Topological Koszulity for Category Algebras
David Favero, Pouya Layeghi
TL;DR
This work develops a topological perspective on Ext groups and Koszulity for category algebras, showing that for indiscretely based categories, Koszulity of the category algebra is equivalent to the local bouquet property of the nerve. It introduces almost discrete fibrations as Koszulity-preserving functors and uses them to recover Reiner–Stamate-type equivalence relations in posets, linking category-theoretic and combinatorial frameworks. The theory is then applied to homotopy path algebras and stratified spaces, giving topological criteria for Koszulity in Bondal–Thomsen HPAs and Beilinson-type constructions, and connecting to the endomorphism algebras of line bundle collections on toric varieties. Overall, the results yield a unifying topological interpretation of Koszulity, quadraticity, and dual exceptional collection properties in several algebraic-geometry contexts, with explicit criteria involving local Cohen–Macaulay conditions on path posets and open intervals of induced stratifications.
Abstract
We give a topological description of Ext groups between simple representations of categories via a nerve type construction. We use it to show that the Koszulity of indiscretely based category algebras is equivalent to the locally bouquet property of this nerve. We also provide a class of functors which preserve the Koszulity of category algebras called almost discrete fibrations. Specializing from categories to posets, we show that the equivalence relations of V. Reiner and D. Stamate in arXiv:0904.1683 [math.AC] are exactly almost discrete fibrations and recover their results. As an application, we classify when a shifted dual collection to a full strong exceptional collection of line bundles on a toric variety is strong.