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It seems to me that the "indization" process of a category can be formulated in the language of sketches (by sketch I mean what is defined in [LPAC].2.F, Def. 2.55); in particular, see this answer by T. Johnson-Freyd.

Expressing the ind-completion of $\bf C$ as the category of models for a sketch $({\bf C},\{\text{filtered categories}\}, \varnothing, \sigma)$ would be extremely useful to generalize the construction of $\text{Ind-}\bf C$ to the case of other partial free-(co)completion of $\bf C$, which add certain rescribed "shapes" of (co)limits, and leave the rest unchanged: I can easily imagine the sifted-, discrete-, connected-, empty-completion and cocompletion of $\bf C$, but I would like to fit this prcedure in a general framework. In this vein, sketches are perfect.

My question is: am I right in doing this? Caan you point me to somewhere in the literature where this is explained in full detail?

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fosco
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2 Answers2

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Quoting the nLab article on accessible categories:

$C$ is accessible iff one of the following equivalent conditions holds:

  • it is the category of models (in $\mathbf{Set}$) of some small sketch.

  • it is of the form $\text{Ind}_\kappa(S)$ for $S$ small, i.e. the $\kappa$-ind-completion of a small category, for some regular cardinal $\kappa$.

  • it is of the form $\kappa\,\text{Flat}(S)$ for $S$ small and some $\kappa$, i.e. the category of $\kappa$-flat functors from some small category to $\mathbf{Set}$.

  • it is the category of models (in $\mathbf{Set}$) of a suitable type of logical theory.

If you are interested in the classical filtered colimit completion, that is the case where $\kappa = \omega$. This material is covered in Locally Presentable and Accessible Categories; see specifically the chapter on accessible categories.

Todd Trimble
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  • It seems possible to generalize this result: let A, B collections of small categories. Then Ind_A(C) (the completion of C with respect to A-shaped diagrams) is the category of models for the sketch having B as class of colimits iff any A-colimit commutes with any B-limit (I hope notations are clear with a little effort: comments force to a really dry exposition!) Are you aware of something at this level of generality? – fosco Oct 26 '13 at 15:55
  • Consider for example, aside to Ind_k(C), the case where A=sifted categories. Then B is the class of all discrete categories, and the completion of C with respect to sifted colimits is the category of models for the sketch having discrete cats as set of colimits (hope I didn't dualize too much times!) – fosco Oct 26 '13 at 15:58
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    The paper Adámek, Borceux, Lack, Rosický, A classification of accessible categories addresses exactly these types of questions. In particular, Section 3 is about the Ind construction and Section 4 is about sketches. Everything is relative to some nice class of small categories $\mathbb{D}$ and these colimits that commute with $\mathbb{D}$-limits. – Karol Szumiło Oct 26 '13 at 18:02
  • @KarolSzumiło Thank you; I wasn't quite sure what the question was when I first answered. If you post your comment as an answer, I'd gladly upvote it. Here is a link: http://www.sciencedirect.com/science/journal/00224049/175/1 – Todd Trimble Oct 26 '13 at 18:25
  • @Karol and Todd: thank you for your answer. The spirit of my question was exactly that of this other topic http://mathoverflow.net/questions/93262/which-colimits-commute-with-which-limits-in-the-category-of-sets so I think this question can be closed as a duplicate. – fosco Oct 26 '13 at 21:50
  • @tetrapharmakon I disagree it should be closed as a duplicate! The context is a little different, so we should let this stand. I for one learned of a paper I was unaware of before, even though the paper was mentioned in the other thread, and I think the other question has not been addressed yet to 100% satisfaction -- still waiting to hear back from Marie Bjerrum!!. – Todd Trimble Oct 26 '13 at 21:58
  • In this case I will be glad to accept Karol's answer if he decides to rewrite his comment. :) I was strongly encouraged by O. Caramello to contact Marie via email to ask for a copy of its thesis. Can't wait for the answer! – fosco Oct 27 '13 at 13:02
  • @tetrapharmakon I already did that. Her viva (thesis examination) should be coming up within the next few months, and she said that after submitting any corrections after the examination and gaining final acceptance, she'd be happy to make the thesis available to interested people. – Todd Trimble Oct 27 '13 at 13:32
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As suggested I repost my comment as an answer.

The paper Adámek, Borceux, Lack, Rosický, A classification of accessible categories addresses exactly these types of questions. In particular, Section 3 is about the Ind construction and Section 4 is about sketches. Everything is relative to some nice class of small categories $\mathbb{D}$ and these colimits that commute with $\mathbb{D}$-limits.

Karol Szumiło
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