equivalences in/of $(\infty,1)$-categories
$Pr(\infty,1)Cat$ is the (∞,1)-category of locally presentable (∞,1)-categories and (∞,1)-colimit-preserving (∞,1)-functors between them (Lawvere distributions).
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 $(\infty,1)$-categories is bilinear if it respects colimits in both variables.
It turns out that there is a universal such bilinear functor
which thereby defines a tensor product of presentable (∞,1)-categories. This defines a monoidal structure on presentable $(\infty,1)$-categories, which is in fact symmetric.
The collection $Pr(\infty,1)Cat$ of presentable $(\infty,1)$-cateories with colimit-preserving (∞,1)-functors between them (i.e. with ”linear” functors between them!), is an $(\infty,1)$-generalization of the category $Set Mod$ 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 $Pr(\infty,1)Cat$ with its notion of “linearity” one obtains a very general notion of $\infty$-linear algebra. This is described at geometric ∞-function theory.
Write $Pr(\infty,1)Cat_1$ for the sub-(∞,1)-category of the (∞,1)-category of (∞,1)-categories whose
objects are presentable (∞,1)-categories;
morphisms are (∞,1)-colimit-preserving (∞,1)-functors.
The symmetric monoidal structure on presentable $(\infty,1)$-categories restricts to one on presentable stable (∞,1)-categories.
The tensor unit of stable presentable $(\infty,1)$-categories is the stable (∞,1)-category of spectra.
For $C, D \in Pr(\infty,1)Cat$, let
be the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve all small (∞,1)-colimits.
For all $C,D$ we have that $Func^L(C,D)$ is itself locally presentable.
This is HTT, prop. 5.5.3.8.
All small limits and colimits exists in $Pr(\infty,1)Cat$. The limits are preserved by the embedding $Pr(\infty,1)Cat \hookrightarrow$ (∞,1)Cat.
This is HTT, prop. 5.5.3.13.
Let $C_1, \cdots, C_n$ be a finite collection of locally presentable $(\infty,1)$-categories. There exists a locally presentable $(\infty,1)$-category $C_1 \otimes \cdots \otimes C_n$ and an (∞,1)-functor
(the tensor product) such that
it preserves (∞,1)-colimits in each variable;
for every $D \in Pr(\infty,1)Cat$, composition with $f$ 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 $Pr(\infty,1)Cat$ a symmetric monoidal (∞,1)-category.
In some context it makes good sense to think of $Pr(\infty,1)Cat$ as a model for an $(\infty,1)$-category of ”$\infty$-vector spaces”. More on this is at integral transforms on sheaves.
Here a small $(\infty,1)$-category $S$ is to be thought of as a basis and the locally presentable $(\infty,1)$-category $C \hookrightarrow PSh_{(\infty,1)}(C)$ as the $\infty$-vector space spanned by this basis. The colimits in $C$ play the role of addition of vectors and the fact that morphisms in $Pr(\infty,1)Cat$ 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 $\otimes : Pr(\infty,1)Cat \times Pr(\infty,1)Cat \to Pr(\infty,1)Cat$ plays the role of the tensor product of vector spaces, with a morphism out of $C \otimes D$ being a bilinear morphism out of $C \times D$, and the fact that $Pr(\infty,1)Cat$ is closed monoidal reflects the fact that Vect is closed monoidal.
Combined with the fact that the embedding $Pr(\infty,1)Cat \hookrightarrow (\infty,1)Cat$ preserves limits and colimits, this yields some useful statements.
For instance with $Pr(\infty,1)Cat$ regarded as $\infty Vect$, for any ∞-group $G$ with delooping ∞-groupoid $\mathbf{B}G$, we may think of an (∞,1)-functor $\rho : \mathbf{B}G \to Pr(\infty,1)Cat$ as a linear representation of $G$: the single object of $\mathbf{B}G$ is sent to a presentable $(\infty,1)$-category $V$ and the morphisms in $\mathbf{B}G$ then define an action of $G$ on that.
The corresponding action groupoid $V//G$ (see there for more) is then the colimit over the action, in $Pr(\infty,1)Cat$
The $(\infty,1)$-category $Pr(\infty,1)Cat$ is introduced in section 5.5.3 of
The monoidal structure on $Pr(\infty,1)Cat$ 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.