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:

- 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.
- 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].
- The tree category $\Omega$, whose free cocompletion $\mathsf{PSh}(\Omega)$ leads to dendroidal sets and can be used to model $\infty$-operads [3].
- 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].
- 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].
- 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].
- 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 $\mathsf{Orb}_{G}$?