Topological Koszulity for Category Algebras

TL;DR

This paper characterizes the Koszulity of category algebras using topological methods, linking it to nerve properties and introducing functors that preserve Koszulity, with applications to toric varieties.

Contribution

It provides a topological criterion for Koszulity of category algebras and identifies functors that preserve this property, extending results to posets and toric varieties.

Findings

Koszulity is equivalent to the locally bouquet property of the nerve.

Almost discrete fibrations preserve Koszulity.

Classifies when shifted dual collections are strong on toric varieties.

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.