# Ind-completion of a 2-category

If $$\mathcal{C}$$ is a category, there is a well known construction called the Ind-completion of $$\mathcal{C}$$, indicated by $$\text{Ind}(\mathcal{C})$$. This can be equivalently defined in several ways: for example, $$\text{Ind}(\mathcal{C})$$ may be seen as the full subcategory of $$[\mathcal{C}^{op}, \mathbf{Set}]$$ whose objects are those presheaves which are isomorphic to filtered colimits of representables. This construction has a universal property: namely, each functor $$\mathcal{C}\rightarrow\mathcal{D}$$, with $$\mathcal{D}$$ a category with filtered colimits, factors uniquely through the inclusion $$\mathcal{C}\hookrightarrow\text{Ind}(\mathcal{C})$$.

Now let $$\mathcal{C}$$ be a 2-category, and consider the 2-category of pseudofunctors $$F:\mathcal{C}^{op}\rightarrow\mathbf{Cat}$$ which are isomorphic to filtered pseudocolimits of representables, with the usual pseudonatural transformations and modifications.

My question is: has this 2-category ever been studied? Does it have a universal property similar to the one of the Ind-completion of a category?

You can find discussion of this in Lack's 2-categories companion. The original reference in a more general context is Kelly's Structures defined by finite limits in the enriched context. See more discussion in Lack and Rosicky's Notions of Lawvere Theory.

The theory is a bit more delicate because often one really wants to work up to 2-equivalence, but the most "automatic" way to develop the theory views 2-categories as categories enriched in $$Cat$$, so the natural notion of equivalence is stricter.