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.
A functor between locally presentable $(\infty,1)$-categories:
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 (equivalently, are left adjoints).
Evidently $Func^L(C,D)$ would be the hom-$(\infty,1)$-category of an enhancement of $Pr(\infty,1)Cat$ to an $(\infty,2)$-category; its maximal sub-$\infty$-groupoid is the hom-$\infty$-groupoid $Pr(\infty,1)Cat(C,D)$.
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. Indeed it is even closed, since the hom-objects $Func^L$ supply adjoints to the tensor product.
The definition of the tensor product is easy: $C \otimes D = Cts(C^{op},D)$, where $Cts$ denotes the category of continuous (small-limit-preserving) functors. Equivalently, this is $Cocts(C,D^{op})^{op}$, where $Cocts$ denotes the category of cocontinuous (small-colimit-preserving) functors; it is tempting to write this as $Func^L(C,D^{op})^{op}$, but note that $D^{op}$ is not itself locally presentable. However, $D^{op}$ is nevertheless cocomplete, so any cocontinuous functor $C \to D^{op}$ has a right adjoint.
To show that $Cts(C^{op},D)$ is locally presentable, recall that $C \simeq Ind_\kappa A$ for some small $\kappa$-cocomplete $A$. Thus $Cts(C^{op},D) \simeq \kappa Cts (A^{op},D)$, where $\kappa Cts$ denotes the category of $\kappa$-limit-preserving functors; this is an accessible localization of $Fun(A^{op},D)$ and hence locally presentable.
To show that $Cts(C^{op},D)$ has the right universal property, we compute
To justify the step $(*)$, note that both sides are equivalent to some subcategory of functors $A^{op} \times E \to D$, and in both cases part of the requirement is that $\kappa$-limits be preserved in the $A^{op}$-variable and all small limits be preserved in the $E$-variable. The remaining condition on the left is that for each $a\in A$ the corresponding functor $E\to D$ is accessible, i.e. preserves $\lambda_a$-filtered colimits for some cardinal $\lambda_a$. On the right, the remaining condition is that the curried functor $E\to \kappa Cts (A^{op},D)$ is accessible, i.e. preserves $\mu$-filtered colimits for some cardinal $\mu$. Since $\mu$-filtered colimits commute with $\kappa$-limits as long as $\mu\ge \kappa$, such colimits in $\kappa Cts (A^{op},D)$ are pointwise, so we are at least talking about the same colimits, so the right-hand condition implies the left (take each $\lambda_a = \mu$). On the other hand, because $A$ is small, if the left-hand condition holds we can take $\mu = \sup_{a\in A} \lambda_a$ to make the right-hand condition hold.
Above we remarked that the forgetful functor $Pr(\infty,1)Cat \to (\infty,1)Cat$ preserves limits. It does not preserve colimits, and while $Pr(\infty,1)Cat$ has colimits they are in general not explicit. However, there are some cases in which they can be described explicitly, such as when they coincide with limits. For instance:
Small coproducts in $Pr(\infty,1)Cat$ coincide with small products.
Let $\prod_i C_i$ be a product in $Pr(\infty,1)Cat$. Then we have
To be precise, the next result requires an enhancement of $Pr(\infty,1)Cat$ to an $(\infty,2)$-category so that we can talk about powers and copowers by small $(\infty,1)$-categories. However, the proof gives an explicit universal property that makes sense without having to define that whole $(\infty,2)$-category.
The copower of $C\in Pr(\infty,1)Cat$ by a small $(\infty,1)$-category $A$ is $C^{A^{op}}$.
Let $P A$ denote the presheaf (∞,1)-category? on $A$. This is its free cocompletion, so we have $Func^L(P A,C) \simeq C^A$. Dually, $(P A)^{op}$ is the free completion of $A^{op}$, so we have $Cts((P A)^{op},C) \simeq C^{A^op}$. Now we have:
In general, this sort of argument should work for all lax colimits of lax functors; for instance, Kleisli objects should also coincide with Eilenberg-Moore objects? (though this also requires enhancing $Pr(\infty,1)Cat$ to an $(\infty,2)$-category). The corresponding 1-categorical fact is that in any bicategory with local colimits? (colimits in each hom-category distributed over by composition), lax colimits of lax functors are also lax limits.
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”, or at least “$\infty$-abelian groups”. For instance, the fact that the $(\infty,2)$-category $Pr(\infty,1)Cat$ has local colimits, making certain limits and colimits coincide, is a sort of categorification of the fact that of Vect and Ab are additive categories in which finite products and coproducts coincide.
More on this analogy 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. The construction of the tensor product as $Cts(C^{op},D)$ corresponds to the fact that, at least for finite-dimensional $V$, we have $V\otimes W \cong Hom(V^\ast,W)$.
(A related decategorification of $Pr(\infty,1)Cat$ is the category Sup of suplattices, which can also be thought of as analogous to abelian groups or vector spaces.)
Combined with the fact that the embedding $Pr(\infty,1)Cat \hookrightarrow (\infty,1)Cat$ preserves limits (prop. 2), 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 $(\infty,1)$-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). This monoidal structure is exhibited as a restriction of a monoidal structure on all cocomplete $(\infty,1)$-categories in section 4.8 of