monoidal categories
4. (strict) linear orders with the join (aka ordinal sum) as tensor product

Does anybody know more examples?

Matteo Capucci (he/him) (Jul 14 2023 at 14:31):

Wait, don't these axioms imply $\otimes$ is cartesian?

Matteo Capucci (he/him) (Jul 14 2023 at 14:32):

At least (2) does, whereas the first seem to imply $I$ is a [[strict terminal object]]

Reid Barton (Jul 14 2023 at 14:55):

The examples would seem to indicate otherwise.

Reid Barton (Jul 14 2023 at 14:56):

In 1, for example, $\otimes$ is the coproduct. And in 3, it is not determined by any kind of universal property.

Kevin Arlin (Jul 14 2023 at 17:57):

$M/I$ is trivial here, not $M,$ so $I$ doesn't look like a terminal in (1).

Joe Moeller (Jul 14 2023 at 18:08):

If $I$ was terminal, $M/I$ would be $M$. This condition is saying there's "no" maps into $I$. I thought that there was a term for this in the usual extensive setup that went along with "disjoint coproduct", but a quick nLab search isn't helping me. I thought it was "empty initial object" or something.

Mike Shulman (Jul 14 2023 at 18:19):

[[strict initial object]]

Joe Moeller (Jul 14 2023 at 18:21):

oh duh. I was too focused on "empty".

Matteo Capucci (he/him) (Jul 15 2023 at 08:19):

Mike Shulman said:

[[strict initial object]]

of course :face_palm: initial not terminal

Matteo Capucci (he/him) (Jul 15 2023 at 08:23):

Reid Barton said:

In 1, for example, $\otimes$ is the coproduct. And in 3, it is not determined by any kind of universal property.

Yeah I confused terminal with initial and cartesian with cocartesian... but you're right anyway