The free completion of a category under sifted colimits

Question

The free completion of a category under sifted colimit is denoted by $$Sind \ \cal K.$$ If $\cal K$ has sifted colimits we have the functor as in the snippet below:$$\text{colim}:Sind\ \cal K\to K.$$

My question is easy: how formally (technically) look the category $Sind\ \cal K$ and the functor $\text{colim}$?

I have some intuition which is, however, not precise.

Answer 1

You can see this paper

Adamek, Rosicky, Vitale, *What are sifted colimits? (2010)

which states a slogan (and conditions for theorem) that:

the free completion of $\mathcal C$ under sifted colimits is the free completion of $Ind\ \mathcal C$ under reflexive coequalizers.

The category $Ind \mathcal\ C$ of Ind-objects you may already know, which can be constructed for example as the full subcategory of presheaves on $\mathcal C$ of objects which are colimits of representables (cf. nlab).

As for the completion under reflexive coequalizers, this is mentioned in:

Adamek, Rosicky; On sifted colimits and generalized varieties (2001)

which in turn points to

Bunge, Carboni; The symmetric topos (1995)

There it deals with with the construction of completion under equaliers, mentions this is equivalent to completion under coreflexive equalizers, and also mentions the dual case of coequalizers.

I won't dig any further, but this is probably enough information to get to a formal definition of $Sind \mathcal\ C$.