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Relative tensor products and Koszul duality in monoidal oo-categories

Ishai Dan-Cohen, Asaf Horev

TL;DR

The paper develops a comprehensive framework for relative tensor products in monoidal ∞-categories and extends Koszul duality to modules, explaining the coalgebra structure on the Koszul dual as $\mathbb{1} \otimes_A \mathbb{1}$ and providing functorial characterizations via twisted arrow categories and Morita-type double ∞-categories. It introduces LM- and BM-monoidal structures to organize bimodules and left modules, constructs an LM-bar-cobar-style Koszul duality functor, and analyzes external relative tensor products through generalized tensorizations $\operatorname{Tens}^{\otimes}_{\vec{\nabla}}$ and related operads. The work culminates in a robust LM-monoidal pairing framework that yields left representable duality functors for modules, enabling a conceptual and systematic approach to Koszul duality beyond algebras. Potential applications include motivic contexts and the study of torsors and fundamental groups in derived settings, illustrating the utility of the formalism in highly structured ∞-categorical settings.

Abstract

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).

Relative tensor products and Koszul duality in monoidal oo-categories

TL;DR

The paper develops a comprehensive framework for relative tensor products in monoidal ∞-categories and extends Koszul duality to modules, explaining the coalgebra structure on the Koszul dual as and providing functorial characterizations via twisted arrow categories and Morita-type double ∞-categories. It introduces LM- and BM-monoidal structures to organize bimodules and left modules, constructs an LM-bar-cobar-style Koszul duality functor, and analyzes external relative tensor products through generalized tensorizations and related operads. The work culminates in a robust LM-monoidal pairing framework that yields left representable duality functors for modules, enabling a conceptual and systematic approach to Koszul duality beyond algebras. Potential applications include motivic contexts and the study of torsors and fundamental groups in derived settings, illustrating the utility of the formalism in highly structured ∞-categorical settings.

Abstract

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).
Paper Structure (37 sections, 34 theorems, 466 equations)

This paper contains 37 sections, 34 theorems, 466 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Review: Koszul duality and bimodules
  4. Review: functorial characterization of Koszul duality for algebras
  5. Review: construction of Koszul duality for algebras
  6. Review: comparison with the relative tensor product
  7. LM-variant: functorial characterization of Koszul duals
  8. LM-variant: construction of Koszul duality functors
  9. Review of $\infty$-operads
  10. Restricted double infinity-categories of bimodules in a monoidal infinity-category
  11. Multimodules
  12. The Morita pre-double infinity-category
  13. Review of the bar construction
  14. The Morita double infinitycategory of A-multimodules
  15. Action of endomorphism categories on morphism categories in a double infinity-category
  16. ...and 22 more sections

Key Result

Lemma 4.2.3

Let $q \colon \mathcal{C}^\otimes \to \operatorname{Assoc^\otimes}$ be a monoidal $\infty$-category, and as above. Then $p$ is an inner fibration.

Theorems & Definitions (116)

  • Definition 3.11.2
  • Definition 4.1.1
  • Remark 4.1.3
  • Definition 4.1.4
  • Example 4.1.5
  • Definition 4.2.1
  • Remark 4.2.2
  • Lemma 4.2.3
  • proof
  • Definition 4.2.4
  • ...and 106 more