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 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 March 19, 2018 at 02:37:44. See the history of this page for a list of all contributions to it.