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1. • CommentRowNumber1.
   • CommentAuthorJohn Baez
   • CommentTimeJan 31st 2020
   • (edited Jan 31st 2020)
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   Author: John Baez
   Format: MarkdownItexCan 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 $CocompleteCAT \to Cat$. 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.

   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 $CocompleteCAT \to Cat$.

   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.

2. • CommentRowNumber2.
   • CommentAuthorJohn Baez
   • CommentTimeJan 31st 2020
   • (edited Jan 31st 2020)
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   Author: John Baez
   Format: MarkdownItexI decided to ask Steve Lack: <hr/> 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.

   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.

3. • CommentRowNumber3.
   • CommentAuthorRichard Williamson
   • CommentTimeJan 31st 2020
   • (edited Jan 31st 2020)
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   Author: Richard Williamson
   Format: MarkdownItexSurely 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?

   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?

4. • CommentRowNumber4.
   • CommentAuthorvarkor
   • CommentTimeJan 31st 2020
   • (edited Jan 31st 2020)
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   Author: varkor
   Format: MarkdownItexThere's a relative pseudoadjunction between $\mathbf{Cat}$ and $\mathbf{COC}$ (or $\mathbf{CocompleteCAT}$), where the presheaf category construction is a relative left pseudoadjoint to the forgetful functor $U : \mathbf{COC} \to \mathbf{CAT}$. This is Example 3.9 in [_Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures_](https://arxiv.org/pdf/1612.03678.pdf).

   There’s a relative pseudoadjunction between $\mathbf{Cat}$ and $\mathbf{COC}$ (or $\mathbf{CocompleteCAT}$), where the presheaf category construction is a relative left pseudoadjoint to the forgetful functor $U : \mathbf{COC} \to \mathbf{CAT}$. This is Example 3.9 in Relative pseudomonads, Kleisli bicategories, and substitution monoidal structures.

5. • CommentRowNumber5.
   • CommentAuthorjesuslop
   • CommentTimeFeb 1st 2020
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   Author: jesuslop
   Format: MarkdownItexAlso 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 $\mathbf{Set}$-enrichment.

   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 $\mathbf{Set}$-enrichment.

6. • CommentRowNumber6.
   • CommentAuthorJohn Baez
   • CommentTimeFeb 3rd 2020
   • (edited Feb 3rd 2020)
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   Author: John Baez
   Format: MarkdownItexI 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.

   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.