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Pr(infinity,1)Cat

Idea

$\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ 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 $(\mathrm{\infty },1)$-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 $(\mathrm{\infty },1)$-categories, which is in fact symmetric.

The collection $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ of presentable $(\mathrm{\infty },1)$-cateories with colimit-preserving (∞,1)-functors between them (i.e. with ”linear” functors between them!), is an $(\mathrm{\infty },1)$-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 $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ with its notion of “linearity” one obtains a very general notion of $\mathrm{\infty }$-linear algebra. This is described at geometric ∞-function theory.

Definition

Unstable version

Write $\mathrm{Pr}(\mathrm{\infty },1){\mathrm{Cat}}_1$ 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.

Stable version

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

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

Properties

Hom-objects

Definition

For $C,D\in \mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$, let

$${\mathrm{Func}}^L(C,D)\subset \mathrm{Func}(C,D)$$Func^L(C,D) \subset Func(C,D)

be the full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors on those that preserve all small (∞,1)-colimits.

Proposition

For all $C,D$ we have that ${\mathrm{Func}}^L(C,D)$ is itself locally presentable.

This is HTT, prop. 5.5.3.8.

Embedding into $(\mathrm{\infty },1)\mathrm{Cat}$

Proposition

All small limits and colimits exists in $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$. The limits are preserved by the embedding $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}\hookrightarrow$ (∞,1)Cat.

This is HTT, prop. 5.5.3.13.

Tensor product

Proposition

Let $C_1,\cdots ,C_n$ be a finite collection of locally presentable $(\mathrm{\infty },1)$-categories. There exists a locally presentable $(\mathrm{\infty },1)$-category $C_1\otimes \cdots \otimes C_n$ and an (∞,1)-functor

$$C_1\times \cdots \times C_n\to C_1\otimes \cdots \otimes C_n$$C_1 \times \cdots \times C_n \to C_1 \otimes \cdots \otimes C_n

(the tensor product) such that

1. it preserves (∞,1)-colimits in each variable;

2. for every $D\in \mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$, composition with $f$ produces an equivalence of (∞,1)-categories

$${\mathrm{Func}}_{(\mathrm{\infty },1)}^L(C_1\otimes \cdots \otimes C_n)\stackrel{\simeq }{\to }{\mathrm{Func}}_{(\mathrm{\infty },1)}^L(C_1\times \cdots \times C_n)\hookrightarrow {\mathrm{Func}}_{(\mathrm{\infty },1)}^{}(C_1\times \cdots \times C_n)$$Func_{(\infty,1)}^L(C_1 \otimes \cdots \otimes C_n) \stackrel{\simeq}{\to} Func^{L}_{(\infty,1)}(C_1 \times \cdots \times C_n) \hookrightarrow Func^{}_{(\infty,1)}(C_1 \times \cdots \times C_n)

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 $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ a symmetric monoidal (∞,1)-category.

As $\mathrm{\infty }$-vector spaces

In some context it makes good sense to think of $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ as a model for an $(\mathrm{\infty },1)$-category of ”$\mathrm{\infty }$-vector spaces”. More on this is at integral transforms on sheaves.

Here a small $(\mathrm{\infty },1)$-category $S$ is to be thought of as a basis and the locally presentable $(\mathrm{\infty },1)$-category $C\hookrightarrow {\mathrm{PSh}}_{(\mathrm{\infty },1)}(C)$ as the $\mathrm{\infty }$-vector space spanned by this basis. The colimits in $C$ play the role of addition of vectors and the fact that morphisms in $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ 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 $\otimes :\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}\times \mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}\to \mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ 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 $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ is closed monoidal reflects the fact that Vect is closed monoidal.

Combined with the fact that the embedding $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}\hookrightarrow (\mathrm{\infty },1)\mathrm{Cat}$ preserves limits and colimits, this yields some useful statements.

For instance with $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ regarded as $\mathrm{\infty }\mathrm{Vect}$, for any ∞-group $G$ with delooping ∞-groupoid $BG$, we may think of an (∞,1)-functor $\rho :BG\to \mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ as a linear representation of $G$: the single object of $BG$ is sent to a presentable $(\mathrm{\infty },1)$-category $V$ and the morphisms in $BG$ 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 $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$

$$\begin{array}{ccc}V//G& \to & Z\\ \downarrow & & \downarrow \\ BG& \stackrel{\rho }{\to }& \mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}\end{array}.$$\array{ V//G &\to& Z \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Pr(\infty,1)Cat } \,.

Related concepts

• 2Ab?, 2Ring?

References

The $(\mathrm{\infty },1)$-category $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ is introduced in section 5.5.3 of

• Jacob Lurie, Higher Topos Theory

The monoidal structure on $\mathrm{Pr}(\mathrm{\infty },1)\mathrm{Cat}$ 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.

category: category