Koszul Monoids in Quasi-abelian Categories

Suppose that we have a bicomplete closed symmetric monoidal quasi-abelian category \(\mathcal {E}\) with enough flat projectives, such as the category of complete bornological spaces \({{\textbf {CBorn}}}_k\) or the category of inductive limits of Banach spaces \({{\textbf {IndBan}}}_k\). Working with monoids in \(\mathcal {E}\), we can generalise and extend the Koszul duality theory of Beilinson, Ginzburg, Soergel. We use an element-free approach to define the notions of Koszul monoids, and quadratic monoids and their duals. Schneiders’ embedding of a quasi-abelian category into an abelian category, its left heart, allows us to prove an equivalence of certain subcategories of the derived categories of graded modules over Koszul monoids and their duals.