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Paolo Perrone (Apr 11 2024 at 17:50):Using the definition of Day convolution as a coend it's pretty clear that they exist for all presheaves on a small categories.
Are some results known outside the small case? For example, has the case of accessible categories been worked out?
Nathanael Arkor (Apr 11 2024 at 18:25):Do you mean that you are taking presheaves on a large monoidal category, and you are wondering when this is also monoidal?
Nathanael Arkor (Apr 11 2024 at 18:25):If you're willing to restrict to small presheaves (which tends to be appropriate when dealing with large categories), then the resulting category of small presheaves will be monoidal.
Paolo Perrone (Apr 11 2024 at 18:26):Nathanael Arkor said:
Do you mean that you are taking presheaves on a large monoidal category, and you are wondering when this is also monoidal?
Yes exactly. Or at least if there are known conditions for where this holds.
Paolo Perrone (Apr 11 2024 at 18:26):Nathanael Arkor said:
If you're willing to restrict to small presheaves (which tends to be appropriate when dealing with large categories), then the resulting category of small presheaves will be monoidal.
Right, very nice. Is there a place where I can learn more about this?
Nathanael Arkor (Apr 11 2024 at 18:27):It follows more generally for a cocompletion under a class of weights: the reference is §3 of Johnson's Monoidal Morita Equivalence.
Paolo Perrone (Apr 11 2024 at 18:28):Thank you! (The link is broken though.)
Nathanael Arkor (Apr 11 2024 at 18:28):I would be surprised if anything nice can be said about arbitrary presheaves, since Set doesn't have many large colimits.
Nathanael Arkor (Apr 11 2024 at 18:29):Paolo Perrone said:
Thank you! (The link is broken though.)
Ah, it's working for me; core.ac.uk is very buggy, unfortunately. You can probably find a copy elsewhere, but I can share a copy if not.
Paolo Perrone (Apr 11 2024 at 18:30):I'll find it. Thank you!
Dylan Braithwaite (Apr 11 2024 at 18:31):For what it's worth the first time I clicked that link I got a 404 page, but when I clicked it again it worked
Kevin Carlson (Arlin) (Apr 12 2024 at 01:08):It seems reasonable to ask when the Day convolution exists for all presheaves on a total category. Maybe it always does.
Mike Shulman (Apr 12 2024 at 01:10):Why would a cocompleteness property of the domain category affect the existence of colimits of presheaves on that domain?
Kevin Carlson (Arlin) (Apr 12 2024 at 01:20):Hmm, just an instinctual reaction…maybe I had nothing deeper than “those are the categories where reasonable large colimits behave reasonably”
Mike Shulman (Apr 12 2024 at 01:23):Well, colimits of presheaves are pointwise, coming from colimits in Set, and Set is total.