Finn Lawler
free 2-cocompletion

We want to show that if $K$ is a bicategory then $PK=[K^{\mathrm{op}},\mathrm{Cat}]$ is the free 2-cocompletion of $K$.

There are several kinds of 2-colimit that we’ll need to talk about. Let $D:J\to K$ and $W:J^{\mathrm{op}}\to \mathrm{Cat}$ be pseudofunctors. Then

1. The 2-colimit $W\star D$ satisfies

   $$K(W\star D,X)\simeq [J^{\mathrm{op}},\mathrm{Cat}](W,K(D-,X))$$K(W \star D, X) \simeq [J^{op}, Cat](W, K(D-, X))

   This is what in the literature is often called a bilimit.

2. If $K$ is a strict 2-category, the pseudocolimit $W{\star }_{p}D$ satisfies the same property up to isomorphism.

3. If $K$ is strict and $W$ and $D$ are strict 2-functors, then the strict pseudocolimit $W{\star }_p^{s}D$ satisfies

   $$K(W{\star }_p^{s}D,X)\cong \mathrm{Ps}(J^{\mathrm{op}},\mathrm{Cat})(W,K(D-,X))$$K(W \star^s_p D, X) \cong Ps(J^{op}, Cat)(W, K(D-, X))

   where on the right the functor category is that of strict 2-functors, pseudonatural transformations and modifications.

4. Under the same hypotheses, the strict colimit $W{\star }_s^{s}D$ satisfies

   $$K(W{\star }_s^{s}D,X)\cong \mathrm{Str}(J^{\mathrm{op}},\mathrm{Cat})(W,K(D-,X))$$K(W \star^s_s D, X) \cong Str(J^{op}, Cat)(W, K(D-, X))

   where now $\mathrm{Str}$ denotes the category of strict 2-functors and strict transformations (and modifications).

We need to show that $PK$ has all small 2-colimits:

1. $\mathrm{Cat}$ is strictly 2-cocomplete: its underlying 1-category has small colimits, and $\mathrm{Cat}$ is enriched and tensored over itself, so that it has strict $\mathrm{Cat}$-weighted colimits.

2. Pseudocolimits, strict or otherwise, are a fortiori 2-colimits, and strict pseudocolimits are just strict colimits whose weights are ‘cofibrant’ in a suitable sense. Moreover, if $K$ is a strict 2-category, then for any index bicategory $J$ there is a strict 2-category $J\prime$ such that strict functors $J\prime \to K$ are the same thing as pseudofunctors $J\to K$, and the 2-colimit of pseudofunctors $W\star D$ is equivalent to the strict pseudocolimit of the strictified functors. So a strictly 2-cocomplete strict 2-category is also 2-cocomplete.

3. $\mathrm{Cat}$ therefore has non-strict 2-colimits. We can now try to compute colimits pointwise in $PK$ as for strictly-enriched functor categories: if now $D:J\to PK$ and $W:J^{\mathrm{op}}\to \mathrm{Cat}$ then set $(W\star D)a=W\star D(-,a)$, and its universal property follows:

   $$\begin{array}{cc}PK(W\star D,F)& \simeq {\int }_a\mathrm{Cat}(W\star D(-,a),Fa)\\ & \simeq {\int }_a[J^{\mathrm{op}},\mathrm{Cat}](W,\mathrm{Cat}(D(-,a),Fa))\\ & \simeq [J^{\mathrm{op}},\mathrm{Cat}](W,{\int }_a\mathrm{Cat}(D(-,a),Fa))\\ & \simeq [J^{\mathrm{op}},\mathrm{Cat}](W,PK(D-,F))\end{array}$$\array{ P K(W \star D, F) & \simeq \int_a Cat(W \star D(-,a), F a) \\ & \simeq \int_a [J^{op}, Cat](W, Cat(D(-,a), F a)) \\ & \simeq [J^{op}, Cat](W, \int_a Cat(D(-,a), F a)) \\ & \simeq [J^{op}, Cat](W, P K(D-, F)) }

   So $PK$ has 2-colimits.

Finally, we need to show that if $L$ is a cocomplete bicategory, then there is a 2-equivalence

$$\mathrm{Cocont}(PK,L)\sim [K,L]$$Cocont(P K, L) \sim [K, L]

For this we simply follow the usual reasoning: from left to right we compose with the Yoneda embedding $y:K\to PK$, and given a functor $F:K\to L$ we get a cocontinuous $PK\to L$ sending $W:K^{\mathrm{op}}\to \mathrm{Cat}$ to $W\star F$.

The co-Yoneda lemma shows that every $W\simeq W\star y$, and if $H$ is cocontinuous then $H(W)\simeq H(W\star y)\simeq W\star Hy$, showing that the functor $F\mapsto -\star F$ is essentially surjective. It is 2-fully-faithful by the universal property of colimits: a transformation $F\to G$ gives rise to an essentially unique transformation $-\star F\to -\star G$.