Conservative cocompletion of categories of geometric shapes for homotopy theory

Question

The recent paper

Calin Tataru, Partial orders are the free conservative cocompletion of total orders. arXiv:2404.12924

has shown that the conservative cocompletion of the simplex category $\Delta$ is the category $\mathsf{Pos}$ of posets.

There are a number of other important "categories of geometric shapes" whose (non-conservative) free cocompletion $\mathcal{C}\mapsto\mathsf{PSh}(\mathcal{C})$ lead to models of higher categories or other important constructions in homotopy theory. Some of these include:

1. The the globe category $\mathbb{G}$, whose free cocompletion $\mathsf{PSh}(\mathbb{G})$ leads to globular sets, which are the basis of Grothendieck–Maltsiniotis $\infty$-categories.
2. The cube category $\square$ and its variant $\square_{\mathsf{c}}$ with connections, whose free cocompletions $\mathsf{PSh}(\mathbb{\square})$ and $\mathsf{PSh}(\square_\mathsf{c})$ lead to cubical sets with or without connections, both of which and can be used to model $\infty$-categories [1], [2].
3. The tree category $\Omega$, whose free cocompletion $\mathsf{PSh}(\Omega)$ leads to dendroidal sets and can be used to model $\infty$-operads [3].
4. Joyal's $\Theta_{n}$ category, whose free cocompletion $\mathsf{PSh}(\Theta_{n})$ leads to cellular sets and can be used to model $(\infty,n)$-categories [4].
5. Segal's $\Gamma$ category, whose free cocompletion $\mathsf{PSh}(\Gamma)$ and it's $\mathsf{Top}$- or $\mathsf{sSets}$-enriched cousins provide a model for homotopy-coherent monoids [5].
6. The cycle category $\Lambda$ and the paracycle category $\Lambda_\infty$, whose free cocompletions $\mathsf{PSh}(\Lambda)$ and $\mathsf{PSh}(\Lambda_\infty)$ are related to topological cyclic homology [6].
7. The orbit category $\mathsf{Orb}_{G}$ of a topological group $G$, whose free cocompletion $\mathsf{PSh}(\mathsf{Orb}_{G})$ is a model for $G$-spaces [7].

There are other such categories (e.g. the quaternionic simplex category $\Delta Q$), but I believe the above ones seem to be the main such "categories of shapes" that have been consistently appearing in homotopy theory recently.

The following is perhaps too broad a question, and most likely partially answering it might be worth of a paper, but I believe it's better than asking 7 separate questions and also it would be good to record it as an MO question:

Question. What is the conservative cocompletion of the categories $\mathbb{G}$, $\square$, $\square_{\mathsf{c}}$, $\Omega$, $\Theta_{n}$, $\Gamma$, $\Lambda$, $\Lambda_{\infty}$, and $\mathsf{Orb}_{G}$?

Answer 1

I don't really see a way to give a single answer here, these are 7 different questions (well actually a lot more than 7 if we count all the different flavours of cubes, and of the other ones, probably more like 30 differents questions).

Though, a common pattern I think is going to hold to answer any of these question (and I'll give the answer in one case to illustrate it), is that the first step is to understand what are colimits in the small category. In the case of $\Delta$, this is done by the following (from the paper of Calin Tataru you cite)

Prop: The embedding of $\Delta \to \text{Pos}$ preserves colimits.

Indeed as it is a fully faithful embedding this completely characterizes colimits in $\Delta$: Given any diagram in $\Delta$, we can compute its colimits in $\text{Pos}$, if the colimit is in $\Delta$ then it is a colimit in $\Delta$, otherwise there is no colimit in $\Delta$. Once you know what are the colimits in the small category, you can think of it as a colimit sketch, and the completion is the category of models of that sketch.

To put it another way, it is generally easy to understand the free cocompletion that preserves an explicitely given set of colimits. What is hard with these conservative cocompletion is that the class of colimits isn't explicitly given, so the first step is to figure out what are the colimits that need to be preserved.

Let's do the example of the category of $G$-Orbit, because it is fairly straightforward how to do it. I claim that:

Prop: A diagram in the category of $G$-Orbits has a colimit if and only if it is connected.

Proof: This is fairly easy to check that a colimit of a connected diagram of $G$-orbits taken in the category of $G$-sets is again a $G$-orbit. Conversely, it is also easy to check that coproducts and initial objects never exists in the category of $G$-orbits.

It then follows that the conservative cocompletion of $G$-Orbits is the free cocompletion that presverse all these connected colimits, so it is actually just the completion under coproduct, in particular we get:

Theorem: The conservative cocompletion of the category of $G$-orbits is the category of $G$-set.

I feel like using well behaved embeddings of the small category onto a cocomplete category is a good way to start studying the colimits in the small category (like the embedding $\Delta \to \text{Pos}$ in the paper you cite, or the embedding $G$-Orbit to $G$-set I'm using here to understand colimits) the fact that these embedding end-up being the cocompletion is I think not strictly neccessary.