(infinity,n)-category of correspondences



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

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

  • objects are ∞-groupoids;

  • morphisms XYX \to Y are correspondences

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

    in ∞Grpd

  • 2-morphisms are correspondences of correspondences

    Z X Q Y Z \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>nk \gt n-morphisms are equivalences of order (kn)(k-n) of higher correspondences.

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

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

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


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 nn 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(H)E nAlg b(H op) 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 H\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(H)Corr_n(\mathbf{H}) is a self-fully dualizable object.)

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

EE×X E \to E \times X

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

Then for

E 1 E 2 X Y Z \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 YY as a coalgebra is the limit over

E 1×E 2E 1×Y×E 2... 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×YE 2(p 1,p 2)E 1×E 2 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


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

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

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

[X 1 L 1 𝒞 0][X 2 L 2 𝒞 0][X 1×X 2 (L 1,L 2) 𝒞 0×𝒞 0 𝒞 𝒞 0]. \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)


If 𝒞 0\mathcal{C}_0 is (or is regarded as) a moduli stack for some kind of bundles forming a linear homotopy type theory over H\mathbf{H}, then the phased tensor product is what is also called the external tensor product.


Examples of phased tensor products include


Full dualizability


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

More generally, if 𝒞\mathcal{C} is a symmetric monoidal (,n)(\infty,n)-category with duals, then so is Corr n(Grpd,𝒞) 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).


The canonical O(n)O(n)-∞-action on Corr n(Grpd)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:


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

And generally, for 𝒞SymmMon(,n)Cat(H)\mathcal{C} \in SymmMon (\infty,n)Cat(\mathbf{H}) a symmetric monoidal (∞,n)-category internal to C\mathbf{C}, then Corr n(H /C)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 nBord_n be the (∞,n)-category of cobordisms.


The following data are equivalent

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

    Bord nCorr n(Grpd) Bord_n \to Corr_n(\infty Grpd)
  2. Pairs (X,V)(X,V), where XX is a topological space and VXV \to X a vector bundle of rank nn.

This appears as (Lurie, claim 3.2.4).


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

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

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

More detailed discussion is given in

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

The generalization to an (,n)(\infty,n)-category Span n((,1)Cat Adj)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

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

Discussion of Span n(H)Alg E n(H op)Span_n(\mathbf{H}) \simeq Alg_{E_n}(\mathbf{H}^{op}) is around remark 1.17 of

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

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

See also

Last revised on September 30, 2018 at 05:31:33. See the history of this page for a list of all contributions to it.