# nLab (infinity,n)-category of spans

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

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

• objects are ∞-groupoids;

• morphisms $X\to Y$ are spans

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

in ∞Grpd

• 2-morphisms are spans of spans

$\begin{array}{ccc}& & Z\\ & ↙& ↑& ↘\\ X& & Q& & Y\\ & ↖& ↓& ↗\\ & & Z\prime \end{array}$\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 $\left(k-n\right)$ 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 $\left(\infty ,n\right)$-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 $C$

$\begin{array}{ccc}& & Z\\ & ↙& & ↘\\ X& & ⇙& & Y\\ & ↘& & ↙\\ & & C\end{array}$\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

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

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

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 $\left(\infty ,n\right)$-functors

${\mathrm{Bord}}_{n}\to {\mathrm{Span}}_{n}\left(\infty \mathrm{Grpd}\right)$Bord_n \to Span_n(\infty Grpd)
2. Pairs $\left(X,V\right)$, 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

${\mathrm{Bord}}_{n}\simeq {\mathrm{Bord}}_{n}^{O\left(n\right)}$Bord_n \simeq Bord_n^{O(n)}

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

## 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

An inductive definition of the symmetric monoidal (∞,n)-category ${\mathrm{Span}}_{n}\left(\infty \mathrm{Grpd}\right)/C$ of spans of ∞-groupoid over a symmetric monoidal $\left(\infty ,n\right)$-category $C$ is in section 3.2 of

there denoted ${\mathrm{Fam}}_{n}\left(C\right)$. Notice the heuristic discussion on page 59.

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

The application of ${\mathrm{Span}}_{n}\left(\infty \mathrm{Grpd}/C\right)$ to the construction of FQFTs is further discussed in section 3 of

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

Revised on May 14, 2013 11:25:30 by Urs Schreiber (82.169.65.155)