nLab Crans-Gray tensor product

Idea

The Crans-Gray tensor product is a tensor product on the category of strict omega-categories which is analogous to (the lax version of) the Gray tensor product for 2-categories.

This tensor product makes the category $Str\omega Cat$ of strict omega-categories into a (non-symmetric) biclosed monoidal structure; thus in particular it has two internal-homs $Hom_r$ and $Hom_l$. Both contain strict $\omega$-functors as their objects, and their $k$-cells for $k\gt 0$ are k-transfors which are lax or oplax in all dimensions (one internal-hom contains lax transfors and the other the oplax ones).

Construction from cubical sets

One abstract way to construct the Crans-Gray tensor product is as follows.

There is an obvious monoidal structure on the cube category obtained from the obvious product of cellular cubes which is analogous to the cartesian product of the topological cubes $[0,1]^n$. By Day convolution this naturally induces a monoidal structure on cubical sets.

Now strict $\omega$-categories, although usually defined as globular sets with extra structure, can also be regarded as cubical sets with extra structure. More precisely, there is a forgetful functor from $Str\omega Cat$ to $CubSet$ which is monadic, and the monad is a monoidal monad. It follows by further general results of Day that the tensor product on $CubSet$ lifts to $Str\omega Cat$.

It is also possible to give a direct construction of this tensor product; see Crans’ thesis below.

Relation to the Gray tensor product

Strict $n$-categories for $n\lt\omega$ can of course be regarded as strict $\omega$-categories with all $k$-cells identities for $k\gt n$. In general, the Crans-Gray tensor product of an $n$-category with an $m$-category will be an $(n+m)$-category, since the tensor product of the $n$-cube and the $m$-cube is the $(n+m)$-cube. (Compare, for example, the fact that the Gray tensor product of two 1-categories is no longer, in general, a 1-category.)

However, we can obtain a tensor product on strict $n$-categories for any $n$ by applying the reflector (or “cotruncation”) functor for the inclusion of strict $n$-categories into strict $\omega$-categories. When $n=1$, this produces the cartesian product of categories, while when $n=2$ it reproduces the lax version of the Gray tensor product.

Sjoerd Crans has also defined a tensor product of Gray-categories, but the relationship between it and the tensor product of strict $\omega$-categories is not entirely clear, since neither $Str\omega Cat$ nor $GrayCat$ contains the other (their “intersection” is $Str3Cat$).

Properties

The Crans-Gray tensor product makes $Str\omega Cat$ into a biclosed monoidal category. So in particular for $X, Y \in Str\omega Cat$ two strict $\omega$-categories, there is an $\omega$-functor category between them defined by

$$[X,Y] = Str\omega Cat( X \otimes G^\bullet, Y) \,,$$

where $\otimes$ is the CG-tensor product and $G^\bullet : G \to \omega Cat$ is the globular object of standard globes. The objects in $[X,Y]$ are the (strict) $\omega$-functors, while the k-morphisms are a lax sort of $k$-transfors between these. A dual $\omega$-functor category can be defined by $\langle X,Y\rangle = Str\omega Cat(G^{\bullet}\otimes X, Y)$; this has the same objects but its k-morphisms are oplax $k$-transfors.

Or maybe lax and oplax should be switched here? Can someone verify?

Further Remarks

• The Crans-Gray tensor product extends the tensor product on strict $\omega$-groupoids given by R. Brown and P.J. Higgins (see below); they construct a symmetric monoidal closed structure on these $\omega$-groupoids. It is used there to construct a monoidal closed structure on the category of crossed complexes.

• The article Al-Agl/Brown/Steiner sets up an equivalence between strict globular $\omega$-categories and certain cubical $\omega$-categories with connections, to which the monoidal closed structure in the previous paper is easily extended.

• The article Steiner gives a new construction of this tensor product in terms of augmented directed chain complexes, where he shows that it can be defined explicitly and computed efficiently on all strict ω-categories admitting a certain kind of loop-freeness assumption on their bases. In particular, it generalizes the approaches in all earlier papers.

• There is the Verity-Gray tensor product on stratified simplicial sets (as described there). Via the omega-nerve strict omega-categories form the complicial sets within all stratified simplicial sets. According to observation 60 of Verity06 section 11.4 of Verity04 proves that the Crans-Gray tensor product is the reflection of the Verity-Gray tensor product under this inclusion.

References

A detailed discussion is in

• Sjoerd Crans, Pasting schemes for the monoidal biclosed structure on $\omega$-Cat (ftp, gzipped PostScript)

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

• 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. ArXiv

• R. Brown, P.J.Higgins, R. Sivera, Nonabelian algebraic topology: filtered spaces, crossed complexes, cubical homotopy groupoids, EMS Tracts in Mathematics vol 15, 650 pages (autumn 2010 to appear). (Nonabelian algebraic topology)

Some helpful remarks and diagrams are in

• Sjoerd Crans, A tensor product for $Gray$-categories (tac)

which is however mainly concerned with a slightly different topic.

A collection of Crans’s papers, including those on teisi, can be found here:

• Papers of Sjoerd Crans (the website seems down at the moment, but there is an archived version)

The modern version, incorporating many new primitives is given in

• Richard Steiner, Omega-categories and chain complexes, Homology, Homotopy and Applications 6(1) (2004), 175-200 (journal, pdf)

A general theory of lax tensor products is in

• Michael Batanin, Denis-Charles Cisinski, Mark Weber, Multitensor lifting and strictly unital higher category theory (arXiv:1209.2776)