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.
Martti Karvonen (May 01 2023 at 17:41):I'm in a situation where I have a (lax monoidal) functor into Set and equivalence relations on each . I can write down conditions under which becomes a lax monoidal functor (so that the quotient maps form a monoidal natural transformation), but I'm wondering if there's some standard terminology for this kind of thing. Has anyone seen this before? What would you call it?
John Baez (May 01 2023 at 18:09):I don't know, but it sounds like your functor might actually be a functor into [[setoids]], where a setoid is a set with an equivalence relation.
John Baez (May 01 2023 at 18:10):(As you can see from the link, people are still fighting a bit about what setoids actually are.)
Martti Karvonen (May 01 2023 at 21:58):That's a good way of looking at the situation!
Morgan Rogers (he/him) (May 02 2023 at 09:39):An alternative take is that you can view as a (co)presheaf and as an equivalence relation on it in the category of (co)presheaves. I don't know what the monoidal product is doing, though (which monoidal structure on is it hitting?) but Day convolution might be useful to handle the monoidal structure.
Martti Karvonen (May 02 2023 at 14:24):I'm using the cartesian monoidal structure on Set. I guess in this take becomes an internal monoid (wrt Day convolution), and is then a congruence on this monoid.