# Is the category of computads for a finitary monad on $n$-globular sets cocomplete?

Context

Given a finitary monad $$T:\operatorname{gSet}_n\to\operatorname{gSet}_n$$ we can define categories $$\operatorname{Comp}_k^T$$ of $$k$$-computads for $$T$$, for any $$k=0,\cdots,n+1$$. This is nicely explained in Schommer-Pries' thesis, for example and the original source is this paper of Batanin. Essentially a $$k$$-computad $$C$$ is defined inductively as a tuple $$(C_k,C_{\leq k -1},s,t)$$ where $$C_k$$ is a set (the set of $$k$$-cells), $$C_{\leq k-1}$$ is a $$(k-1)$$-computad and $$s,t$$ are maps $$C_k\to [F_{k-1}(C_{\leq k-1})]_{k-1}$$ where $$F_{k-1}$$ is the functor that constructs the $$T$$-algebra generated by a $$(k-1)$$-computad. These are required to satisfy the usual globularity conditions $$ss=st$$ and $$ts=tt$$. Then one completes the induction by defining $$F_k$$ by a certain pushout diagram in $$\operatorname{Alg_T}$$ (using the fact that $$\operatorname{Alg_T}$$ is cocomplete, because $$T$$ is finitary).

Questions

1. Is the category $$\operatorname{Comp}_k^T$$ always cocomplete?
2. Does the functor $$\operatorname{Comp}_k^T\to\operatorname{Set}$$ that takes a computad to its set of $$i$$-cells for some $$i\leq k$$ preserve colimits?
3. Is there a reference for these facts?

If there is no reference, then

1. Is the sketch of proof below correct, or am I missing something?

Sketch of proof of 1. and 2.

One constructs the colimit of some diagram $$C(i)$$ of $$k$$-computads by induction on $$k$$: one constructs a $$k$$-computad whose set of $$k$$-cells is the colimit of the diagram on $$k$$-cells and whose underlying $$(k-1)$$-computad is the colimit of the diagram of underlying $$(k-1)$$-computads. Source and target maps $$\operatorname{colim}_i C(i)_k\to [F_{k-1}(\operatorname{colim}_i C(i)_{\leq k-1})]_{k-1}$$ can be defined by the composite $$C(i)_k\to [F_{k-1}(C(i)_{\leq k-1})]_{k-1}\to\operatorname{colim}_i [F_{k-1}(C(i)_{\leq k-1})]_{k-1}\to[\operatorname{colim}_iF_{k-1}(C(i)_{\leq k-1})]_{k-1}=[F_{k-1}(\operatorname{colim}_i C(i)_{\leq k-1})]_{k-1}$$ where the equality comes from the fact that $$F_{k-1}$$ is left adjoint and the final arrow is induced by the maps $$[F_{k-1}(C(i)_{\leq k-1})]_{k-1}\to[\operatorname{colim}_iF_{k-1}(C(i)_{\leq k-1})]_{k-1}$$ which are the maps of underlying $$(k-1)$$-morphisms associated to the canonical maps of $$T$$-algebras $$F_{k-1}(C(i)_{\leq k-1})\to\operatorname{colim}_iF_{k-1}(C(i)_{\leq k-1})$$.

Now one needs to check that $$s,t$$ satisfy globularity and that the construction has the right universal property. This also seems completely straightforward.

What I have found in the literature

In the above cited paper of Batanin, it is proved that when the monad is truncable and preserves finite pullbacks, the category $$\operatorname{Comp}_n^T$$ is an elementary topos, so it particular has finite colimits. It is also proved that if additionally $$T$$ preserves wide pullbacks and the unit is cocartesian then $$\operatorname{Comp}_n^T$$ is a presheaf topos, so in particular is cocomplete. This appears as Theorem 4.1.

I am hoping, however, that if one only needs cocompleteness then the hypotheses are not necessary and there is the above direct proof.

The answer to $$1$$ and $$2$$ are both yes. I don't know if this appears in the literature. The argument you give seems reasonable - I don't completely follow your notation but the general idea is that in the "inductive" definition of computads, you can show that colimits of "k-computads" are computed by taking the colimit of the (k-1)-computads and the sets of k-cells separately, which seems to be exactly what you are saying. Colimits of general computads are then computed by taking the colimit of their "underlying k-computads" for all k.
I would consider this as a folklore results in the area, and I wouldn't be surprised if it is written somewhere, at least for the special case of the strict $$\infty$$-category monads, but I couldn't find a reference to give you.
Note however that the results of Batanin you quote is unfortunately false - the category of computads for the free strict $$\infty$$-category (which satisfies all the assumption of Batamin's theorem) is not a topos. In fact it is not cartesian closed (or here).
A latter paper of Batanin gives a finer criterion for such categories of computads to indeed be presheaf categories which I think* is correct. This second paper applies for example to the computads for Batanin's definition of weak $$\infty$$-categories.