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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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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.

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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.

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  • $\begingroup$ 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? $\endgroup$
    – fosco
    Commented Oct 26, 2013 at 15:55
  • $\begingroup$ 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!) $\endgroup$
    – fosco
    Commented Oct 26, 2013 at 15:58
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    $\begingroup$ 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. $\endgroup$
    – Karol Szumiło
    Commented Oct 26, 2013 at 18:02
  • $\begingroup$ @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: sciencedirect.com/science/journal/00224049/175/1 $\endgroup$
    – Todd Trimble
    Commented Oct 26, 2013 at 18:25
  • $\begingroup$ @Karol and Todd: thank you for your answer. The spirit of my question was exactly that of this other topic mathoverflow.net/questions/93262/… so I think this question can be closed as a duplicate. $\endgroup$
    – fosco
    Commented Oct 26, 2013 at 21:50

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