# Strong colimits of categories.

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 $\mathcal F:\mathcal C\to\mathcal C\textrm{at}$?

• There is no tricategory or bicategory of bicategories and functors whose 2-cells are lax transformations (try to define the whiskering ($F \circ \alpha$). Nov 24, 2010 at 12:16
• Tank you. This was a point i did not really consider. Nov 24, 2010 at 12:39
• This does not make the question obsolete, right? (I fixed the question, btw.) Nov 25, 2010 at 16:11
• Have a look at Tim Porter's Crossed Menagerie ncatlab.org/timporter/show/crossed+menagerie section 8.2.10 and also arxiv.org/abs/math/0408298 chapter 3. Nov 26, 2010 at 0:28
• FWIW, I think what you have called a "lax colimit" should more properly be called an "oplax colimit", as confusing as it may be; see the discussion at ncatlab.org/nlab/show/2-limit#lax . Oct 11, 2012 at 12:28

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.
• Yes, this is the right answer. In order to prove it one can either follow the construction given by Tom Fiore or split it into two parts: First, we establish the grothendieck construction as a lax colimit. Then, we realize that among all cones the strong cones correspond to functors that send every morphism $X\to Pm(X)$ and thus every (op)cartesian morphism to an isomorphism. These functors however correspond to functors out of the localisation of the grothendieck construction along all (op)cartesian morphisms. Dec 5, 2010 at 18:23