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
in ∞Grpd
2-morphisms are correspondences of correspondences
(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 $C$
and so on.
Even more generally one can allow the ∞-groupoids $X, Y, \cdots$ to be (∞,n)-categories themselves.
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).
In (BenZvi-Nadler 13, remark 1.17) it is observed that
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
and hence is canonically given by just a map $E \to X$.
Then for
two consecutive correspondences, now interpreted as two bi-comodules, their tensor product of comodules over $Y$ as a coalgebra is the limit over
This is indeed the fiber product
as it should be for the composition of correspondences.
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
whose k-morphisms are $k$-fold correspondence in $\mathbf{H}$ over $k$-fold correspondences in $\mathcal{C}$, and whose monoidal structure is given by
This is (Haugseng 14, def. 4.6, corollary 7.5)
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.
Examples of phased tensor products include
$Corr_n(\infty Grpd)$ 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).
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:
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
Let $Bord_n$ be the (∞,n)-category of cobordisms.
The following data are equivalent
Symmetric monoidal $(\infty,n)$-functors
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).
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 $Span_n(\infty Grpd)/C$ of spans of ∞-groupoid over a symmetric monoidal $(\infty,n)$-category $C$ is sketched in section 3.2 of
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
The application of $Span_n(\infty Grpd/C)$ to the construction of FQFTs is further discussed in section 3 of
Discussion of $Span_n(\mathbf{H}) \simeq Alg_{E_n}(\mathbf{H}^{op})$ is around remark 1.17 of
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
A discussion that $Span_2(-)$ in a 2-category with weak finite limits is a compact closed 2-category:
See also
Coisotropic orrespondences for derived Poisson stacks:
Last revised on April 26, 2019 at 07:18:48. See the history of this page for a list of all contributions to it.