Enriched Koszul duality

Eurenius, Björn and Lazarev, Andrey (2024) Enriched Koszul duality. PhD thesis, Lancaster University.

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Abstract

We investigate the monoidal and enriched category properties of Koszul duality between the category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras. We find that the category of non-counital conilpotent dg-coalgebras has a non-unital monoidal structure compatible with its standard model structure. We then show that the category of non-unital dg-algebras carries a non-unital module category structure, over the category of non-counital conilpotent dg-coalgebras, compatible with its standard model structure. Furthermore we show that the Quillen equivalence between these two model categories extends to a non-unital module category Quillen equivalence. We also show the analogous results in the case of Koszul duality between the category of non-counital cocommutative conilpotent dg-coalgebras and the category of dg-Lie algebras. Thus we establish what we call an enriched form of Koszul duality. We then proceed to show that the homotopy category of non-counital conilpotent dg-coalgebras and the category of non-unital dg-algebras inherit a semi-module structure over the homotopy category of reduced simplicial sets with the Quillen model structure. We also consider how our results can be used to possibly compute simplicial mapping spaces of non-unital dg-algebras and dg-Lie algebras and reach some partial results in this direction.