arXiv subjects

Guille Carrión Santiago

Publications and source records attributed to Guille Carrión Santiago.

4 recordsLinked to original sources

arXiv2026

Cohomology of non-finite CL-shellable posets

Shellable complexes are homotopy equivalent to a wedge of spheres of possibly different dimensions, so that the (co)homology of the constant functor over the complex is concentrated in those degrees. In this work, we introduce the concept of a stable functor -a local weakening of fibrancy- over a shellable poset, which ensures the vanishing of the (co)homology of such a functor in specific degrees. The methods are based on a model category structure on the category of functors indexed by a filtered poset and the combinatorial structure of shellable posets. Our techniques work over non-finite and non-pure posets and employ a description of (co)homology via explicit fibrant replacements. Applications include acyclicity criteria for Mackey functors, computation of cohomology of $j$-th exterior powers over arrangement lattices, and homological decompositions for Bianchi groups $Γ_d$ for $d=1,2$.

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arXiv2026

A homotopical algebra approach to the computation of higher limits

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).

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arXiv2025

The plus construction with respect to subrings of the rationals

We construct explicit models of universal $H \mathbb{Z}[J^{-1}]$-acyclic spaces $\mathcal M$, for any subset $J$ of the prime numbers. The corresponding nullification functors provide thus plus construction functors for ordinary homology with $\mathbb{Z}[J^{-1}]$ coefficients. Motivated by classical results about Quillen's plus construction for integral homology, we prove that the $H \mathbb{Z}[J^{-1}]$-acyclization functor and the $\mathcal M$-cellularization functor coincide. We show that the acyclization-plus construction fiber sequence is always a cofiber sequence for simply connected spaces, but almost never so when the plus construction is not simply connected, unlike in the classical case.

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arXiv2024

Mackey functors for posets

We characterize cofibrant objects in the category of functors indexed in a filtered poset and we show that these objects are acyclic. As a consequence, we show that Mackey functors over posets are also acyclic, where we define this type of Mackey functors mimicking the classical notion. As application, we study homotopy colimits over posets and we give a homology decomposition for the classifying space of the Bianchi group $Γ_1$.

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