**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.



Is every category a localization of a poset?
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](https://arxiv.org/abs/1011.1691) and [ArXiv:1101.0772](https://arxiv.org/abs/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.


