[Paper Review] Gray tensor products and lax functors of $(\infty,2)$-categories

Andrea Gagna, Yonatan Harpaz+1 more|arXiv (Cornell University)|Jun 25, 2020

Homotopy and Cohomology in Algebraic Topology8 references8 citations

TL;DR

This paper introduces a well-behaved Gray tensor product for $(\infty,2)$-categories using scaled simplicial sets, proving it is associative and a left Quillen bifunctor. It characterizes the product via a universal property using oplax functors, providing a foundational step toward comparing with Gaitsgory-Rozenblyum's construction and enabling homotopy-theoretic applications in higher category theory.

ABSTRACT

We give a definition of the Gray tensor product in the setting of scaled simplicial sets which is associative and forms a left Quillen bifunctor with respect to the bicategorical model category of Lurie. We then introduce a notion of oplax functor in this setting, and use it in order to characterize the Gray tensor product by means of a universal property. A similar characterization was used by Gaitsgory and Rozenblyum in their definition of the Gray product, thus giving a promising lead for comparing the two settings.

Motivation & Objective

• To define a well-behaved Gray tensor product for $(\infty,2)$-categories in the setting of scaled simplicial sets.
• To establish that this product is associative and forms a left Quillen bifunctor with respect to Lurie's bicategorical model structure.
• To introduce a notion of oplax functors in the context of $(\infty,2)$-categories and use it to characterize the Gray product via a universal property.
• To provide a pathway for comparing the authors' construction with Gaitsgory-Rozenblyum's Gray product by aligning on the universal property of oplax functors.

Proposed method

• The authors work within the category of scaled simplicial sets equipped with Lurie's bicategorical model structure.
• They define the Gray tensor product as a bifunctor that is associative on-the-nose and preserves cofibrations and weak equivalences in each variable.
• They introduce oplax functors between $(\infty,2)$-categories as maps that preserve thin triangles in a controlled way, generalizing 2-categorical oplax functors.
• They construct a universal property by showing that the functor of oplax functors from $\mathcal{C} \times \mathcal{D}$ to $\mathcal{E}$ is equivalent to the functor of functors from $\mathcal{C} \otimes \mathcal{D}$ to $\mathcal{E}$, under specific conditions on the image of 2-simplices.
• They use a saturated closure argument involving a 4-simplex $\mathfrak{P}$ and its faces to show that all thin triangles in the target are captured by the image of the oplax functors.
• They establish that the restriction functor from $\mathrm{Fun}(\mathcal{C} \otimes \mathcal{D}, \mathcal{E})$ to $\mathrm{Fun}_{\mathrm{oplax}}(\mathcal{C} \times \mathcal{D}, \mathcal{E})$ is fully faithful with a specific essential image condition.

Experimental results

Research questions

• RQ1 How can a Gray tensor product for $(\infty,2)$-categories be defined in the scaled simplicial set model so that it is associative and respects the bicategorical model structure?
• RQ2 Can the Gray tensor product be characterized by a universal property involving oplax functors in the context of $(\infty,2)$-categories?
• RQ3 To what extent does the proposed construction align with the Gray product used by Gaitsgory and Rozenblyum in their work on higher category theory?
• RQ4 What conditions on oplax functors ensure that they factor through the Gray tensor product?
• RQ5 Is the universal property of the Gray product equivalent to that of the construction in Gaitsgory-Rozenblyum’s framework?

Key findings

• The Gray tensor product defined on scaled simplicial sets is associative on-the-nose and forms a left Quillen bifunctor with respect to Lurie’s bicategorical model structure.
• The restriction map $\mathrm{Fun}(\mathcal{C} \otimes \mathcal{D}, \mathcal{E}) \to \mathrm{Fun}_{\mathrm{oplax}}(\mathcal{C} \times \mathcal{D}, \mathcal{E})$ is fully faithful, with the essential image consisting of oplax functors satisfying two conditions: preservation of maps on slices and mapping of specific 2-simplices to thin triangles.
• The proof relies on showing that every thin 2-simplex in the target is in the saturated closure of a specific set $G_{\mathcal{C} \times \mathcal{D}}$, using a 4-simplex $\mathfrak{P}$ and its face decompositions.
• The construction provides a universal property that mirrors the one used by Gaitsgory and Rozenblyum, thus paving the way for a comparison between their Gray product and the authors’ construction.
• The result implies that the underlying $(\infty,2)$-category of $(\infty,2)$-categories carries a presentably monoidal structure via the Gray product.
• The authors establish that their Gray product is equivalent to the one defined via stratified sets in Verity’s framework, via the known equivalence between the models.