nLab
Pr(infinity,1)Cat

Contents

Idea

Pr(,1)CatPr(\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×DEC \times D \to E from the cartesian product of two presentable (,1)(\infty,1)-categories is bilinear if it respects colimits in both variables.

It turns out that there is a universal such bilinear functor

C×DCD, C \times D \to C \otimes D \,,

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

The collection Pr(,1)CatPr(\infty,1)Cat of presentable (,1)(\infty,1)-cateories with colimit-preserving (∞,1)-functors between them (i.e. with “linear” functors between them!), is an (,1)(\infty,1)-generalization of the category SetModSet 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(,1)CatPr(\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.

Definition

Unstable version

Write Pr(,1)Cat 1Pr(\infty,1)Cat_1 for the sub-(∞,1)-category of the (∞,1)-category of (∞,1)-categories whose

Stable version

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

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

Properties

Hom-objects

Definition

For C,DPr(,1)CatC, D \in Pr(\infty,1)Cat, let

Func L(C,D)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,DC,D we have that Func L(C,D)Func^L(C,D) is itself locally presentable.

This is HTT, prop. 5.5.3.8.

Embedding into (,1)Cat(\infty,1)Cat

Proposition

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

This is HTT, prop. 5.5.3.13.

Tensor product

Proposition

Let C 1,,C nC_1, \cdots, C_n be a finite collection of locally presentable (,1)(\infty,1)-categories. There exists a locally presentable (,1)(\infty,1)-category C 1C nC_1 \otimes \cdots \otimes C_n and an (∞,1)-functor

C 1××C nC 1C 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 DPr(,1)CatD \in Pr(\infty,1)Cat, composition with ff produces an equivalence of (∞,1)-categories

Func (,1) L(C 1C n)Func (,1) L(C 1××C n)Func (,1) (C 1××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 Pr(,1)CatPr(\infty,1)Cat a symmetric monoidal (∞,1)-category.

As \infty-vector spaces

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

Here a small (,1)(\infty,1)-category SS is to be thought of as a basis and the locally presentable (,1)(\infty,1)-category CPSh (,1)(C)C \hookrightarrow PSh_{(\infty,1)}(C) as the \infty-vector space spanned by this basis. The colimits in CC play the role of addition of vectors and the fact that morphisms in Pr(,1)CatPr(\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 :Pr(,1)Cat×Pr(,1)CatPr(,1)Cat\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 CDC \otimes D being a bilinear morphism out of C×DC \times D, and the fact that Pr(,1)CatPr(\infty,1)Cat is closed monoidal reflects the fact that Vect is closed monoidal.

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

For instance with Pr(,1)CatPr(\infty,1)Cat regarded as Vect\infty Vect, for any ∞-group GG with delooping ∞-groupoid BG\mathbf{B}G, we may think of an (∞,1)-functor ρ:BGPr(,1)Cat\rho : \mathbf{B}G \to Pr(\infty,1)Cat as a linear representation of GG: the single object of BG\mathbf{B}G is sent to a presentable (,1)(\infty,1)-category VV and the morphisms in BG\mathbf{B}G then define an action of GG on that.

The corresponding action groupoid V//GV//G (see there for more) is then the colimit over the action, in Pr(,1)CatPr(\infty,1)Cat

V//G Z BG ρ Pr(,1)Cat. \array{ V//G &\to& Z \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Pr(\infty,1)Cat } \,.
  • 2Ab?, 2Ring?

References

The (,1)(\infty,1)-category Pr(,1)CatPr(\infty,1)Cat is introduced in section 5.5.3 of

The monoidal structure on Pr(,1)CatPr(\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.

category: category

Revised on February 4, 2013 11:26:43 by Urs Schreiber (82.113.99.102)