A homotopical algebra approach to the computation of higher limits

Abstract

We develop a new technique for computing higher limits of functors over filtered posets by constructing explicit fibrant replacements within a suitable model category structure. We apply this procedure to develop two systematic vanishing bounds for the higher limits. For the first one, we define a combinatorial bound derived from the poset's Hasse diagram. This bound systematically outperforms classical poset-length bounds and provides critical stopping criteria for computing the sheaf cohomology of hypergraphs and higher-order interaction networks. The second one is an inductive bound driven by the local algebraic information of the functor. We demonstrate the robustness of this theoretical framework by applying it to two distinct domains: establishing vanishing bounds for Mackey functors over posets with local quasi-units, and bounding the unnormalised Khovanov homology of the torus knot T(3,4).