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Tim Campion
Tim Campion
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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.

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).

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.
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Tim Campion
Tim Campion
  • 63.9k
  • 13
  • 143
  • 384

Is every category a localization of a poset?

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).