Finn Lawler free 2-cocompletion

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

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

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

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

   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 $W \star^s_{p} D$ satisfies

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

   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

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

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

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

1. $Cat$ is strictly 2-cocomplete: its underlying 1-category has small colimits, and $Cat$ is enriched and tensored over itself, so that it has strict $Cat$-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 $J'$ such that strict functors $J' \to K$ 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. $Cat$ therefore has non-strict 2-colimits. We can now try to compute colimits pointwise in $P K$ as for strictly-enriched functor categories: if now $D \colon J \to P K$ and $W \colon J^{op} \to Cat$ then set $(W \star D) a = W \star D(-,a)$, and its universal property follows:

   $$\array{ P K(W \star D, F) & \simeq \int_a Cat(W \star D(-,a), F a) \\ & \simeq \int_a [J^{op}, Cat](W, Cat(D(-,a), F a)) \\ & \simeq [J^{op}, Cat](W, \int_a Cat(D(-,a), F a)) \\ & \simeq [J^{op}, Cat](W, P K(D-, F)) }$$

   So $P K$ has 2-colimits.

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

$$Cocont(P K, L) \sim [K, L]$$

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

The co-Yoneda lemma shows that every $W \simeq W \star y$, and if $H$ is cocontinuous then $H(W) \simeq H(W \star y) \simeq W \star H y$, 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 $-\star F \to -\star G$.