nLab (infinity,n)-category of correspondences

Idea

The generalization of the bicategory Span to (∞,n)-categories:

An $(\infty,n)$-category of correspondences in ∞-groupoid is an (∞,n)-category whose

• objects are ∞-groupoids;

• morphisms $X \to Y$ are correspondences

  $$\array{ && Z \\ & \swarrow && \searrow \\ X &&&& Y }$$

  in ∞Grpd

• 2-morphisms are correspondences of correspondences

  $$\array{ && Z \\ & \swarrow &\uparrow& \searrow \\ X &&Q&& Y \\ & \nwarrow &\downarrow& \nearrow \\ && Z' }$$

  (where the triangular sub-diagrams are filled with 2-morphisms in ∞Grpd which we do not display here)

• and so on up to n-morphisms

• $k \gt n$-morphisms are equivalences of order $(k-n)$ of higher correspondences.

Using the symmetric monoidal structure on ∞Grpd this becomes a symmetric monoidal (∞,n)-category.

More generally, for $C$ some symmetric monoidal (∞,n)-category, there is a symmetric monoidal $(\infty,n)$-category of correspondences over $C$, whose

• objects are ∞-groupoids $X$ equipped with an (∞,n)-functor $X \to C$;

• morphisms $X \to Y$ are correspondences in (∞,1)Cat over $C$

  $$\array{ && Z \\ & \swarrow && \searrow \\ X &&\swArrow&& Y \\ & \searrow && \swarrow \\ && C }$$

• and so on.

Even more generally one can allow the ∞-groupoids $X, Y, \cdots$ to be (∞,n)-categories themselves.

Definition

Direct definition

The (∞,2)-category of correspondences in ∞Grpd is discussed in some detail in (Dyckerhoff-Kapranov 12, section 10). A sketch of the definition for all $n$ was given in (Lurie, page 57). A fully detailed version of this definition is in (Haugseng 14).

Definition via coalgebras

In (BenZvi-Nadler 13, remark 1.17) it is observed that

$$Corr_n(\mathbf{H}) \simeq E_n Alg_b(\mathbf{H}^{op})$$

is equivalently the (∞,n)-category of En-algebras and (∞,1)-bimodules between them in the opposite (∞,1)-category of $\mathbf{H}$ (since every object in a cartesian category is uniquely a coalgebra by its diagonal map).

(This immediately implies that every object in $Corr_n(\mathbf{H})$ is a self-fully dualizable object.)

To see how this works, consider $X \in \mathbf{H}$ any object regarded as a coalgebra in $\mathbf{H}$ via its diagonal map (here). Then a comodule $E$ over it is a co-action

$$E \to E \times X$$

and hence is canonically given by just a map $E \to X$.

Then for

$$\array{ && E_1 &&&& E_2 \\ & \swarrow && \searrow && \swarrow && \searrow \\ X && && Y && && Z }$$

two consecutive correspondences, now interpreted as two bi-comodules, their tensor product of comodules over $Y$ as a coalgebra is the limit over

$$E_1 \times E_2 \stackrel{\to}{\to} E_1 \times Y \times E_2 \stackrel{\to}{\stackrel{\to}{\to}} ...$$

This is indeed the fiber product

$$E_1 \underset{Y}{\times} E_2 \stackrel{(p_1, p_2)}{\to} E_1 \times E_2$$

as it should be for the composition of correspondences.

With the phased tensor product

Proposition

For $\mathbf{H}$ an (∞,1)-topos and $\mathcal{C} \in Cat_{(\infty,n)}(\mathbf{H})$ a symmetric monoidal internal (∞,n)-category then there is a symmetric monoidal (∞,n)-category

$$Corr_n(\mathbf{H}_{/\mathcal{C}})^\otimes \in SymmMon (\infty,n)Cat$$

whose k-morphisms are $k$-fold correspondence in $\mathbf{H}$ over $k$-fold correspondences in $\mathcal{C}$, and whose monoidal structure is given by

$$\left[ \array{ X_1 \\ \downarrow^{\mathrlap{\mathbf{L}_1}} \\ \mathcal{C}_0 } \right] \otimes \left[ \array{ X_2 \\ \downarrow^{\mathrlap{\mathbf{L}_2}} \\ \mathcal{C}_0 } \right] \coloneqq \left[ \array{ X_1 \times X_2 \\ \downarrow^{\mathrlap{(\mathbf{L}_1, \mathbf{L}_2)}} \\ \mathcal{C}_0 \times \mathcal{C}_0 \\ \downarrow^{\mathrlap{\otimes_{\mathcal{C}}}} \\ \mathcal{C}_0 } \right] \,.$$

This is (Haugseng 14, def. 4.6, corollary 7.5)

Remark

If $\mathcal{C}_0$ is (or is regarded as) a moduli stack for some kind of bundles forming a linear homotopy type theory over $\mathbf{H}$, then the phased tensor product is what is also called the external tensor product.

Example

Examples of phased tensor products include

• the Cauchy product of species;

• some external tensor products in indexed monoidal categories;

Properties

Full dualizability

Proposition

$Corr_n(\infty Grpd)$ is a symmetric monoidal (∞,n)-category with duals.

More generally, if $\mathcal{C}$ is a symmetric monoidal $(\infty,n)$-category with duals, then so is $Corr_n(\infty Grpd,\mathcal{C})^\otimes$ equipped with the phased tensor product of prop. .

In particular every object in these is a fully dualizable object.

This appears as (Lurie, remark 3.2.3). A proof is written down in (Haugseng 14, corollary 6.6).

Conjecture

The canonical $O(n)$-∞-action on $Corr_n(\infty Grpd)$ induced via prop. by the cobordism hypothesis (see there at the canonical O(n)-action) is trivial.

This statement appears in (Lurie, below remark 3.2.3) without formal proof. For more see (Haugseng 14, remark 9.7).

More generally:

Proposition

For $\mathbf{H}$ an (∞,1)-topos, then $Corr_n(\mathbf{H})$ is an (∞,n)-category with duals.

And generally, for $\mathcal{C} \in SymmMon (\infty,n)Cat(\mathbf{H})$ a symmetric monoidal (∞,n)-category internal to $\mathbf{C}$, then $Corr_n(\mathbf{H}_{/\mathbf{C}})$ equipped with the phased tensor product of prop. is an (∞,n)-category with duals

(Haugseng 14, cor. 7.8)

Let $Bord_n$ be the (∞,n)-category of cobordisms.

Claim

The following data are equivalent

1. Symmetric monoidal $(\infty,n)$-functors

   $$Bord_n \to Corr_n(\infty Grpd)$$

2. Pairs $(X,V)$, where $X$ is a topological space and $V \to X$ a vector bundle of rank $n$.

This appears as (Lurie, claim 3.2.4).

Related concepts

• symplectic category

• span trace

• relations, bicategory of relations

• sheaf with transfer, Mackey functor

References

For references on 1- and 2-categories of spans see at correspondences.

An explicit definition of the (∞,2)-category of spans in ∞Grpd is in section 10 of

• Tobias Dyckerhoff, Mikhail Kapranov, Higher Segal spaces I, (arxiv:1212.3563)

An inductive definition of the symmetric monoidal (∞,n)-category $Span_n(\infty Grpd)/C$ of spans of ∞-groupoid over a symmetric monoidal $(\infty,n)$-category $C$ is sketched in section 3.2 of

• Jacob Lurie, On the Classification of Topological Field Theories

there denoted $Fam_n(C)$. Notice the heuristic discussion on page 59.

More detailed discussion is given in

• Rune Haugseng, Iterated spans and “classical” topological field theories (arXiv:1409.0837)

• Yonatan Harpaz, Ambidexterity and the universality of finite spans (arXiv:1703.09764)

Both articles comment on the relation to Local prequantum field theory.

The generalization to an $(\infty,n)$-category $Span_n((\infty,1)Cat^Adj)$ of spans between (∞,n)-categories with duals is discussed on p. 107 and 108.

The extension to the case when the ambient $\infty$-topos is varied is in

• David Li-Bland, The stack of higher internal categories and stacks of iterated spans, (arXiv:1506.08870)

The application of $Span_n(\infty Grpd/C)$ to the construction of FQFTs is further discussed in section 3 of

• Dan Freed, Mike Hopkins, Jacob Lurie, Constantin Teleman, Topological Quantum Field Theories from Compact Lie Groups

Discussion of $Span_n(\mathbf{H}) \simeq Alg_{E_n}(\mathbf{H}^{op})$ is around remark 1.17 of

• David Ben-Zvi, David Nadler, Nonlinear traces (arXiv:1305.7175)

A discussion of a version $Span(B)$for $B$ a 2-category with $Span(B)$ regarded as a tricategory and then as a 1-object tetracategory is in

• Alex Hoffnung, Spans in 2-Categories: A monoidal tricategory (arXiv:1112.0560)

A discussion that $Span_2(-)$ in a 2-category with weak finite limits is a compact closed 2-category:

• Mike Stay, Compact Closed Bicategories (arXiv:1301.1053)

See also

• David Ayala, John Francis, Fibrations of $\infty$-Categories (arXiv:1702.02681)

Coisotropic orrespondences for derived Poisson stacks:

• Rune Haugseng, Valerio Melani, Pavel Safronov, Shifted Coisotropic Correspondences (arXiv:1904.11312)