Is every category a localization of a poset?

Question

Question 1: Let $C$ be a small category. Does there exist a poset $P$, a set of $W$ of morphisms in $P$, and an equivalence $P[W^{-1}] \simeq C$?

Here $P[W^{-1}]$ is the universal category receiving a functor from $P$ which carries each morphism of $W$ to an isomorphism.

I'm hoping for an affirmative answer. I'm also interested in the following variation:

Question 2: Let $C$ be a small category with finite colimits. Does there exist a join-semilattice $P$, a set $W$ of morphisms in $P$, and an equivalence $P[W^{-1}] \simeq C$?

Version control:

• There are actually two versions of question 2 -- in one version we require that $P \to P[W^{-1}]$ preserves finite colimits, and in the other we don't. As I'm hoping for an affirmative answer, it should be easier to do this without requiring the preservation of the finite colimits, and I'd be happy with an answer to that version.

• On top of that, I am interested in two versions of these questions: the 1-categorical version and the $\infty$-categorical version (the term "poset" means the same thing in both versions).

Other Notes:

• I'm thinking a good way to try to construct such a $P$ in general may be via some sort of of subdivision of $C$. But I'm a bit unclear as to when the barycentric subdivision, say, of a category is a poset.

Answer 1

Yes, this is true. It follows form the work of Barwick and Kan on relative categories as a model for $\infty$-categories.

The idea is similar to how Thomason's work shows that every homotopy type can be modeled by a poset (by taking subdivisions of the category of simplicies of the simplicial set).

Specifically in ArXiv:1011.1691 and ArXiv:1101.0772 they construct a Quillen equivalence between the Joyal model structure and the model structure of "Relative Categories". This latter is a model structure on the category of small ordinary categories equipped with a collection of "weak equivalences". Importantly they show that the cofibrant objects in the latter are relative posets - relative categories whose underlying category is a poset.

This means that every $\infty$-category can be modeled by a relative poset.

If you are just interested in ordinary categories, then you don't even need the Quillen equivalence. View the ordinary category $C$ as a relative category with trivial weak equivalences. Then its cofibrant replacement in the Barwick-Kan model structure will be a relative poset $(P,W)$, and it will satisfy $C \simeq P[W^{-1}]$.

Added for clarification

From the comments to the OP is seems that people want to see a bit more about how this works. In particular how can we get a cofibrant replacement? Is it functorial?

In the first paper Barwick and Kan construct an adjunction which they then show is a Quillen equivalence:

$$K_\xi: ssSet \leftrightarrows RelCat: N_\xi$$

Here $N_\xi$ is a sort of nerve functor. Claim: for any relative category $C$, $K_\xi N_\xi(C) \to C$ is a cofibrant replacement of $C$ (hence a relative poset modeling $C$). This is clearly functorial in $C$ by construction.

Proof: $N_\xi(C)$ will automatically be a cofibrant object in bisimplicial sets in the Reedy(=injective) model structure, and so $K_\xi$, being a left Quillen functor, will send it to a cofibrant relative category.

We just need to know that (1) the counit $\epsilon_\xi :K_\xi N_\xi C \to C$ is a weak equivalence in Relative categories.

Weak equivalences in the model category of relative categories are detected by $N_\xi$ (by construction - it is a transferred model structure). Thus (1) will be true if the map $N_\xi\epsilon_\xi: N_\xi K_\xi N_\xi C \to N_\xi C$ is a weak equivalence.

Prop 10.3 in 1011.1691 states that the unit map $\eta_\xi: id \to N_\xi K_\xi$ is always a weak equivalence. Thus $\eta_\xi N_\xi$ is a weak equivalence. This is a left inverse of $N_\xi \epsilon_\xi$, so by two-out-of-three $N_\xi \epsilon_\xi$, and hence $\epsilon_\xi$ are weak equivalences. $\square$

You can also get an easier description using the functor "$N$" rather than the more cumbersome $N_\xi$. See section 7-8 of that same paper.