# Weak colimits of weak and strict presheaves in groupoids

Let $C$ be a small category, and for this question, let groupoid mean an (essentially small) groupoid. There are two 2-categories in question: the 2-category of strict presheaves in groupoids and strict transformations and modifications, and the 2-category of weak presheaves in groupoids with lax transformations and modifications. There is an obvious inclusion from the former to the latter which is 2-categorically essentially surjective and faithful, but not full: there are more lax natural transformations than strict ones. Does this functor none-the-less preserve weak colimits? Naively, I would like to argue that in both 2-categories, weak colimits are computed "pointwise"- however, this is not rigorous as it does not take into account how the weak colimit presheaf behaves on arrows.

In my situation, I am given a diagram of stacks, but with each stack represented by a groupoid object in sheaves. I would like to argue that the weak colimit of this diagram of stacks is represented by the weak colimit of this diagram of groupoid objects (using the equivalence between the 2-category of groupoid objects in sheaves and the 2-category of strict presheaves in groupoids). Is this true?

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Regarding the weak colimit: surely you'd need to stackify after taking the colimit of the presheaves in any case? – David Roberts Oct 28 '10 at 20:19
@David, yes, but stackifying is a left adjoint so it preserves these colimits :). – David Carchedi Oct 28 '10 at 21:31
For a while, I had an answer here posted that wasn't quite right, that I had sinced fixed on my own and forgot to update. The correct proof is now below. – David Carchedi Nov 26 '10 at 9:22

It suffices to show this for psuedo-colimits, i.e. ones for which the induced equivalence of categories is in fact an isomorphism of categories (such a strict model for the weak colimit exists, since we can explicitly calculate one in the 2-category of groupoids). Denote pseudo-colimits by $pcolim$.

Some notation:

If $\mu:F \Rightarrow \Delta_X$ is a cocone, and $f:X \to Y$, denote by $\hat{\mu}(f)$ the cocone $\Delta_{f} \circ \mu:F \Rightarrow \Delta_{Y}$.

Let $j$ denote the canonical $2$-functor from strict presheaves in groupoids into weak ones.

Let $F:J \to Psh\left(\mathcal{C},Gpd\right)$ be any pseudo-functor. Let $$\mu_{S}:F \Rightarrow \Delta_{pcolim F}$$ be a pseudo-colimiting cocone for $F$ and $$\mu_{W}:j \circ F \Rightarrow \Delta_{pcolim j \circ F}$$ be a pseudo-colimiting cocone for $j \circ F$. To simplify notation, let $S:=pcolim F$ and $W:=pcolim j\circ F$. Then $$j\mu_{S}:j\circ F \Rightarrow \Delta{j\left(S\right)}$$ is a cocone for $j \circ F$ with vertex $j\left(S\right)$. Hence there exists a morphism $\phi:W \to S$ such that $j\mu_{S}=\Delta_{\phi} \circ \mu_{W}$. We claim that $\phi$ is an isomorphism.

It suffices to show that for each $C \in \mathcal{C}_0$, the map $\phi\left(C\right):W\left(C\right) \to S\left(C\right)$ is an isomorphism of groupods. Consider the inclusion of the object $C$ as a functor $$1\stackrel{\imath}{\rightarrow} \mathcal{C}$$ from the terminal category. This induces two $2$-functors, and by abuse of notation, we will denote both by $$\imath^{*}:Psh\left(C,Gpd\right) \to Gpd$$ $$\imath^{*}:Gpd^{C^{op}} \to Gpd.$$ Both of these $2$-functors are pseudo left adjoints and are given by evaluation at the objet $C$, so clearly $\imath^{*}j=\imath^{*}$.

We want to show that $$\phi\left(C\right)=\imath^{*}\left(\phi\right)$$ is an isomorphism of groupoids. Since $\imath^{*}$ is a pseudo left adjoint, it follows that $$\imath^{*}\mu_{W}:\imath^{*}\circ j \circ F=F\left(C\right) \Rightarrow \imath^{*} \circ \Delta_{W}= \Delta_{W\left(C\right)}$$ is pseudo-colimiting. Now, since pseudo-colimits $Psh\left(C,Gpd\right)$ are computed pointwise, it follows that $\imath^{*}\mu{S}$ is pseudo-colimiting. So, there exists a functor $$\psi:S\left(C\right) \to W\left(C\right)$$ such that $$\Delta{\psi}\circ\imath^{*}\mu_{S}=\imath^{*}\mu_{W}.$$

Notice that $$\imath^{*}\mu_{S}=\imath^{*}\left(\Delta_{\phi}\right)\circ \mu_{W})=\Delta_{\phi\left(C\right)}\circ \imath^{*}\mu_{W}.$$ So

$$\Delta_{\psi}\circ \Delta{\phi\left(C\right)}\circ \imath^{*}\mu_{W}=\imath^{*}\mu_{W}.$$ The left hand side of this equation is equal to $\hat{\imath^{*}\mu_{W}}\left(\psi \circ \phi\left(C\right)\right)$ whereas the right hand side is equal to $\hat{\imath^{*}\mu_{W}}\left(id_{W\left(C\right)}\right).$ But $\imath^{*}\mu_{W}$ is pseudo-colimiting, so $\hat{\imath^{*}\mu_{W}}$ is an isomorphism of categories, hence $\psi \circ \phi\left(C\right)=id_{W\left(C\right)}.$

Notice further that

\begin{eqnarray*} \imath^{*}\mu_{S}\left(\phi\left(C\right) \circ \psi\right)&=&\Delta_{\phi\left(C\right)} \circ \Delta{\psi} \circ \Delta{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\\ &=&\Delta_{\phi\left(C\right)} \circ \imath^{*}\mu_{W}\\ &=&\imath^{*}\mu_{S}\\ &=&\hat{\imath^{*}\mu_{S}}\left(id_{S\left(C\right)}\right).\\ \end{eqnarray*}

But $\imath^{*}\mu_{S}$ is pseudo-colimiting, so $\hat{\imath^{*}\mu_{S}}$ is an isomorphism, hence $$\phi\left(C\right) \circ \psi=id_{S\left(C\right)}.$$ Therefore, $\phi$ is an isomorphism. Since pseudo-colimits are stable under isomorphisms, it follows that $j\left(S\right)$ is a pseudo-colimit for $j\circ F$.

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I apologize for the compiling error, but I can't get it to listen to me. – David Carchedi Nov 26 '10 at 9:21
You want to use backticks (`) to escape misbehaving TeX, not apostrophes ('). – Charles Rezk Nov 26 '10 at 15:02
Thanks, I was wondering what was going on. – David Carchedi Nov 26 '10 at 18:22