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 which has two objects, 0 and 1, one non-identity morphism , and no nonidentity 2-cells. Then the cartesian product is a commuting square, while the Gray tensor product is a square which commutes up to isomorphism.
More generally, for any 2-categories and , a 2-functor consists of two 2-functors and a strict 2-natural transformation between them, while a 2-functor consists of two 2-functors and a pseudonatural transformation between them.
Following up on the last comment, can be defined by
where is the 2-category of 2-functors, pseudonatural transformations, and modifications . In other words, the category 2Cat of strict 2-categories and strict 2-functors is a closed symmetric monoidal category, whose tensor product is and whose internal hom is .
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.
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 is “completely filled in” (i.e. it commutes). There is another “white” tensor product in which the square 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.
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 -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