Relative tensor products and Koszul duality in monoidal oo-categories

Technical Summary: Relative Tensor Products and Koszul Duality in Monoidal $\infty$-Categories

Problem Statement
The paper addresses the foundational theory of relative tensor products and their application to Koszul duality within the setting of monoidal $\infty$-categories. Specifically, it seeks to generalize the Koszul duality framework established by Lurie in Higher Algebra [15] to include modules, not just algebras. While Lurie's construction provides an adjunction between augmented algebras and augmented coalgebras, extending this to an adjunction between pairs of (augmented algebra, module) and (augmented coalgebra, comodule) requires a robust understanding of how bimodules act on modules via external relative tensor products.

A central technical hurdle identified by the authors is the construction of a left action of the monoidal $\infty$-category of $A$-$A$-bimodules on the $\infty$-category of left $A$-modules. Previous literature (including [15]) often assumed the unit object is terminal or relied on blanket assumptions that tensor products commute with all geometric realizations. The authors aim to refine these constructions to handle cases where the unit is not terminal and where compatibility with realizations is restricted to specific classes of algebras (those "bar compatible").

Methodology
The authors employ a direct, constructive approach rooted in the theory of $\infty$-operads and double $\infty$-categories. Their methodology proceeds through several stages:

1. Refinement of Double $\infty$-Categories of Bimodules:
   The paper refines Lurie's construction of the Morita double $\infty$-category. Instead of assuming the monoidal category $\mathcal{C}$ admits all geometric realizations compatible with the tensor product, the authors restrict attention to a class of algebras $\mathcal{A}$ that are "bar compatible." They construct a double $\infty$-category $\text{BMod}_{\mathcal{A}}(\mathcal{C})^{\boxtimes}$ whose fibers over $[n]$ consist of $\mathcal{A}$-multimodules. This allows for the construction of relative tensor products without requiring global existence of realizations.

2. Construction of LM-Monoidal Actions:
   A key technical innovation is the construction of an LM-monoidal $\infty$-category $\text{BLMod}_{\mathcal{A}}(\mathcal{C})^{\otimes}$. This category witnesses a left action of the monoidal $\infty$-category of $A$-$A$-bimodules ($\text{BMod}_{A}^{A}(\mathcal{C})$) on the $\infty$-category of left $A$-modules ($\text{LMod}_{A}(\mathcal{C})$). This is achieved by defining the $\infty$-category of horizontal morphisms in a double $\infty$-category and constructing an action of horizontal endomorphisms on these morphisms. The action is explicitly given by the relative tensor product $N \otimes_A M$.

3. Generalization to BM-Monoidal Categories:
   To handle cases where the unit object is not terminal, the authors work within the more general framework of BM-monoidal $\infty$-categories (bimodules over two potentially different monoidal categories). They construct a generalized $\infty$-operad $\text{Tens}^{\otimes}_{\vec{\nabla}}$ and use it to define "external" relative tensor products. This allows them to treat the twisted arrow category $\text{TwArr}(\mathcal{C})$ as an LM-monoidal category, even when the unit of $\mathcal{C}$ is not terminal, by working with the slice category $\mathcal{C}_{/1}$.

4. Koszul Duality via Representability:
   The core of the Koszul duality construction relies on the concept of "left representable pairings." The authors define an LM-monoidal pairing $\lambda$ and prove that under specific assumptions (existence of bar construction realizations, triviality of certain algebras, and totalization capabilities), the induced pairing on module categories is left representable. This yields a duality functor $D_{\text{Ksz}}$.

Key Contributions and Results

• External Relative Tensor Products: The paper constructs an action of $A$-$A$-bimodules on left $A$-modules via an external relative tensor product $N \otimes_A M$. The authors note that even the "internal" case ($\mathcal{C} = \mathcal{M}$) appears to have been absent from the literature prior to this work.
• Koszul Duality for Modules: The main result (Theorem 9.4.3) establishes an adjunction between the $\infty$-category of augmented modules $\text{LMod}_{\text{aug}}(\mathcal{C})$ and augmented comodules $\text{coLMod}_{\text{aug}}(\mathcal{C})$.
  • The functor $D_{\text{Ksz}}$ maps a pair $(A, X)$ (an augmented algebra and a module) to a pair $(1 \otimes_A 1, 1 \otimes_A X)$.
  • The object $1 \otimes_A 1$ is the Koszul dual coalgebra, and $1 \otimes_A X$ is the underlying object of the dual comodule.
• Relaxation of Assumptions: The construction generalizes Lurie's results by:
  • Allowing the unit object to be non-terminal (by passing to the LM-monoidal category associated with the slice $\mathcal{C}_{/1}$).
  • Requiring compatibility of tensor products with realizations only for a specific class of algebras (bar-compatible algebras) rather than all algebras.
• Twisted Arrow Category Realization: The paper demonstrates that the twisted arrow category $\text{TwArr}(\mathcal{C})$ admits realizations of bar constructions compatible with the tensor product, provided the underlying category does. This validates the use of the twisted arrow category as the setting for the Koszul duality functor.

Significance and Claims
The authors position this work as a "semi-expository" expansion of Lurie's Higher Algebra [15], specifically addressing the "rather condensed account" of relative tensor products and their application to modules.

• Conceptual Clarity: The paper claims to provide a "coherent and conceptually satisfying picture" of the Koszul duality for modules. By constructing the action of bimodules on modules explicitly, the authors aim to demystify the upgrade from algebra duality to module duality.
• Foundational Utility: The authors state that their foundational work on the action of endomorphism categories on morphism categories in double $\infty$-categories is of independent interest.
• Modesty on Generality: The authors acknowledge that their "straightforward approach" requires assumptions regarding the compatibility of tensor products with limits. They note that subsequent work by Brantner–Campos–Nuiten [4] has shown these assumptions to be unnecessary in a more indirect approach. However, the authors maintain that their direct construction offers a clearer conceptual framework for the specific case of modules, even if it is less general in its technical assumptions.
• Application to Motives: The paper briefly illustrates the potential application of these results to the study of commutative algebras in motives, specifically relating the Koszul dual of an augmented algebra to the unipotent fundamental group and torsors of paths in a scheme. This is presented as a direction for future work rather than a completed application.

In summary, the paper provides a rigorous, step-by-step construction of Koszul duality for modules in monoidal $\infty$-categories, filling a gap in the literature regarding the interaction between bimodules and modules, and refining the necessary technical assumptions to accommodate non-terminal unit objects.