# On the barycentric subdivision of a poset

Hi everybody,

I'm interested in the barycentric subdivision of a poset $P$, defined as the face poset of the corresponding order complex $\mathcal F(\Delta(P))$ (or alternatively, the poset of chains in $P$ ordered by inclusion). Actually, I'm interested in the subdivision of small categories, but I'm also happy understanding the poset case.

I was wondering if the functor $sd:\mbox{Poset} \to \mbox{Poset}$ has a right adjoint. I am aware of the Ex functor, which is right adjoint to the corresponding functor of simplicial sets. Can one define a similar object for posets?

This functor has no right adjoint, since it doesn't preserve colimits. Let $[m]$ denote the linearly ordered set $\{0,\ldots,m\}$ and consider the pushout of $\overset{0}{\to}$ and $\overset{1}{\to}$, which can be identified with $$. Now $\mathrm{sd}$ is the poset of chains in $$, but the pushout of $\mathrm{sd}\leftarrow\mathrm{sd}\to\mathrm{sd}$ has only five elements, in fact it can be identified with the subset of $\mathrm{sd}$ consisting of chains $\{0\},\{1\},\{2\},\{0,1\},\{1,2\}$.