nLab Gray tensor product

Idea

The Gray tensor product is a “better” replacement for the cartesian product of strict 2-categories. 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, for any 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.

Definition

Following up on the last comment, $B\otimes C$ can be defined by

$$2Cat(B\otimes C, D) \cong 2Cat(B, Ps(C,D))$$

where $Ps(C,D)$ is the 2-category of 2-functors, pseudonatural transformations, and modifications $C\to D$. In other words, the category $2Cat$ of strict 2-categories and strict 2-functors is a closed symmetric monoidal category, whose tensor product is $\otimes$ and whose internal hom is $Ps(-,-)$.

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

• see also generalized Gray tensor product

References

Theorem 1,4.14 of:

• John W. Gray, Formal category theory: adjointness for 2-categories, Lecture Notes in Mathematics, Vol. 391. Springer-Verlag, Berlin-New York, 1974. xii+282 pp doi:10.1007/BFb0061280 (see also Adjointness for 2-Categories)

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

• Robert Gordon, John Power, Ross Street. Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558, vi+81 pp. doi:10.1090/memo/0558 (AMS bookstore incl. free sample chapter)

• Stephen Lack, A Quillen model structure for 2-categories, K-Theory, 26(2) (2002) pp171-205, (gzipped .ps) (doi:10.1023/A:1020305604826 - requires Portico subscription)

• Stephen Lack, A Quillen model structure for bicategories, K-theory, 33(3) (2004) pp185-197, (gzipped .ps) (doi:10.1007/s10977-004-6757-9 - requires Portico subscription)

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

• 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

The Gray tensor product as the left Kan extension of a tensor product on the full subcategory $Cu$ of $2Cat$ is on page 16 of

• Ross Street, Gray’s tensor product of 2-categories, 22-page handwritten note, (1988) (PDF at Macquarie)

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