Completing 2-categories under lax colimits

Given a class of colimits, $\mathrm{\Phi }$, one can freely add $\mathrm{\Phi }$-colimits to a category to obtain its free completion under $\mathrm{\Phi }$-colimits. Examples include the (small) presheaf category (when $\mathrm{\Phi }=$ all small colimits), the Fam construction ($\mathrm{\Phi }=$ coproducts), the Ind-completion ($\mathrm{\Phi }=$ filtered colimits) and the Karoubi envelope ($\mathrm{\Phi }=$ splitting of idempotents).

Higher-dimensional categories provide richer notions of colimit for which we can describe corresponding completions. For example, both the Kleisli completion for 2-categories and the map sending a bicategory $𝒱$ to $𝒱\mathrm{𝖢𝖺𝗍}$ can be viewed as free completions under certain classes of lax colimits for 2-dimensional categories, as shown in [2] and [1] respectively.

This talk will provide descriptions first for the completion of a 2-category under the class of all small lax colimits, and then for certain subclasses of lax colimits. In particular, we will observe that completing a 2-category under lax colimits of lax functors from small 1-categories yields a construction analogous to the Fam construction for ordinary categories, both in its structure and properties. This completion extends the Kleisli completion (i.e. the completion under lax colimits of lax functors from the terminal category) and is related to enrichment: the objects of the lax-functor lax-colimit completion of the delooping $𝔹𝒱$ of a monoidal category $𝒱$ can be viewed as categories enriched in $\mathrm{𝖥𝖺𝗆}(𝒱)$.

Notable examples include the lax-functor lax-colimit completions of ${\mathrm{𝖢𝖺𝗍}}_{\cong }$ (with 2-cells natural isomorphisms), $\mathrm{𝖦𝗋𝗉𝖽}$ and $\mathrm{𝖲𝖾𝗍}$, which yield respectively the 2-categories of fibrations $\mathrm{𝖥𝗂𝖻}$, $\mathrm{𝖨𝗌𝗈𝖥𝗂𝖻}$ and $\mathrm{𝖣𝗂𝗌𝖼𝖥𝗂𝖻}$, whereas the completion of the arrow category $\mathrm{𝟐}$ yields a certain 2-category of profunctors.

If time permits, we will also observe how free completions under classes of lax colimits naturally extend from 2-functors on $\mathrm{𝟤}𝖢𝖺𝗍$ to ${\mathrm{𝖦𝗋𝖺𝗒}}_{\mathrm{𝗅𝖺𝗑}}$-functors, i.e. functors which additionally act on lax transformations and modifications between them.

This talk is based on work from my PhD thesis [3] supervised by Richard Garner.