On the barycentric subdivision of a poset

Question

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?

Answer 1

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 $[0]\overset{0}{\to}[1]$ and $[0]\overset{1}{\to}[1]$, which can be identified with $[2]$. Now $\mathrm{sd}[2]$ is the poset of chains in $[2]$, but the pushout of $\mathrm{sd}[1]\leftarrow\mathrm{sd}[0]\to\mathrm{sd}[1]$ has only five elements, in fact it can be identified with the subset of $\mathrm{sd}[2]$ consisting of chains $\{0\},\{1\},\{2\},\{0,1\},\{1,2\}$.