nLab Gray tensor product

Idea

The Gray tensor product is a “better” replacement for the cartesian product of strict 2-categories with respect to pseudonatural transformations. To get the idea it suffices to consider the 2-category $\mathbf{2}$ which has two objects, 0 and 1, one non-identity morphism $0\to 1$, and no nonidentity 2-cells. Then the cartesian product $\mathbf{2}\times\mathbf{2}$ is a commuting square, while the Gray tensor product $\mathbf{2}\otimes\mathbf{2}$ is a square which commutes up to isomorphism.

More generally, given 2-categories $C$ and $D$, a 2-functor $C\times\mathbf{2} \to D$ consists of two 2-functors $C\to D$ and a strict 2-natural transformation between them, while a 2-functor $C\otimes\mathbf{2} \to D$ consists of two 2-functors $C\to D$ and a pseudonatural transformation between them.

Preliminaries

Throughout, write

(1)$$2Cat \in CAT$$

for the very large category whose

• objects are strict 2-categories

• morphisms are strict 2-functors.

For $X, Y \in 2Cat$, we write

$$Hom_{2Cat}(X,Y) \in Set$$

for the hom set of this 2-category, hence for the set of strict 2-functors $X \to Y$.

Of course, there is not just a set but a strict 2-category of such functors. Specifically, we write

(2)$$Ps(Y,Z) \in 2Cat$$

for the strict 2-functor 2-category whose

• objects are strict 2-functors $Y \to Z$,

• 1-morphisms are pseudonatural transformations between these,

• 2-morphisms are modifications between those,

with composition operations the canonical horizontal composition and vertical composition of this data.

(cf. Johnson & Yau 2020, Ntn. 12.2.24)

The point of the Gray tensor product may be regarded as bringing out the enrichment of $2Cat$ by these hom 2-categories $Ps(-,-)$ (2):

Definition

Definition

The Gray tensor product is the functor

(3)$$\otimes_{Gray} \;\colon\; 2Cat \times 2Cat \longrightarrow 2Cat$$

given by (…).

(cf. Johnson & Yau 2020, Def. 12.2.5)

Properties

Hom-adjunction and closure

First of all:

Proposition

The Gray tensor product (3) makes $2Cat$ (1) into a symmetric monoidal category

(4)$$\big( 2Cat, \otimes_{Gray}, \ast \big) \in MonCAT \,.$$

(cf. Johnson & Yau 2020, Thm. 12.2.23)

As such, the characteristic property of the Gray tensor product is that:

Proposition

There are natural bijections

(5)$$Hom_{2Cat}(X \otimes_{Gray} Y, Z) \simeq Hom_{2Cat}\big(X, Ps(Y,Z)\big) \,,$$

natural in $X, Y, Z \in 2 Cat$.

(cf. Johnson & Yau 2020, Prop. 12.2.27 with Thm. 12.2.20)

Since (5) are the hom isomorphism for an internal hom-adjunction

$$\big( (-) \otimes_{Gray} Y \big) \;\dashv\; Ps(Y,-) \,,$$

Prop. says that the symmetric monoidal category $(2Cat, \otimes_{Gray}, \ast)$ (4) is in a fact closed, hence is a symmetric closed monoidal category (with tensor product being the Gray tensor and) with internal hom given by $Ps(-,-)$ (2).

Further properties

Proposition

For $X, Y \in 2Cat$, the canonical comparison 2-functor from their Gray tensor product to their Cartesian product is an equivalence of 2-categories:

(6)$$X \otimes_Gray Y \xrightarrow{\phantom{--} \sim \phantom{--}} X \times Y \,.$$

(cf. Lack 2002, top of p. 7)

Proposition

Under passage to simplicial nerves of strict 2-groupoids

(7)$$N \;\colon\; 2Grpd \longrightarrow sSet \mathrlap{\,,}$$

1. the nerve of a Gray tensor product of 2-groupoids,

2. the Cartesian product (cf. product of simplices) of the nerves of the separate 2-groupoids

are related by a simplicial weak equivalence, in fact by an acyclic Kan fibration:

$$\mathllap{ X, Y \in 2Grpd \;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\; } N\big( X \otimes_{Gray} Y \big) \xrightarrow{ \in Fib \cap W } N(X) \times N(Y) \mathrlap{\,.}$$

Proof

The comparison map (6) is indeed an acyclic fibration (Lack 2002, top of p. 7) in the canonical model structure on 2-groupoids, and on that the nerve (7) is a right Quillen functor to the classical model structure on simplicial sets (Lack 2002, end of p. 29, following Moerdijk & Svensson 1993).

Remark

Related to Prop. : There is a natural comparison map

$$N\big( Ps(X,Y) \big) \longrightarrow N(Y)^{N(X)}$$

(from the simplicial nerve of (2) to the simplicial function complex between the separate nerves of the arguments) and this is a simplicial weak equivalence when $X$ is cofibrant in the canonical model structure on 2-groupoids.

This is due to Alexander Campbell, see discussion here.

Remarks

• When considered with this monoidal structure, $2Cat$ is often called $Gray$. Gray-categories, or categories enriched over $Gray$, are a model for semi-strict 3-categories. Categories enriched over $2Cat$ with its cartesian product are strict 3-categories, which are not as useful. This is one precise sense in which the Gray tensor product is “more correct” than the cartesian product.

• $Gray$ is an example of a semicartesian monoidal category, i.e. a non-cartesian monoidal category whose unit object is nevertheless the terminal object.

• There are also versions of the Gray tensor product in which pseudonatural transformations are replaced by lax or oplax ones. (In fact, these were the ones originally defined by Gray.)

• $Gray$ is actually a monoidal model category (that is, a model category with a monoidal structure that interacts well with the model structure), which $2Cat$ with the cartesian product is not. In particular, the cartesian product of two cofibrant 2-categories need not be cofibrant. This is another precise sense in which the Gray tensor product is “more correct” than the cartesian product.

• The cartesian monoidal structure is sometimes called the “black” product, since the square $2\times 2$ is “completely filled in” (i.e. it commutes). There is another “white” tensor product in which the square $2\Box 2$ is “not filled in at all” (doesn’t commute at all), and the “gray” tensor product is in between the two (the square commutes up to an isomorphism). This is a pun on the name of John Gray, for whom the Gray tensor product is named. The “white” tensor product is also called the funny tensor product.

• There are generalizations to higher categories of the Gray tensor product. In particular there is a tensor product on strict omega-categories – the Crans-Gray tensor product – which is such that restricted to strict 2-categories it reproduces the Gray tensor product.

• A closed monoidal structure on strict omega-categories is introduced by Al-Agl, Brown and Steiner. This uses an equivalence between the categories of strict (globular) omega categories and of strict cubical omega categories with connections; the construction of the closed monoidal structure on the latter category is direct and generalises that for strict cubical omega groupoids with connections established by Brown and Higgins.

• The Gray tensor product cannot be extended to a 2-functor (or 3-functor) on the 2-category of 2-categories, 2-functors, 2-natural transformations, and modifications. See this MathOverflow answer.

Related entries

• Crans-Gray tensor product

• generalized Gray tensor product

• Verity-Gray tensor product

References

In 2-category theory

Original discussion of the Gray tensor product in 2-category theory:

• John W. Gray, Theorem 1, 4.14 of: Formal category theory: Adjointness for 2-categories, Lecture Notes in Mathematics 391, Springer (1974) [doi:10.1007/BFb0061280]

• John W. Gray, Coherence for the Tensor Product of 2-Categories, and Braid Groups, in: Algebra, Topology, and Category Theory, Academic Press (1976) 62-76

In the context of the canonical model structure on 2-categories:

• Stephen Lack: A Quillen model structure for 2-categories, K-Theory 26 2 (2002) 171-205 [doi:10.1023/A:1020305604826, pdf, zbmath]

Further discussion:

• Ronnie Brown, Philip Higgins: Tensor products and homotopies for $\omega$-groupoids and crossed complexes,J. Pure Appl. Alg. 47 (1987) 1-33 lbrack;doi:10.1016/0022-4049(87)90099-5, (pdf]

• Robert Gordon, John Power, Ross Street: Coherence for tricategories, Mem. Amer. Math. Soc. 117 558 (1995) [doi:10.1090/memo/0558, ams:memo-117-558]

• F.A. Al-Agl, R. Brown and R. Steiner, Multiple categories: the equivalence between a globular and cubical approach, Advances in Mathematics, 170 (2002) 71-118. doi:10.1006/aima.2001.2069, arXiv:math/0007009

Comprehensive review:

• Niles Johnson, Donald Yau, section 12.2 of: 2-Dimensional Categories, Oxford University Press (2021) [arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001]

On the Gray tensor product as the left Kan extension of a tensor product on the full subcategory $Cu$ of $2Cat$:

• Ross Street, p. 16 of: Gray’s tensor product of 2-categories, 22-page handwritten note, (1988) [pdf]

A general theory of lax tensor products, unifying Gray tensor products with the Crans-Gray tensor product is in

• Michael Batanin, Denis-Charles Cisinski, Mark Weber, Multitensor lifting and strictly unital higher category theory (arXiv:1209.2776)

A proof that the Gray tensor product does form a monoidal structure, based only on its universal property, is in

• John Bourke, Nick Gurski, The Gray tensor product via factorisation, Appl Categor Struct 25 (2017) p603-624, doi:10.1007/s10485-016-9467-6, arXiv:1508.07789

On its generalization to Gray-categories

• Sjoerd Crans. “A tensor product for $Gray$-categories”. Theory and applications of categories, 5(2), 12-69. (tac) (pdf)

In enhanced 2-category theory

For enhanced 2-categories:

• Nathanael Arkor, John Bourke, Joanna Ko: Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation [arXiv:2412.07475]

In $(\infty,2)$-category theory

Discussion of the Gray tensor product in $(\infty,2)$-category theory:

• Dennis Gaitsgory, Nick Rozenblyum: Lax functors and the Gray product, Chapter 10.3 in the Appendix of: A study in derived algebraic geometry, Mathematical Surveys and Monographs 221, Americal Mathematical Society (2017) [ams:SURV/221, book webpage]

• Félix Loubaton, Jaco Ruit: On the squares functor and the Gaitsgory-Rozenblyum conjectures [arXiv:2507.07807]

• Félix Loubaton, answer to: The missing proofs in Gaitsgory–Rozenblyum’s derived algebraic geometry book (Jul 2025) [MO:a/497541]