nLab
(infinity,n)-category of spans

Idea

An $(\mathrm{\infty },n)$-category of spans in ∞-groupoid is an (∞,n)-category whose

• objects are ∞-groupoids;

• morphisms $X\to Y$ are spans

  $$\begin{array}{ccc}& & Z\\ & \swarrow & & \searrow \\ X& & & & Y\end{array}$$\array{ && Z \\ & \swarrow && \searrow \\ X &&&& Y }

  in ∞Grpd

• 2-morphisms are spans of spans

  $$\begin{array}{ccc}& & Z\\ & \swarrow & \uparrow & \searrow \\ X& & Q& & Y\\ & \nwarrow & \downarrow & \nearrow \\ & & Z\prime \end{array}$$\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

• $k>n$-morphisms are equivalences of order $(k-n)$ of higher spans.

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 $(\mathrm{\infty },n)$-category of spans over $C$, whose

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

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

  $$\begin{array}{ccc}& & Z\\ & \swarrow & & \searrow \\ X& & ⇙& & Y\\ & \searrow & & \swarrow \\ & & C\end{array}$$\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

The (∞,2)-category of spans in ∞Grpd is discussed in some detail in (Dyckerhoff-Kapranov 12, section 10). For a sketch of the definition for all $n$ see (Lurie, page 57).

Properties

Claim

${\mathrm{Span}}_n(\mathrm{\infty }\mathrm{Grpd})$ is a symmetric monoidal (∞,n)-category with duals.

More generally If $C$ is any symmetric monoidal $(\mathrm{\infty },n)$-category with duals, then so is ${\mathrm{Span}}_n(\mathrm{\infty }\mathrm{Grpd},C)$.

This appears as (Lurie, remark 3.2.3).

Let ${\mathrm{Bord}}_n$ be the (∞,n)-category of cobordisms.

Claim

The following data are equivalent

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

   $${\mathrm{Bord}}_n\to {\mathrm{Span}}_n(\mathrm{\infty }\mathrm{Grpd})$$Bord_n \to Span_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).

Note

In view of the cobordism hypothesis for cobordisms equipped with extra topological structure and noticing that

$${\mathrm{Bord}}_n\simeq {\mathrm{Bord}}_n^{O(n)}$$Bord_n \simeq Bord_n^{O(n)}

(Lurie, example 2.4.22) this says something like that in ${\mathrm{Span}}_n(\mathrm{\infty }\mathrm{Grpd})$ every object of ∞Grpd becomes fully dualizable.

Related concepts

• span trace

References

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

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 ${\mathrm{Span}}_n(\mathrm{\infty }\mathrm{Grpd})/C$ of spans of ∞-groupoid over a symmetric monoidal $(\mathrm{\infty },n)$-category $C$ is in section 3.2 of

• Jacob Lurie, On the Classification of Topological Field Theories

there denoted ${\mathrm{Fam}}_n(C)$. Notice the heuristic discussion on page 59.

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

The application of ${\mathrm{Span}}_n(\mathrm{\infty }\mathrm{Grpd}/C)$ 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

A discussion of a version $\mathrm{Span}(B)$for $B$ a 2-category with $\mathrm{Span}(B)$ 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)