nLab Gray tensor product




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\mathbf{2} which has two objects, 0 and 1, one non-identity morphism 010\to 1, and no nonidentity 2-cells. Then the cartesian product 2×2\mathbf{2}\times\mathbf{2} is a commuting square, while the Gray tensor product 22\mathbf{2}\otimes\mathbf{2} is a square which commutes up to isomorphism.

More generally, for any 2-categories CC and DD, a 2-functor C×2DC\times\mathbf{2} \to D consists of two 2-functors CDC\to D and a strict 2-natural transformation between them, while a 2-functor C2DC\otimes\mathbf{2} \to D consists of two 2-functors CDC\to D and a pseudonatural transformation between them.


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

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

where Ps(C,D)Ps(C,D) is the 2-category of 2-functors, pseudonatural transformations, and modifications CDC\to D. In other words, the category 2Cat2Cat 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(,)Ps(-,-).


  • When considered with this monoidal structure, 2Cat2Cat is often called GrayGray. Gray-categories, or categories enriched over GrayGray, are a model for semi-strict 3-categories. Categories enriched over 2Cat2Cat 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.

  • GrayGray 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.)

  • GrayGray is actually a monoidal model category (that is, a model category with a monoidal structure that interacts well with the model structure), which 2Cat2Cat 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 2×22\times 2 is “completely filled in” (i.e. it commutes). There is another “white” tensor product in which the square 222\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.


Theorem 1,4.14 of:

The Gray tensor product as the left Kan extension of a tensor product on the full subcategory CuCu of 2Cat2Cat is on page 16 of

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

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

Last revised on November 29, 2023 at 08:16:44. See the history of this page for a list of all contributions to it.