부산대학교 도서관

This semi-expository work covers central aspects of the theory of relative tensor products as developed in Higher Algebra, as well as their application to Koszul duality for algebras in monoidal oo-categories. Part of our goal is to expand on the rather condensed account of loc. cit. Along the way, we generalize various aspects of the theory. For instance, given a monoidal oo-category Cc, an oo-category Mm which is left-tensored over Cc, and an algebra A in Cc, we construct an action of A-A-bimodules N in Cc on left A-modules M in Mm by an "external relative tensor product" N \otimes_A M. (Up until now, even the special ("internal") case Cc = Mm appears to have escaped the literature. As an application, we generalize the Koszul duality of loc. cit. to include modules. Our straightforward approach requires that we at this point assume certain compatibilities between tensor products and limits; these assumptions have recently been shown to be unnecessary in work by Brantner, Campos and Nuiten (arXiv:2104.03870).
Comment: Some of the assumptions needed for our approach to Koszul duality for left-modules in (LM-)monoidal oo-categories have been shown to be unnecessary by Brantner, Campos and Nuiten in arXiv:2104.03870. We've reworked the introduction to reflect this fact, and to highlight our foundational work on external relative tensor products