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Can we think of free cocompletion as a left adjoint? In what follows, by ’cocomplete’ I mean ’having all small colimits’.
Free cocompletion can’t be a left adjoint functor from Cat (the category of small categories) to CocompleteCAT (the category of locally small cocomplete categories), because there’s no right adjoint CocompleteCAT→Cat.
Also surely not from Cat to CocompleteCat (the category of small cocomplete categories), because there just aren’t enough small cocomplete categories: only posets.
The article on free cocompletion mentions a free cocompletion functor from CAT (the category of locally small categories) to CocompleteCAT, studied by Day and Lack. There’s an obvious candidate right adjoint from CocompleteCAT to CAT. So this might work!
Does it?
Day and Lack work in the enriched case, and I’m a bit confused, but they seem to imply that free cocompletion is a pseudomonad on CAT, which would be a step in the right direction.
I decided to ask Steve Lack:
Together with Brian Day you discussed the “free cocompletion” of large categories. You did it in the enriched setting, but I have a question about the plain old setting.
By “cocomplete” I mean the usual thing, “small-cocomplete”.
By CAT I will mean the category of locally small categories.
By CocompleteCAT I will mean the category of locally small cocomplete categories.
Is there a (pseudo)adjunction between CAT and CocompleteCAT, where the right adjoint is the obvious forgetful functor?
That’s what I want to know! But if I’ve slipped up a bit on my size assumptions, feel free to fix them.
Surely the (2-)adjunction exists between CAT and CocompleteCAT, agreeing with free co-completion on small categories, and being given on large categories by freely adding a colimit for all small diagrams, i.e. by small presheaves?
There’s a relative pseudoadjunction between Cat and COC (or CocompleteCAT), where the presheaf category construction is a relative left pseudoadjoint to the forgetful functor U:COC→CAT. This is Example 3.9 in Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures.
Also a bit indirectly, Shen and Tholen in arXiv:1501.00703, after thm. 7.4, mention a related adjunction involving total categories, though they are focusing in quantaloid enriched ones. Their adjunction is between the category of small total Q-categories with weighted colimit preserving enriched functors, and Q-categories (for Q a quantaloid). Perhaps the situation is the same for Set-enrichment.
I got an email from Steve Lack confirming that I was right (and Richard Williamson was right). I used this to polish up the nLab page free cocompletion.
I had written:
Is there a (pseudo)adjunction between CAT and CoComCAT, where the right adjoint is the obvious forgetful functor?
Steve answered:
Yes, that’s right. And the left biadjoint is precisely the thing we discuss. We may not have mentioned local smallness: in the enriched context this is automatic: a Set-category is a category whose homs lie in Set.
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