Finn Lawler free 2-cocompletion (Rev #3, changes)

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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 $W\star \mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; D</ins></span>}$ W \star J D satisfies

   $$K(W\star \mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; D</ins></span>},X)\simeq [J^{\mathrm{op}},\mathrm{Cat}](W,K(D-,X))$$ K(W \star J, 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 $W{\star }_p\mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; D</ins></span>}$ W \star_{p} J 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 $W{\star }_p^s\mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; D</ins></span>}$ W \star^s_{p} J D satisfies

   $$K(W{\star }_p^s\mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; D</ins></span>},X)\cong \mathrm{Ps}(J^{\mathrm{op}},\mathrm{Cat})(W,K(D-,X))$$ K(W \star^s_p J, 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 $W{\star }_s^s\mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; D</ins></span>}$ W \star^s_s J D satisfies

   $$K(W{\star }_s^s\mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; D</ins></span>},X)\cong \mathrm{Str}(J^{\mathrm{op}},\mathrm{Cat})(W,K(D-,X))$$ K(W \star^s_s J, 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 the bicategory in question is a strict 2-category, then for any index bicategory$\mathrm{<span><del\; class="diffmod">\; J</del><ins\; class="diffmod">\; K</ins></span>}$ J K there is a strict 2-category 2-category, then for any index bicategory$J\prime$ J' J such there that is a strict functors 2-category$J\prime \to K$ J' \to K are such the that same strict thing functors as pseudofunctors$J\prime \to K$ J J' \to K , and are the 2-colimit same of thing as pseudofunctors$\mathrm{<span><del\; class="diffmod">\; W</del><ins\; class="diffmod">\; J</ins></span>}<span><del\; class="diffmod">\; \star </del><ins\; class="diffmod">\; \to </ins></span>\mathrm{<span><del\; class="diffmod">\; D</del><ins\; class="diffmod">\; K</ins></span>}$ W J \star \to D 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. $P K = [K^{op}, Cat]$ is a strict 2-category, and it has all strict ($Cat$-weighted) colimits, which are computed pointwise as usual for enriched functor categories: if now $D \colon J \to P K$ and $W \colon J^{op} \to Cat$ are strict, then we can define $(W \star^s_s D) a = W \star^s_s D(-,a)$, and verify its universal property:

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

So $P K$ has strict 2-colimits and hence also non-strict 2-colimits.

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

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 $H(W)\simeq H(W\star y)\simeq W\star Hy$ H(W) \simeq H(W \star y) \simeq W \star H y, showing that the functor $F\mapsto -<span><del\; class="diffmod">\; \circ </del><ins\; class="diffmod">\; \star </ins></span>\mathrm{<span><del\; class="diffmod">\; y</del><ins\; class="diffmod">\; F</ins></span>}$ F \mapsto - \circ \star y 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 $-\star F \to -\star G$.