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.
Asad Saeeduddin (Apr 05 2020 at 02:45):Given a category that is closed under an internal hom functor , is the category also closed under some internal hom?
Asad Saeeduddin (Apr 05 2020 at 02:51):Sorry, let me amend the question. Given a category that is closed monoidal (i.e. having a monoidal structure, an internal hom functor, and an adjunction between the tensor and the hom), is the opposite category also a) closed, and b) closed-monoidal?
Asad Saeeduddin (Apr 05 2020 at 02:53):The opposite category is definitely monoidal under the same tensor, and I believe
Christian Williams (Apr 05 2020 at 04:49):Yes, that's right. Nice, I've never thought about this before. Any particular motivation?
sarahzrf (Apr 05 2020 at 04:57):wait, what's the hom
sarahzrf (Apr 05 2020 at 04:57):wouldnt it be coclosed
Christian Williams (Apr 05 2020 at 05:09):Oh! Good point, now the "hom" is a left adjoint. Where have you come across the notion of coclosed?
sarahzrf (Apr 05 2020 at 05:59):one time i was questioning whether the opposite of a topos could be a topos and "cocartesian coclosed" came up
John Baez (Apr 05 2020 at 06:29):The only place I've seen it is in co-Heyting algebras, where "or" has a left adjoint.
James Wood (Apr 05 2020 at 09:40):Cocartesian closed categories really suck (https://ncatlab.org/nlab/show/cocartesian+closed+category), but yeah, cocartesian coclosed categories make sense.
Asad Saeeduddin (Apr 06 2020 at 12:14):@Christian Williams In programming people like working with the "closed functor" representations more frequently than the "monoidal functor" representation for whatever reason, and I was trying to port some facts to the opposite category