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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

Adjoint functor theorem

A functor between locally presentable (,1)(\infty,1)-categories:

  • is a left adjoint if and only if it preserves small colimits (hence, in particular, is accessible), and
  • is a right adjoint if and only if it preserves small limits and is accessible.

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 (equivalently, are left adjoints).

Evidently Func L(C,D)Func^L(C,D) would be the hom-(,1)(\infty,1)-category of an enhancement of Pr(,1)CatPr(\infty,1)Cat to an (,2)(\infty,2)-category; its maximal sub-\infty-groupoid is the hom-\infty-groupoid Pr(,1)Cat(C,D)Pr(\infty,1)Cat(C,D).

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. Indeed it is even closed, since the hom-objects Func LFunc^L supply adjoints to the tensor product.

The definition of the tensor product is easy: CD=Cts(C op,D)C \otimes D = Cts(C^{op},D), where CtsCts denotes the category of continuous (small-limit-preserving) functors. Equivalently, this is Cocts(C,D op) opCocts(C,D^{op})^{op}, where CoctsCocts denotes the category of cocontinuous (small-colimit-preserving) functors; it is tempting to write this as Func L(C,D op) opFunc^L(C,D^{op})^{op}, but note that D opD^{op} is not itself locally presentable. However, D opD^{op} is nevertheless cocomplete, so any cocontinuous functor CD opC \to D^{op} has a right adjoint.

To show that Cts(C op,D)Cts(C^{op},D) is locally presentable, recall that CInd κAC \simeq Ind_\kappa A for some small κ\kappa-cocomplete AA. Thus Cts(C op,D)κCts(A op,D)Cts(C^{op},D) \simeq \kappa Cts (A^{op},D), where κCts\kappa Cts denotes the category of κ\kappa-limit-preserving functors; this is an accessible localization of Fun(A op,D)Fun(A^{op},D) and hence locally presentable.

To show that Cts(C op,D)Cts(C^{op},D) has the right universal property, we compute

Func L(C,Func L(D,E)) κCoCts(A,Func L(D,E)) (by the universal property of C=Ind κ(A)) κCts(A op,Func L(D,E) op) op (by taking opposites) κCts(A op,CtsAcc(E,D)) op (by the adjoint functor theorem) CtsAcc(E,κCts(A op,D)) op (*) Func L(κCts(A op,D),E) (by the adjoint functor theorem) \begin{aligned} Func^L(C,Func^L(D,E)) &\simeq \kappa CoCts (A,Func^L(D,E)) & \quad \text{(by the universal property of }\; C = Ind_\kappa(A))\\ &\simeq \kappa Cts (A^{op},Func^L(D,E)^{op})^{op} &\quad \text{(by taking opposites)}\\\\ &\simeq \kappa Cts (A^{op},CtsAcc(E,D))^{op} &\quad \text{(by the adjoint functor theorem)}\\ &\simeq CtsAcc(E,\kappa Cts (A^{op},D))^{op} &\quad (*)\\ &\simeq Func^L(\kappa Cts (A^{op},D),E) &\quad \text{(by the adjoint functor theorem)} \end{aligned}

To justify the step (*)(*), note that both sides are equivalent to some subcategory of functors A op×EDA^{op} \times E \to D, and in both cases part of the requirement is that κ\kappa-limits be preserved in the A opA^{op}-variable and all small limits be preserved in the EE-variable. The remaining condition on the left is that for each aAa\in A the corresponding functor EDE\to D is accessible, i.e. preserves λ a\lambda_a-filtered colimits for some cardinal λ a\lambda_a. On the right, the remaining condition is that the curried functor EκCts(A op,D)E\to \kappa Cts (A^{op},D) is accessible, i.e. preserves μ\mu-filtered colimits for some cardinal μ\mu. Since μ\mu-filtered colimits commute with κ\kappa-limits as long as μκ\mu\ge \kappa, such colimits in κCts(A op,D)\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 λ a=μ\lambda_a = \mu). On the other hand, because AA is small, if the left-hand condition holds we can take μ=sup aAλ a\mu = \sup_{a\in A} \lambda_a to make the right-hand condition hold.

Colimits

Above we remarked that the forgetful functor Pr(,1)Cat(,1)CatPr(\infty,1)Cat \to (\infty,1)Cat preserves limits. It does not preserve colimits, but Pr(,1)CatPr(\infty,1)Cat does have colimits; they are constructed by passing to right adjoints and taking the limit there.

In particular, this implies there are some cases in which colimits coincide with limits. For instance:

Proposition

Small coproducts in Pr(,1)CatPr(\infty,1)Cat coincide with small products.

Proof

Let iC i\prod_i C_i be a product in Pr(,1)CatPr(\infty,1)Cat. Then we have

Func L( iC i,D) Cts(D, iC i) iCts(D,C i) iFunc L(C i,D) \begin{aligned} Func^L(\prod_i C_i, D) &\simeq Cts(D, \prod_i C_i)\\ &\simeq \prod_i Cts(D,C_i)\\ &\simeq \prod_i Func^L(C_i,D) \end{aligned}

To be precise, the next result requires an enhancement of Pr(,1)CatPr(\infty,1)Cat to an (,2)(\infty,2)-category so that we can talk about powers and copowers by small (,1)(\infty,1)-categories. However, the proof gives an explicit universal property that makes sense without having to define that whole (,2)(\infty,2)-category.

Proposition

The copower of CPr(,1)CatC\in Pr(\infty,1)Cat by a small (,1)(\infty,1)-category AA is C A opC^{A^{op}}.

Proof

Let PAP A denote the presheaf (∞,1)-category? on AA. This is its free cocompletion, so we have Func L(PA,C)C AFunc^L(P A,C) \simeq C^A. Dually, (PA) op(P A)^{op} is the free completion of A opA^{op}, so we have Cts((PA) op,C)C A opCts((P A)^{op},C) \simeq C^{A^op}. Now we have:

Func L(C,D) A Func L(C,D A) Func L(C,Func L(PA,D)) Func L(CPA,D) Func L(Cts((PA) op,C),D) Func L(C A op,D) \begin{aligned} Func^L(C,D)^A &\simeq Func^L(C,D^A)\\ &\simeq Func^L(C,Func^L(P A,D))\\ &\simeq Func^L(C \otimes P A, D)\\ &\simeq Func^L(Cts((P A)^{op},C),D)\\ &\simeq Func^L(C^{A^op},D) \end{aligned}

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(,1)CatPr(\infty,1)Cat to an (,2)(\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.

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”, or at least “\infty-abelian groups”. For instance, the fact that the (,2)(\infty,2)-category Pr(,1)CatPr(\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 (,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. The construction of the tensor product as Cts(C op,D)Cts(C^{op},D) corresponds to the fact that, at least for finite-dimensional VV, we have VWHom(V *,W)V\otimes W \cong Hom(V^\ast,W).

(A related decategorification of Pr(,1)CatPr(\infty,1)Cat is the category Sup of suplattices, which can also be thought of as analogous to abelian groups or vector spaces.)

Applications

Combined with the fact that the embedding Pr(,1)Cat(,1)CatPr(\infty,1)Cat \hookrightarrow (\infty,1)Cat preserves limits (prop. 2), 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.

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). This monoidal structure is exhibited as a restriction of a monoidal structure on all cocomplete (,1)(\infty,1)-categories in section 4.8 of

category: category

Revised on August 25, 2017 01:33:31 by Mike Shulman (76.167.222.204)