span trace

In the context of integral transforms on sheaves one thinks of a span as a generalized linear map. The *span trace* is the corresponding generalization of the notion of a trace of a linear map.

This is just the general trace of an endomorphism which is definable in any compact/autonomous symmetric monoidal (2-)category, of which $Span$ is an example (as described below).

In the context of FQFT a useful aspect of the span trace is that it is manifestly dual to the co-span co-trace, which, as described there, corresponds under the interpretation of spans as cobordisms to gluing of the two ends of a cobordism.

For

$\array{
&& R
\\
& {}^x\swarrow && \searrow^{y}
\\
X &&&& X
}$

a span with identical left and right index object $X$, the simplest way to define its **span trace** $tr(R)$ is as by regarding it as a map $R\to X\times X$, then pulling back along the diagonal morphism $X\to X\times X$.

This can be expressed in terms of the bicategory Span in several ways. For instance, we can regard it as the composite of the result

$\array{
&& R
\\
& {}^{x \times y}\swarrow && \searrow
\\
X \times X &&&& pt
}$

of dualizing one leg of the span with the span

$\array{
&& X
\\
& {}^{}\swarrow && \searrow^{Id \times Id}
\\
pt &&&& X \times X
}$

i.e. the pullback

$\array{
&&&&
\mathrm{tr}R
\\
&&& \swarrow && \searrow
\\
&& X &&&& R
\\
& {}^{}\swarrow && \searrow^{Id \times Id}
&& {}^{x \times y}\swarrow && \searrow
\\
pt &&&& X \times X &&&& pt
}$

regarded as a span from the point to the point

$\array{
&& tr(R)
\\
& {}^{}\swarrow && \searrow
\\
pt &&&& pt
}
\,.$

More generally, the trace of a multispan over $n$ identical of its index objects $X$ is the composite with the multispan

$\array{
& X
\\
& {}^{Id}\swarrow \downarrow^{Id} & \cdots
\\
X & X & \cdots & X & \cdots
}$

Let the ambient category be Set, let $X$ be a finite set and $R \to X \times X$ an $|X| \times |X|$-matrix of finite sets, regarded under groupoid cardinality as a groupoidified $|X| \times |X|$-matrix with entries in $\mathbb{N}$.

The trace of the span

$\array{
&& R
\\
& {}^x\swarrow && \searrow^{y}
\\
X &&&& X
}$

is the pullback

$\array{
tr(R) &\to& R
\\
\downarrow && \downarrow^{x,y}
\\
X &\stackrel{Id \times Id}{\to}& X \times X
}$

which is the coproduct set $tr(R) = \sqcup_{x \in X} R_{x,x}$. Under groupoid cardinality this is indeed the trace $|tr(R)| = \sum_{x} |R_{x,x}|$ of the matrix $|R|$ represented by $R$.

Let $C$ be a category of fibrant objects with interval object $I$. Recall that for every object $B$ of $C$ its free loop space object is the part of the path object $B^I = [I,B]$ which consists of closed paths, i.e. the pullback.

$\array{
\Omega B &\to& [I,B]
\\
\downarrow && \downarrow^{d_0 \times d_1}
\\
B &\stackrel{Id \times Id}{\to}& B \times B
}
\,.$

This can be understood as the *homotopy span trace* of the identity span on $B$

$\Omega B = hotr(Id_B) = hotr\left(
\array{
&& B
\\
& {}^{Id}\swarrow && \searrow^{Id}
B &&&& B
}
\right)
\,,$

where the homotopy span trace is computed like the span trace but with the pullback replaced by a homotopy pullback:

$hotr(Id_B) =
holim
\left(
\array{
B &&&& B
\\
& {}_{Id \times Id}\searrow && \swarrow_{Id \times Id}
\\
&&
B \times B
}
\right)
\,.$

According to the example described at homotopy limit and using that we assume that we are in a category of fibrant objects we can compute this homotopy limit, up to weak equivalence, as the ordinary limit of the weakly equivalent pullback diagram

$\array{
F
&&&&
B &\stackrel{Id \times Id}{\to}& B \times B
&\stackrel{Id \times Id}{\leftarrow}& B
\\
\downarrow^{\simeq}
&&&&
\downarrow^{\simeq} && \downarrow^{Id} && \downarrow^{Id}
\\
F'
&&&&
[I,B] &\stackrel{d_0 \times d_1}{\to}&
B \times B &\stackrel{Id \times Id}{\leftarrow}&
B
}$

where we replace $B$ by its path object $B^I = [I,B]$ using the factorization of $B \stackrel{Id \times Id}{\to} B \times B$ as $B \stackrel{\simeq}{\to} [I,B] \stackrel{d_0 \times d_1}{\to} B \times B$ guaranteed to exist in a category of fibrant objects, where $[I, B] \stackrel{d_0 \times d_1}{\to} B \times B$ is a *fibration*:

$holim_D F \stackrel{\simeq}{\to} lim_D F'
\,.$

But by the above $lim_D F' = \Omega B$.

The categorical trace on a 1-endomorphism in a 2-category $C$ is the homotopy trace on the span given by that endomorphism.

This should be true quite generally, but here are the details just for the special case that connects to the above example:

Let $C =$ Grpd with the standard interval object $I = \{a \stackrel{\simeq}{\to} b\}$. This is a category of fibrant objects with respect to the folk model structure.

Notice that natural transformations $\eta : F \to G$ between two functors $F, G : C \to D$ are in bijection with commuting diagrams

$\array{
X &\stackrel{\eta}{\to}& [I,Y]
\\
\downarrow^{Id} && \downarrow^{d_0 \times d_1}
\\
X &\stackrel{F \times G}{\to}& Y \times Y
}
\,.$

Now, the homotopy trace on the span corresponding to an ednofunctor $F : B \to B$ is

$hotr(F)
=
hotr\left(
\array{
&& B
\\
& {}^{Id}\swarrow && \searrow^{F}
\\
B &&&& B
}
\right)
=
holim\left(
\array{
B &&&& B
\\
& {}_{Id \times Id}\searrow && \swarrow{F \times Id}
\\
&& B \times B
}
\right)
\,.$

Since we are in a category of fibrant objects the assumptions of the example discussed at homotopy limit apply and the above homotopy limit is again computed, up to weak equivalence, by the ordinary limit of

$\cdots
\stackrel{\simeq}{\to}
lim
\left(
\array{
[I,B] &&&& B
\\
& {}_{d_0 \times d_1}\searrow && \swarrow{Id \times F}
\\
&& B \times B
}
\right)
\,.$

By the above, every cone over the pullback diagram with a functor $h : Q \to B$, on the right defines a natural transformation $h^*(Id_B \Rightarrow F)$. By the universal property of the limit, it *represents* the collection of these transformations.

- The dual notion is that of co-span co-trace.

That the canonical trace on $Span$ is compatible with the interpretation of spans as linear maps in the context of groupoidification, and that it corresponds under duality (in terms of the co-span co-trace) to the gluing of ends of cobordisms was mentioned in

- Urs Schreiber, (co)-traces (2008)

More discussion is in

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

Revised on June 3, 2013 10:22:19
by Urs Schreiber
(82.113.98.121)