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\begin{document}

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\section*{trace of a category}

\hypertarget{the_trace_of_a_category}{}\section*{{The trace of a category}}\label{the_trace_of_a_category}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{example}{Example}\dotfill \pageref*{example} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The trace of a [[category]] (or more generally of an endo[[bimodule]] or endo[[profunctor]]) is a [[categorification]] of the [[trace]] of a linear [[endomorphism]] on a finite dimensional [[vector space]] (that is a [[matrix]]).

A notion of trace is generally definable for maps in a [[compact closed category]] (even more generally in a [[traced monoidal category]]), and here the idea is to categorify this to the context of compact closed [[bicategories]], in particular the bicategory of bimodules between small categories.

As an instance of the [[microcosm principle]], the trace of a category is the recipient of the \emph{universal trace} function for morphisms \emph{in} that category, and also supplies the necessary structure to define [[trace in a bicategory|bicategorical traces]] in the bicategory [[Prof]].

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

Let $C$ be a compact closed [[symmetric monoidal category]], with monoidal product $\otimes$ and monoidal unit $1$. The \textbf{trace} of an endomorphism $f: c \to c$ is the composite

\begin{displaymath}
1 \overset{\eta}{\to} c^* \otimes c \overset{1_{c^*} \otimes f}{\to} c^* \otimes c \overset{\varepsilon}{\to} 1
\end{displaymath}
where $\eta$ is a unit and $\varepsilon$ is a counit of appropriate [[adjunctions]] (note that the symmetry makes the dual $c^*$ both a right and left adjoint of $c$: the adjunctions are ambidextrous). In the classical case where $C$ is the category of finite-dimensional vector spaces with its usual monoidal structure, this gives the usual trace of an endomorphism; in particular, for $f = 1_c$, this defines $dim(c) \in hom(1, 1)$.

The same idea applies to compact closed symmetric monoidal bicategories. In particular, it applies to the bicategory [[Prof]] whose objects are small categories, whose 1-morphisms are [[profunctors]] $C \to D$, i.e., functors

\begin{displaymath}
R: D^{op} \times C \to Set
\end{displaymath}
and whose 2-morphisms are [[natural transformations]] between profunctors. The bicategory $Prof$ is a [[cartesian bicategory]] and hence symmetric monoidal under $\times$, and is also compact closed: the dual of a category $C$ in this case is just the opposite category $C^{op}$, and the unit and counit profunctors

\begin{displaymath}
\eta: 1 \to C^{op} \times C, \, \varepsilon: C^{op} \times C \to 1
\end{displaymath}
are given by $hom_{C^{op}}$ and $hom_C$. Composing these (according to the [[coend]] formula for profunctor composition) yields

\begin{displaymath}
\int^{c, c' \in Ob(C)} hom(c, c') \times hom(c', c) \cong \int^c hom(c, c)
\end{displaymath}
and this is the trace of the identity $1_C$ in $Prof$ (which as a functor is also given by $hom_C$); this coend is called the \textbf{trace} of the category $C$. It could also reasonably be called, by analogy with the vector space case, the \textbf{dimension} of the category $C$. The trace of a general endoprofunctor $F$ on $C$ is the coend

\begin{displaymath}
\int^{c \in Ob(C)} F(c, c)
\end{displaymath}
which generalizes the trace of linear functions:

\begin{displaymath}
Tr(f) = \sum_i f_{i i}
\end{displaymath}
(where the matrix entries $f_{i j}$ are computed with respect to any basis).

The foregoing discussion can be generalized to the case of bimodules between small categories enriched in a [[cocomplete category|cocomplete]] symmetric [[monoidal closed category]] $V$, where the dimension of a small $V$-category $C$ is the object of $V$ given by the enriched coend

\begin{displaymath}
\int^c hom(c, c)
\end{displaymath}
\hypertarget{example}{}\subsection*{{Example}}\label{example}

We calculate the trace or dimension of [[FinSet]], the category of finite sets. The calculation is quite down-to-earth: the relevant coend is just the quotient of the set of all endofunctions $h: c \to c$ between finite sets, modulo the equivalence relation $\sim$ generated by the stipulation $f \circ g \sim g \circ f$ whenever $f: c \to d$ and $g: d \to c$ are functions between finite sets.

Let $h: c \to c$ be a finite endofunction, and let

\begin{displaymath}
c \overset{p}{\to} d \overset{i}{\to} c
\end{displaymath}
be its epi-mono factorization. Then $h = (i \circ p) \sim (p \circ i)$; if we think of $d$ as the image $h(c)$, then $p \circ i$ can be viewed as the restriction

\begin{displaymath}
{h|}\colon h(c) \to h(c)
\end{displaymath}
and this process iterates. The sequence of epis

\begin{displaymath}
h(c) \overset{h|}{\to} h^{(2)}(c) \overset{h|}{\to} \ldots
\end{displaymath}
eventually stabilizes (after finitely many steps) to a finite set $h^{(\infty)}(c)$ on which $h$ restricts to a [[surjection|surjective]] [[endofunction]], which is a bijection since we are dealing with finite sets. Thus every $h: c \to c$ is $\sim$-equivalent to a permutation $\sigma: d \to d$.

Furthermore, given two permutations $\sigma: c \to c$ and $\tau: d \to d$ such that $\sigma \sim \tau$ is witnessed by a chain of function equalities

\begin{displaymath}
\sigma = g_1 f_1, \; f_1 g_1 = h_1 = g_2 f_2, \; \ldots, \; f_{n-1} g_{n-1} = h_{n-1} = g_n f_n, \; f_n g_n = \tau
\end{displaymath}
one may show (using $g_k h_k^j f_k = g_k (f_k g_k)^j f_k = (g_k f_k)^{j+1} = h_{k-1}^{j+1}$ and similarly $f_k h_{k-1}^j g_k = h_k^{j+1}$) that

\begin{displaymath}
g_1 g_2 \ldots g_{n-1} g_n f_n f_{n-1} \ldots f_2 f_1 = \sigma^n, \qquad f_n f_{n-1} \ldots f_2 f_1 g_1 g_2 \ldots g_{n-1} g_n = \tau^n
\end{displaymath}
which implies that $f = f_n \ldots f_1$ is invertible and $\tau f = f \sigma$, or $\tau = f \sigma f^{-1}$. Conversely, if $\tau = f \sigma f^{-1}$, then for $g = \sigma f^{-1}$ we have $\sigma = g f$ and $f g = \tau$, so $\sigma \sim \tau$.

In other words, the trace of the category of finite sets is isomorphic to the trace of the [[core|underlying groupoid]] of finite sets and bijections, where the equivalence classes with respect to $\sim$ are the conjugacy classes of [[permutation]]s, given by cycle types. In this way, the trace of $FinSet$ is naturally identified with the class of finite [[Young diagram]]s.

\begin{remark}
\label{}\hypertarget{}{}
Furthermore, the functorial operations $\hom(x, x) \times \hom(y, y) \to \hom(x \times x, y \times y)$ and $\hom(x, x) \times \hom(y, y) \to \hom(x + y, x + y)$ induce operations $\cdot: Tr(FinSet) \times Tr(FinSet) \to Tr(FinSet)$ and $+: Tr(FinSet) \times Tr(FinSet) \to Tr(FinSet)$. Since $\times$ distributes over $+$ in $FinSet$, we obtain a [[rig]] structure on $Tr(FinSet)$, namely the [[Burnside rig]] of $\mathbb{Z}$.

\end{remark}
[[!redirects trace of a category]] [[!redirects traces of categories]] [[!redirects universal trace]] [[!redirects shadow]]



\end{document}