[Preprint] An homotopical algebra approach to the computation of higher limits.

We introduce a new methodology for computing higher limits of functors over filtered posets by leveraging explicit fibrant replacements within a suitable model category structure. We apply this new 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 multi-way interaction networks. The second one is an inductive bound driven by the local bounds for the vanishing of the higher limits 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 unnormalized Khovanov homology of links, exhibiting explicit boundary computations for the torus knot (T(3,4)).

The SageMath code for the labeling function can be found in github