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?