Question

Let $\mathcal C$ be a category and let $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$ be a strong bifunctor. Given another category $\mathcal D$, let $\triangle_{\mathcal D}$ denote the constant functor $\mathcal C\to\mathcal C\textrm{at}$. Now define lax/strong limits and colimits as follows:

• A lax limit of $\mathcal F$ is a category $\mathsf{lim}\mathcal F$ together with a natural equivalence $$[\triangle_{(-)},\mathcal F] \cong \mathcal C\textrm{at}(-,\mathsf{lim}\mathcal F).$$ Here $[\triangle_{(-)},\mathcal F]$ denotes the category of lax natural transformations and modifications.

• A lax colimit of $\mathcal F$ is a category $\mathsf{colim}\mathcal F$ together with a natural equivalence $$[\mathcal F,\triangle_{(-)}] \cong \mathcal C\textrm{at}(\mathsf{colim}\mathcal F,-).$$

• We define strong limits and strong colimts by replacing lax natural transormations with strong natural transfomations.

Now, if my calculations are correct, a lax colimit of such a functor $\mathcal F$ is given by the grothendieck construction $\mathcal C\int\mathcal F$ and a lax limit is given by the category of strict sections $s:\mathcal C\to \mathcal C\int\mathcal F$, i.e. the category $\mathcal C\textrm{at}/\mathcal C(\operatorname{id}_\mathcal C,\pi)$, where $\pi:\mathcal C\int\mathcal F\to\mathcal C$ is the opfibration corresponding to $\mathcal F$.

If we consider only the category of opcartesian sections, that is sections that map every morphism in $\mathcal C$ to an opcartesian morphism, we get a strong limit.

Now for the question:

Is there an explicit description of the strong colimit of a functor $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$?

Answer 1

Adapted from the papers by Fiore and by Porter that I referred to above:

To form the usual lax colimit of P we take the disjoint union of the $P_i$ for each $i \in C$ and then adjoin new arrows to represent the action of P: for each $m \colon i \to j$ in C and each $X \in P_i$ there is an arrow $X \to Pm(X)$. Then we quotient by a congruence that ensures that the assignment of $X \to Pm(X)$ to $X$ is lax natural. This lax transformation is the universal cone.

To form the pseudo colimit we simply make sure each $X \to Pm(X)$ is an isomorphism: adjoin a formal inverse along with it and add the requisite equations to the congruence.