nLab Gray-category

Idea

A $Gray$-category (or Gray category) is a certain type of semi-strict 3-category, in which composition is strictly associative and unital, but the interchange law holds only up to coherent isomorphism.

Definition

A $Gray$-category is a category enriched over the symmetric monoidal category $Gray$, which is the category of 2-categories and strict 2-functors with the Gray tensor product.

Properties

Coherence theorem

Gordon, Power, and Street proved that every tricategory (that is, weak 3-category) is equivalent to a $Gray$-category. Not every tricategory is equivalent to a fully strict 3-category; any doubly-degenerate braided monoidal category which is not symmetric is an example. So this is “the best one can do” in one sense, although there are other incomparable paths one can take, such as weakening units but keeping interchange strict.

The inclusion of $Gray$-categories into tricategories is not uniquely determined – there is a left and right-hand version (from a remark in Example 9.3.9 of Leinster’s book cited below). However, the two possible ways are canonically equivalent as tricategories.

Canonical model structure

Gray-categories support a canonical model structure (Lack)

$Gray$-groupoids

A $Gray$-category that is a 3-groupoid is a Gray-groupoid.

Examples

• The prototypical Gray-category is Gray, which consists of strict 2-categories, strict 2-functors, pseudonatural transformations, and modifications.

• A Gray-category with one object is called a Gray-monoid, and is a semi-strict version of a monoidal bicategory.

• A doubly-degenerate Gray-category is the same as a category with two monoidal structures satisfying an exchange law. This is essentially the same as a braided monoidal category. (Gurski & Cheng)

• Since any tricategory is equivalent to a Gray-category, one can obtain examples of Gray-categories in this way. For example, the tricategory Bicat of bicategories, pseudofunctors, pseudonatural transformations, and modifications is equivalent to some Gray-category.

  It is important to note that Bicat is not equivalent to Gray, due to the absence of pseudofunctors in the latter. It is equivalent to the sub-Gray-category of Gray determined by the “flexible” or “cofibrant” 2-categories, however, since between such 2-categories any pseudofunctor is equivalent to a strict one.

• Since pseudofunctors between strict 2-categories compose strictly associatively, and between any 2-categories $A$ and $B$ there is a strict 2-category $Ps(A,B)$ of pseudofunctors, pseudonatural transformations, and modifications, one might hope that there is a Gray-category consisting of strict 2-categories, pseudofunctors, pseudonatural transformations, and modifications, despite the fact that the prototypical example $Gray$ contains only strict 2-functors. However, this is false, because in a Gray-category the whiskering of 2-cells by a 1-cell is strictly functorial relative to composition of 2-cells along 1-cells, but this fails for whiskering of pseudonaturals by a pseudofunctor.

Related pages

• intercategory

References

• Robert Gordon, John Power, Ross Street, Coherence for tricategories, Memoirs of the American Mathematical Society, 117, 1995. (doi:10.1090/memo/0558)

• Nick Gurski, An algebraic theory of tricategories, PhD thesis (2007) [pdf, pdf]

• Tom Leinster, Higher operads, higher categories, Cambridge University Press, 2004. (arXiv:math/0305049,

  doi:10.1017/CBO9780511525896)

• Steve Lack, A Quillen model structure for Gray-categories, Journal of K-Theory, 8, 2011. (arxiv:1001.2366, doi:10.1017/is010008014jkt127)

• Nick Gurski, Eugenia Cheng, The periodic table of n-categories II: degenerate tricategories, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 52, 2011. (link, arxiv:0706.2307)

• Peter Guthmann, The tricategory of formal composites and its strictification, arXiv:1903.05777

• Nicola Di Vittorio, A Gray-categorical pasting theorem, Theory and Applications of Categories 39 5 (2023) 150-171 [tac:39-05]