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\section*{traced monoidal category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories}

[[!include monoidal categories - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{RelationToCompactClosedCategories}{Relation to compact closed categories}\dotfill \pageref*{RelationToCompactClosedCategories} \linebreak
\noindent\hyperlink{in_cartesian_monoidal_categories}{In cartesian monoidal categories}\dotfill \pageref*{in_cartesian_monoidal_categories} \linebreak
\noindent\hyperlink{categorical_semantics}{Categorical semantics}\dotfill \pageref*{categorical_semantics} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

The concept of \emph{traced monoidal category} axiomatizes the structure on a [[monoidal category]] for it to have a sensible notion of [[trace]] the way it exists canonically in [[compact closed categories]].

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

The original definition due to (\hyperlink{JoyalStreetVerity96}{Joyal-Street-Verity 96}) is stated in the general setting of [[balanced monoidal categories]]. Here we give just the slightly simpler formulation for the case of symmetric monoidal categories (\href{Hasegawa97}{Hasegawa 1997}).

A [[symmetric monoidal category]] $(C,\otimes,1,b)$ (where $b$ is the symmetry) is said to be \textbf{traced} if it is equipped with a natural family of functions

\begin{displaymath}
Tr_{A,B}^X : C(A \otimes X, B\otimes X) \to C(A,B)
\end{displaymath}
satisfying three axioms:

\begin{itemize}%
\item \textbf{Vanishing:} $Tr_{A,B}^1(f) = f$ (for all $f : A \to B$) and $Tr_{A,B}^{X\otimes Y}(f) = Tr_{A,B}^X(Tr_{A\otimes X,B\otimes X}^Y(f))$ (for all $f : A \otimes X \otimes Y \to B \otimes X \otimes Y$)


\item \textbf{Superposing:} $Tr_{C\otimes A,C\otimes B}^X(id_C \otimes f) = id_C \otimes Tr_{A,B}^X(f)$ (for all $f : A \otimes X \to B \otimes X$)


\item \textbf{Yanking:} $Tr_{X,X}^X(b_{X,X}) = id_X$



\end{itemize}
In [[string diagrams]], the trace $Tr(f) : A \to B$ of a morphism $f : A \otimes X \to B \otimes X$ is visualized by wrapping the outgoing wire representing $X$ to the incoming wire representing $X$, thus ``tying a loop'' in the diagram of $f$. The three axioms above (as well as the naturality conditions) then all have natural graphical interpretations (see \hyperlink{JoyalStreetVerity96}{Joyal-Street-Verity 96} or \href{Hasegawa97}{Hasegawa 1997}).

\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{RelationToCompactClosedCategories}{}\subsubsection*{{Relation to compact closed categories}}\label{RelationToCompactClosedCategories}

Every [[compact closed category]] is equipped with a canonical trace defined by

\begin{displaymath}
Tr_{A,B}^X(f) = A \overset{id\otimes \eta}{\to} A \otimes X \otimes X^* \overset{f \otimes id}{\to} B \otimes X \otimes X^* \overset{id \otimes \varepsilon'}{\to} B
\end{displaymath}
where $\eta$ is a unit and $\varepsilon'$ is a counit of appropriate [[adjunctions]] (note that the symmetry makes the dual $X^*$ both a right and left adjoint of $X$: the adjunctions are ambidextrous).

Conversely, given a traced monoidal category $\mathcal{C}$, there is a [[free construction]] completion of it to a [[compact closed category]] $Int(\mathcal{C})$ (\hyperlink{JoyalStreetVerity96}{Joyal-Street-Verity 96}):

the objects of $Int(\mathcal{C})$ are pairs $(A^+, A^-)$ of objects of $\mathcal{C}$, a morphism $(A^+ , A^-) \to (B^+ , B^-)$ in $Int(\mathcal{C})$ is given by a morphism of the form $A^+\otimes B^- \longrightarrow A^- \otimes B^+$ in $\mathcal{C}$, and [[composition]] of two such morphisms $(A^+ , A^-) \to (B^+ , B^-)$ and $(B^+ , B^-) \to (C^+ , C^-)$ is given by [[trace|tracing out]] $B^-$ from the composite

\begin{displaymath}
A^+ \otimes B^- \otimes C^- \xrightarrow{f\otimes 1} A^- \otimes B^+ \otimes C^- \xrightarrow{1\otimes g} A^- \otimes B^- \otimes C^+.
\end{displaymath}
Note that $\mathcal{C}$ embeds [[fully faithful functor|fully-faithfully]] in $Int(\mathcal{C})$ by sending $A$ to $(A,I)$, where $I$ is the unit object of the monoidal structure.

\hypertarget{in_cartesian_monoidal_categories}{}\subsubsection*{{In cartesian monoidal categories}}\label{in_cartesian_monoidal_categories}

For a [[cartesian monoidal category]], the existence of a trace operator is equivalent to the existence of a ``parameterized'' [[fixed point]] operator satisfying certain properties (\href{Hasegawa97}{Hasegawa 1997}).

\hypertarget{categorical_semantics}{}\subsubsection*{{Categorical semantics}}\label{categorical_semantics}

Traced monoidal categories serve as an ``operational'' [[categorical semantics]] for [[linear logic]], known as \emph{[[Geometry of Interactions]]}. See there for more.

In this context the free compact closure $Int(\mathcal{C})$ from \href{RelationToCompactClosedCategories}{above} is sometimes called the \emph{Geometry of Interaction construction} and denoted $\mathcal{G}(\mathcal{C})$ (\href{AbramskyHaghverdiScott02}{Abramsky-Haghverdi-Scott 02, def. 2.6}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

The concept was introduced in

\begin{itemize}%
\item [[André Joyal]], [[Ross Street]], [[Dominic Verity]], \emph{Traced monoidal categories}, Math. Proc. Camb. Phil. Soc. (1996), 119, 447 (\href{http://sci-prew.inf.ua/v119/3/S0305004100074338.pdf}{pdf})

\end{itemize}
A characterization of trace structures on cartesian monoidal categories is given in

\begin{itemize}%
\item [[Masahito Hasegawa]], \emph{Recursion from Cyclic Sharing: Traced Monoidal Categories and Models of Cyclic Lambda Calculi}, Proc. 3rd International Conference on Typed Lambda Calculi and Applications (TLCA 1997). Springer LNCS1210, 1997 (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.52.31}{citeseer})

\end{itemize}
The equivalence between traces and parameterized fixed point operators appears as Theorem 3.1 (the author notes that this theorem was also proved independently by [[Martin Hyland]]).

Comprehensive discussion as a source for [[categorical semantics]] of the [[Geometry of Interactions]] is in

\begin{itemize}%
\item [[Samson Abramsky]], [[Esfandir Haghverdi]], [[Philip Scott]], \emph{Geometry of Interaction and Linear Combinatory Algebras}. MSCS, vol. 12(5), 2002, 625-665, CUP (2002) (\href{http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.7818}{citeseer})

\end{itemize}
[[!redirects traced monoidal categories]]



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