In symplectic geometry, a Weinstein surface W can be understood by combinatorial data on its skeleton, which is generically a finite trivalent graph G. Motivated by the microlocal theory of sheaves, there is a naturally associated diagram D of differential-graded categories, defined over the quiver which replaces each edge of G by a cospan. We call such quivers ’graphic’. After adding in appropriate homological shifts, the homotopy limit of D yields an invariant of W, so we are motivated to find explicit presentations for these homotopy limits.
Abstracting the story above, we fix an arbitrary graphic quiver Q. Given any model category M (above we had M = dg-categories), we consider M-valued diagrams over Q. I will present a combinatorial characterization of all such diagrams D which are suitably Reedy-fibrant, which in particular implies that the homotopy limit of D is just its classical limit. This yields an explicit formula for the homotopy limit of any diagram, fibrant or not.