nLab
symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories

Idea

Recall that a presentable (∞,1)-category is a localization of a (∞,1)-category of (∞,1)-presheaves. In particular it has all small colimits. An (∞,1)-functor $C\times D\to E$ from the cartesian product of two presentable $(\mathrm{\infty }\mathrm{,1})$-categories is bilinear if it respects colimits in both variables.

It turns out that there is a universal such bilinear functor

$$C\times D\to C\otimes D,$$C \times D \to C \otimes D \,,

which thereby defines a tensor product of presentable (∞,1)-categories. This defines a monoidal structure on presentable $(\mathrm{\infty }\mathrm{,1})$-categories, which is in fact symmetric.

The collection of presentable $(\mathrm{\infty }\mathrm{,1})$-cateories with colimit-preserving (∞,1)-functors between them (i.e. with ”linear” functors between them!), called

$${\mathrm{Pr}}^L,$$Pr^L \,,

is an $(\mathrm{\infty }\mathrm{,1})$-generalization of the category $\mathrm{Set}\mathrm{Mod}$ of ordinary categories and bimodules or profunctors, or distributors between them. See distributor and in particular the discussion there about the equivalent reformulation in terms of colimit-preserving functors.

Using ${\mathrm{Pr}}^L$ with its notion of “linearity” one obtains a very general notion of $\mathrm{\infty }$-linear algebra. This is described at geometric ∞-function theory.

Definition

Write $\mathrm{Pr}(\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}_1^L$ for the sub-$(\mathrm{\infty }\mathrm{,1})$-category of the (∞,1)-category of (∞,1)-categories whose

• objects are presentable (∞,1)-categories;

• morphisms are colimit-preserving (∞,1)-functors.

With $C\times D\to C\otimes D$ the tensor product obtainmed as the universal colimit-preserving functor, $\mathrm{Pr}(\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}_1^L$ becomes a symmetric monoidal (∞,1)-category.

Stable version

The symmetric monoidal structure on presentable $(\mathrm{\infty }\mathrm{,1})$-categories restricts to one on presentable stable (∞,1)-categories.

The tensor unit of stable presentable $(\mathrm{\infty }\mathrm{,1})$-categories is the stable (∞,1)-category of spectra.

References

The monoidal structure on $\mathrm{Pr}(\mathrm{\infty }\mathrm{,1}){\mathrm{Cat}}_1^L$ is described in section 4.1 of

• Jacob Lurie, Noncommutative algebra.

That this is in fact a symmetric monoidal structure is discussed in section 6 of

• Jacob Lurie, Commutative algebra.

see proposition 6.14 and remark 6.18.