**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](https://arxiv.org/abs/1112.1000), for example and the original source is this [paper](https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.53.2118) 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


4. 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.



Is the category of computads for a finitary monad on $n$-globular sets cocomplete?
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](https://arxiv.org/abs/0710.5202) (or [here](https://arxiv.org/abs/1209.0414)).


A [latter paper](https://arxiv.org/abs/math/0209035) 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.


\*: To be clear, I have to admit I have never been able to fully understand the argument given by Batanin in this second paper, and the paper has never been published, so I can't tell if the proof given there is correct (though I have no reasons to doubt it is). But, this being said I have very good reason to believe the result itself is definitely correct due to my [own](https://arxiv.org/abs/1711.00744) work on the topic which involves similar assumptions.


