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 $2$ which has two objects, 0 and 1, one non-identity morphism $0\to 1$, and no nonidentity 2-cells. Then the cartesian product $2\times 2$ is a commuting square, while the Gray tensor product $2\otimes 2$ is a square which commutes up to isomorphism.

More generally, for any 2-categories $C$ and $D$, a 2-functor $C\times 2\to D$ consists of two 2-functors $C\to D$ and a strict 2-natural transformation between them, while a 2-functor $C\otimes 2\to D$ 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

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

where $\mathrm{Ps}(C,D)$ 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 $\mathrm{Ps}(-,-)$.

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 a rare example of a non-cartesian monoidal category whose unit object is nevertheless the terminal object — that is, a semicartesian monoidal category.

• 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 $2\square 2$ 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 who gave his name to the Gray 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.

Related entries

• see also generalized Gray tensor product

References

• John W. Gray, Formal category theory: adjointness for 2-categories

• Gordon, Power, Street. Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558, vi+81 pp.

• Stephen Lack, A Quillen model structure for 2-categories and A Quillen model structure for bicategories.

• R. Brown and P.J. Higgins, Tensor products and homotopies for $\omega$-groupoids and crossed complexes, J. Pure Appl. Alg. 47 (1987) 1-33.

• 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.

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)