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 from the cartesian product of two presentable -categories is bilinear if it respects colimits in both variables.
It turns out that there is a universal such bilinear functor
The collection of presentable -cateories with colimit-preserving (∞,1)-functors between them (i.e. with ”linear” functors between them!), is an -generalization of the category 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 with its notion of “linearity” one obtains a very general notion of -linear algebra. This is described at geometric ∞-function theory.
The symmetric monoidal structure on presentable -categories restricts to one on presentable stable (∞,1)-categories.
The tensor unit of stable presentable -categories is the stable (∞,1)-category of spectra.
For , let
For all we have that is itself locally presentable.
This is HTT, prop. 126.96.36.199.
This is HTT, prop. 188.8.131.52.
Let be a finite collection of locally presentable -categories. There exists a locally presentable -category and an (∞,1)-functor
(the tensor product) such that
it preserves (∞,1)-colimits in each variable;
for every , composition with produces an equivalence of (∞,1)-categories
onto the full sub-(∞,1)-category of those functors, that preserves colimits in each argument.
This is (Lurie, NA, theorem 4.1.4)
This tensor product makes a symmetric monoidal (∞,1)-category.
In some context it makes good sense to think of as a model for an -category of ”-vector spaces”. More on this is at integral transforms on sheaves.
Here a small -category is to be thought of as a basis and the locally presentable -category as the -vector space spanned by this basis. The colimits in play the role of addition of vectors and the fact that morphisms in are colimit-presserving means that they play the role of linear maps between vector spaces. This is described also at Lawvere distribution.
The monoidal product plays the role of the tensor product of vector spaces, with a morphism out of being a bilinear morphism out of , and the fact that is closed monoidal reflects the fact that Vect is closed monoidal.
Combined with the fact that the embedding preserves limits and colimits, this yields some useful statements.
For instance with regarded as , for any ∞-group with delooping ∞-groupoid , we may think of an (∞,1)-functor as a linear representation of : the single object of is sent to a presentable -category and the morphisms in then define an action of on that.
The -category is introduced in section 5.5.3 of
The monoidal structure on is described in section 4.1 of
That this is in fact a symmetric monoidal structure is discussed in section 6 of
see proposition 6.14 and remark 6.18.