\documentclass[12pt,titlepage]{article} \usepackage{amsmath} \usepackage{mathrsfs} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsthm} \usepackage{mathtools} \usepackage{graphicx} \usepackage{color} \usepackage{ucs} \usepackage[utf8x]{inputenc} \usepackage{xparse} \usepackage{hyperref} %----Macros---------- % % Unresolved issues: % % \righttoleftarrow % \lefttorightarrow % % \color{} with HTML colorspec % \bgcolor % \array with options (without options, it's equivalent to the matrix environment) % Of the standard HTML named colors, white, black, red, green, blue and yellow % are predefined in the color package. Here are the rest. \definecolor{aqua}{rgb}{0, 1.0, 1.0} \definecolor{fuschia}{rgb}{1.0, 0, 1.0} \definecolor{gray}{rgb}{0.502, 0.502, 0.502} \definecolor{lime}{rgb}{0, 1.0, 0} \definecolor{maroon}{rgb}{0.502, 0, 0} \definecolor{navy}{rgb}{0, 0, 0.502} \definecolor{olive}{rgb}{0.502, 0.502, 0} \definecolor{purple}{rgb}{0.502, 0, 0.502} \definecolor{silver}{rgb}{0.753, 0.753, 0.753} \definecolor{teal}{rgb}{0, 0.502, 0.502} % Because of conflicts, \space and \mathop are converted to % \itexspace and \operatorname during preprocessing. % itex: \space{ht}{dp}{wd} % % Height and baseline depth measurements are in units of tenths of an ex while % the width is measured in tenths of an em. \makeatletter \newdimen\itex@wd% \newdimen\itex@dp% \newdimen\itex@thd% \def\itexspace#1#2#3{\itex@wd=#3em% \itex@wd=0.1\itex@wd% \itex@dp=#2ex% \itex@dp=0.1\itex@dp% \itex@thd=#1ex% \itex@thd=0.1\itex@thd% \advance\itex@thd\the\itex@dp% \makebox[\the\itex@wd]{\rule[-\the\itex@dp]{0cm}{\the\itex@thd}}} \makeatother % \tensor and \multiscript \makeatletter \newif\if@sup \newtoks\@sups \def\append@sup#1{\edef\act{\noexpand\@sups={\the\@sups #1}}\act}% \def\reset@sup{\@supfalse\@sups={}}% \def\mk@scripts#1#2{\if #2/ \if@sup ^{\the\@sups}\fi \else% \ifx #1_ \if@sup ^{\the\@sups}\reset@sup \fi {}_{#2}% \else \append@sup#2 \@suptrue \fi% \expandafter\mk@scripts\fi} \def\tensor#1#2{\reset@sup#1\mk@scripts#2_/} \def\multiscripts#1#2#3{\reset@sup{}\mk@scripts#1_/#2% \reset@sup\mk@scripts#3_/} \makeatother % \slash \makeatletter \newbox\slashbox \setbox\slashbox=\hbox{$/$} \def\itex@pslash#1{\setbox\@tempboxa=\hbox{$#1$} \@tempdima=0.5\wd\slashbox \advance\@tempdima 0.5\wd\@tempboxa \copy\slashbox \kern-\@tempdima \box\@tempboxa} \def\slash{\protect\itex@pslash} \makeatother % math-mode versions of \rlap, etc % from Alexander Perlis, "A complement to \smash, \llap, and lap" % http://math.arizona.edu/~aprl/publications/mathclap/ \def\clap#1{\hbox to 0pt{\hss#1\hss}} \def\mathllap{\mathpalette\mathllapinternal} \def\mathrlap{\mathpalette\mathrlapinternal} \def\mathclap{\mathpalette\mathclapinternal} \def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}} \def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}} \def\mathclapinternal#1#2{\clap{$\mathsurround=0pt#1{#2}$}} % Renames \sqrt as \oldsqrt and redefine root to result in \sqrt[#1]{#2} \let\oldroot\root \def\root#1#2{\oldroot #1 \of{#2}} \renewcommand{\sqrt}[2][]{\oldroot #1 \of{#2}} % Manually declare the txfonts symbolsC font \DeclareSymbolFont{symbolsC}{U}{txsyc}{m}{n} \SetSymbolFont{symbolsC}{bold}{U}{txsyc}{bx}{n} \DeclareFontSubstitution{U}{txsyc}{m}{n} % Manually declare the stmaryrd font \DeclareSymbolFont{stmry}{U}{stmry}{m}{n} \SetSymbolFont{stmry}{bold}{U}{stmry}{b}{n} % Manually declare the MnSymbolE font \DeclareFontFamily{OMX}{MnSymbolE}{} \DeclareSymbolFont{mnomx}{OMX}{MnSymbolE}{m}{n} \SetSymbolFont{mnomx}{bold}{OMX}{MnSymbolE}{b}{n} \DeclareFontShape{OMX}{MnSymbolE}{m}{n}{ <-6> MnSymbolE5 <6-7> MnSymbolE6 <7-8> MnSymbolE7 <8-9> MnSymbolE8 <9-10> MnSymbolE9 <10-12> MnSymbolE10 <12-> MnSymbolE12}{} % Declare specific arrows from txfonts without loading the full package \makeatletter \def\re@DeclareMathSymbol#1#2#3#4{% \let#1=\undefined \DeclareMathSymbol{#1}{#2}{#3}{#4}} \re@DeclareMathSymbol{\neArrow}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\neArr}{\mathrel}{symbolsC}{116} \re@DeclareMathSymbol{\seArrow}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\seArr}{\mathrel}{symbolsC}{117} \re@DeclareMathSymbol{\nwArrow}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\nwArr}{\mathrel}{symbolsC}{118} \re@DeclareMathSymbol{\swArrow}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\swArr}{\mathrel}{symbolsC}{119} \re@DeclareMathSymbol{\nequiv}{\mathrel}{symbolsC}{46} \re@DeclareMathSymbol{\Perp}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\Vbar}{\mathrel}{symbolsC}{121} \re@DeclareMathSymbol{\sslash}{\mathrel}{stmry}{12} \re@DeclareMathSymbol{\bigsqcap}{\mathop}{stmry}{"64} \re@DeclareMathSymbol{\biginterleave}{\mathop}{stmry}{"6} \re@DeclareMathSymbol{\invamp}{\mathrel}{symbolsC}{77} \re@DeclareMathSymbol{\parr}{\mathrel}{symbolsC}{77} \makeatother % \llangle, \rrangle, \lmoustache and \rmoustache from MnSymbolE \makeatletter \def\Decl@Mn@Delim#1#2#3#4{% \if\relax\noexpand#1% \let#1\undefined \fi \DeclareMathDelimiter{#1}{#2}{#3}{#4}{#3}{#4}} \def\Decl@Mn@Open#1#2#3{\Decl@Mn@Delim{#1}{\mathopen}{#2}{#3}} \def\Decl@Mn@Close#1#2#3{\Decl@Mn@Delim{#1}{\mathclose}{#2}{#3}} \Decl@Mn@Open{\llangle}{mnomx}{'164} \Decl@Mn@Close{\rrangle}{mnomx}{'171} \Decl@Mn@Open{\lmoustache}{mnomx}{'245} \Decl@Mn@Close{\rmoustache}{mnomx}{'244} \makeatother % Widecheck \makeatletter \DeclareRobustCommand\widecheck[1]{{\mathpalette\@widecheck{#1}}} \def\@widecheck#1#2{% \setbox\z@\hbox{\m@th$#1#2$}% \setbox\tw@\hbox{\m@th$#1% \widehat{% \vrule\@width\z@\@height\ht\z@ \vrule\@height\z@\@width\wd\z@}$}% \dp\tw@-\ht\z@ \@tempdima\ht\z@ \advance\@tempdima2\ht\tw@ \divide\@tempdima\thr@@ \setbox\tw@\hbox{% \raise\@tempdima\hbox{\scalebox{1}[-1]{\lower\@tempdima\box \tw@}}}% {\ooalign{\box\tw@ \cr \box\z@}}} \makeatother % \mathraisebox{voffset}[height][depth]{something} \makeatletter \NewDocumentCommand\mathraisebox{moom}{% \IfNoValueTF{#2}{\def\@temp##1##2{\raisebox{#1}{$\m@th##1##2$}}}{% \IfNoValueTF{#3}{\def\@temp##1##2{\raisebox{#1}[#2]{$\m@th##1##2$}}% }{\def\@temp##1##2{\raisebox{#1}[#2][#3]{$\m@th##1##2$}}}}% \mathpalette\@temp{#4}} \makeatletter % udots (taken from yhmath) \makeatletter \def\udots{\mathinner{\mkern2mu\raise\p@\hbox{.} \mkern2mu\raise4\p@\hbox{.}\mkern1mu \raise7\p@\vbox{\kern7\p@\hbox{.}}\mkern1mu}} \makeatother %% Fix array \newcommand{\itexarray}[1]{\begin{matrix}#1\end{matrix}} %% \itexnum is a noop \newcommand{\itexnum}[1]{#1} %% Renaming existing commands \newcommand{\underoverset}[3]{\underset{#1}{\overset{#2}{#3}}} \newcommand{\widevec}{\overrightarrow} \newcommand{\darr}{\downarrow} \newcommand{\nearr}{\nearrow} \newcommand{\nwarr}{\nwarrow} \newcommand{\searr}{\searrow} \newcommand{\swarr}{\swarrow} \newcommand{\curvearrowbotright}{\curvearrowright} \newcommand{\uparr}{\uparrow} \newcommand{\downuparrow}{\updownarrow} \newcommand{\duparr}{\updownarrow} \newcommand{\updarr}{\updownarrow} \newcommand{\gt}{>} \newcommand{\lt}{<} \newcommand{\map}{\mapsto} \newcommand{\embedsin}{\hookrightarrow} \newcommand{\Alpha}{A} \newcommand{\Beta}{B} \newcommand{\Zeta}{Z} \newcommand{\Eta}{H} \newcommand{\Iota}{I} \newcommand{\Kappa}{K} \newcommand{\Mu}{M} \newcommand{\Nu}{N} \newcommand{\Rho}{P} \newcommand{\Tau}{T} \newcommand{\Upsi}{\Upsilon} \newcommand{\omicron}{o} \newcommand{\lang}{\langle} \newcommand{\rang}{\rangle} \newcommand{\Union}{\bigcup} \newcommand{\Intersection}{\bigcap} \newcommand{\Oplus}{\bigoplus} \newcommand{\Otimes}{\bigotimes} \newcommand{\Wedge}{\bigwedge} \newcommand{\Vee}{\bigvee} \newcommand{\coproduct}{\coprod} \newcommand{\product}{\prod} \newcommand{\closure}{\overline} \newcommand{\integral}{\int} \newcommand{\doubleintegral}{\iint} \newcommand{\tripleintegral}{\iiint} \newcommand{\quadrupleintegral}{\iiiint} \newcommand{\conint}{\oint} \newcommand{\contourintegral}{\oint} \newcommand{\infinity}{\infty} \newcommand{\bottom}{\bot} \newcommand{\minusb}{\boxminus} \newcommand{\plusb}{\boxplus} \newcommand{\timesb}{\boxtimes} \newcommand{\intersection}{\cap} \newcommand{\union}{\cup} \newcommand{\Del}{\nabla} \newcommand{\odash}{\circleddash} \newcommand{\negspace}{\!} \newcommand{\widebar}{\overline} \newcommand{\textsize}{\normalsize} \renewcommand{\scriptsize}{\scriptstyle} \newcommand{\scriptscriptsize}{\scriptscriptstyle} \newcommand{\mathfr}{\mathfrak} \newcommand{\statusline}[2]{#2} \newcommand{\tooltip}[2]{#2} \newcommand{\toggle}[2]{#2} % Theorem Environments \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{lemma}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{dagger category} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory} [[!include category theory - contents]] \hypertarget{_categories}{}\section*{{$\dagger$ categories}}\label{_categories} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{dagger_categories}{Dagger categories}\dotfill \pageref*{dagger_categories} \linebreak \noindent\hyperlink{special_morphisms}{Special morphisms}\dotfill \pageref*{special_morphisms} \linebreak \noindent\hyperlink{CatOfDagCats}{The category of \dag{}-categories}\dotfill \pageref*{CatOfDagCats} \linebreak \noindent\hyperlink{terminology_and_wording}{Terminology and wording}\dotfill \pageref*{terminology_and_wording} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak \noindent\hyperlink{ModelStructure}{Model structure on \dag{}-categories}\dotfill \pageref*{ModelStructure} \linebreak \noindent\hyperlink{simplicial_set}{$\dagger$-simplicial set}\dotfill \pageref*{simplicial_set} \linebreak \noindent\hyperlink{graphs}{$\dagger$-Graphs}\dotfill \pageref*{graphs} \linebreak \noindent\hyperlink{#oo1Version}{$(\infty,1)$-\dag{}-categories}\dotfill \pageref*{#oo1Version} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{relation_to_staralgebras}{Relation to star-algebras}\dotfill \pageref*{relation_to_staralgebras} \linebreak \noindent\hyperlink{applications}{Applications}\dotfill \pageref*{applications} \linebreak \noindent\hyperlink{quantum_mechanics_in_terms_of_compact_categories}{Quantum mechanics in terms of $\dagger$-compact categories}\dotfill \pageref*{quantum_mechanics_in_terms_of_compact_categories} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The definition of a [[category]] effectively enforces an ordering on the ``0-faces'' -- the source and target [[object]]s -- of every 1-cell (every [[morphism]]). In many cases this is essential, in that there is no way to regard the generic morphism $a \stackrel{f}{\to} b$ in the category as a morphism from $b$ to $a$ instead. But there are many categories for which this is not the case, where every morphism naturally only comes with the information of an unordered pair $\{a,b \}$ of objects, without any prejudice on which is to be regarded as source and which as target. An important general example is: \begin{itemize}% \item the category $Spans(C)$ of [[span|spans]] in a category $C$ with pullbacks, or [[duality|dually]], the category $CoSpans(C)$ of [[cospan|cospans]] in a category $C$ with pushouts. \end{itemize} More concrete examples are: \begin{itemize}% \item categories of [[cobordism]]s (but notice that cobordisms are naturally regarded as [[cospan]]s which makes this a special case of the above example); \item the category [[Hilb]] of Hilbert spaces, where for every linear map $f : H_1 \to H_2$ we also have the adjoint map (in the sense of Hilbert spaces, not in the categorical sense) $f^\dagger : H_2 \to H_1$ (but notice that according to [[groupoidification]] this is also essentially to be regarded as a special case of categories of spans). \end{itemize} A \emph{dagger structure} on a category is extra structure which encodes the idea of \emph{removing} the ordering information on the 0-faces of 1-cells in a category: it is a contravariant functor which sends every morphism $f : a \to b$ to a morphims going the other way, $f^\dagger : b \to a$. The notation and terminology here is motivated from the example [[Hilb]] of Hilbert spaces, where $f^\dagger$ is traditionally the notion for the adjoint of a linear map $f$. The canonical \dag{}-structure on [[Hilb]] and on [[nCob]] is crucial in [[FQFT|quantum field theory]] where it is used to encode the idea of \textbf{unitarity}: a \emph{unitary} [[FQFT|functorial QFT]] of dimension $n$ is supposed to be a functor $n Cob \to Hilb$ which respects the \dag{}-structure on both sides. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{dagger_categories}{}\subsubsection*{{Dagger categories}}\label{dagger_categories} A \textbf{dagger category}, or \dag{}-category, is a [[category]] $C$ equipped with a [[contravariant functor|contravariant endofunctor]], hence an ordinary [[functor]] from the [[opposite category]] $C^{op}$ of $C$ to $C$ itself \begin{displaymath} \dagger : C^{op} \to C \end{displaymath} which \begin{enumerate}% \item is the identity on [[objects]], \item is an [[involution]] $\dagger \circ \dagger = \mathrm{id}_C$. \end{enumerate} Note that regarded as an extra structure on categories, the concept of \dag{}-structure violates the [[principle of equivalence]], since it imposes equations on objects. \hypertarget{special_morphisms}{}\subsubsection*{{Special morphisms}}\label{special_morphisms} \begin{udefn} A morphism $f$ in a \dag{}-category is called \textbf{[[unitary morphism]]} if its \dag{}-adjoint equals its [[inverse]]: \begin{displaymath} f^\dagger = f^{-1} \,. \end{displaymath} \end{udefn} For the purpose of considering what makes two objects of a $\dagger$-category [[equivalence|equivalent]], one should not consider all [[isomorphism]]s (invertible morphisms) but rather all unitary isomorphisms. The unitary isomorphisms form a [[groupoid]], which may be regarded as the \emph{dagger-[[core]]} of the $\dagger$-category. For example, in [[Hilb]], there are many invertible linear operators, but only those of norm $1$ (the invertible isometries) are unitary. \begin{udefn} A morphism $f$ in a \dag{}-category is called a \textbf{[[self-adjoint morphism]]} if it equals its \dag{}-adjoint \begin{displaymath} f^\dagger = f \,. \end{displaymath} \end{udefn} \hypertarget{CatOfDagCats}{}\subsubsection*{{The category of \dag{}-categories}}\label{CatOfDagCats} A morphism $F : (C, \dagger) \to (D, \ddagger)$ of \dag{}-categories -- a \textbf{\dag{}-functor} -- is a [[functor]] $F : C \to D$ of the underlying categories, which commutes with the \dag{}-structures in that \begin{displaymath} F \circ \dagger = \ddagger \circ F^{op} \,. \end{displaymath} A [[natural transformation]] between \dag{}-functors is just a natural transformation of the underlying functors. \begin{udefn} The \dag{}-adjoint $\eta^*$ of a natural transformation \begin{displaymath} \eta : F \to G \end{displaymath} between two \dag{}-functors $F, G : (C,\dagger) \to (D,\ddagger)$ is given by the componentwise $\ddagger$-adjoint: \begin{displaymath} (\eta^*)_a := (\eta_a)^\ddagger \,. \end{displaymath} \end{udefn} To check that $\eta^*$ is indeed a natural transformation $\eta^* : G \to F$ consider $f : a \to b$ any morphism in $C$ and $f^\dagger : b \to a$ its $\dagger$-adjoint and let \begin{displaymath} \itexarray{ F(a) &\stackrel{\eta_a}{\to}& G(a) \\ \uparrow^{\mathrlap{F(f^\dagger)}} && \uparrow^{\mathrlap{G(f^\dagger)}} \\ F(b) &\stackrel{\eta_b}{\to}& G(b) } \end{displaymath} be the corresponding naturality square of $\eta$. Taking the $\ddagger$-adjoint of the entire diagram yields \begin{displaymath} \itexarray{ F(a) &\stackrel{\eta_a^\ddagger}{\leftarrow}& G(a) \\ \downarrow^{\mathrlap{F(f^\dagger)^{\ddagger}}} && \downarrow^{\mathrlap{G(f^\dagger)^{\ddagger}}} \\ F(b) &\stackrel{\eta_b^{\ddagger}}{\leftarrow}& G(b) } \;\;\; = \;\;\; \itexarray{ F(a) &\stackrel{\eta_a^\ddagger}{\leftarrow}& G(a) \\ \downarrow^{\mathrlap{F(f)}} && \downarrow^{\mathrlap{G(f)}} \\ F(b) &\stackrel{\eta_b^{\ddagger}}{\leftarrow}& G(b) } \end{displaymath} by the fact that $F$ and $G$ are \dag{}-functors. This is the naturality square over $f$ of $\eta^* : G \to F$. \begin{udef} Write $DagCat$ for the [[category]] whose objects are \dag{}-categories and whose morphisms are \dag{}-functors. For $(C,\dagger)$ and $(D,\dagger)$ two \dag{}-categories, write $([(C,\dagger),(D,\ddagger)]_{dag}, \star) \in DagCat$ for the \dag{}-category whose objects are \dag{}-functors, whose morphisms are natural transformations, with the \dag{}-operation $\star : \eta \mapsto \eta^*$ as above. \end{udef} \begin{uprop} The assignment $((C,\dagger),(D,\ddagger)) \mapsto [(C,\dagger),(D,\ddagger)]_{dag}, \star)$ extends to an [[internal hom]]-functor \begin{displaymath} [-,-] : DagCat^{op} \times DagCat \to DagCat \end{displaymath} that makes $DagCat$ into a [[cartesian closed category]]. \end{uprop} \begin{proof} This follows step-by-step the standard proof that [[Cat]] is cartesian closed, while observing that each step respects the respect for \dag{}-structures. To indicate the main point, let $C, D$ and $E$ be \dag{}-categories and consider a functor $F : C \times D \to E$. For $(f : c_1 \to c_2) \in C$ and $(g : d_1 \to d_2) \in D$ we have natural assignments \begin{displaymath} \itexarray{ (c_1, d_1) &\stackrel{(Id,g)}{\to}& (c_1, d_2) \\ \downarrow^{\mathrlap{(f,Id)}} &\searrow^{(f,g)}& \downarrow^{\mathrlap{(f,Id)}} \\ (c_2, d_1) &\stackrel{(Id,g)}{\to}& (c_2, d_2) } \;\;\;\;\; \mapsto \;\;\;\;\; \itexarray{ F(c_1, d_1) &\stackrel{F(Id,g)}{\to}& F(c_1, d_2) \\ \downarrow^{\mathrlap{F(f,Id)}} && \downarrow^{\mathrlap{F(f,Id)}} \\ F(c_2, d_1) &\stackrel{F(Id,g)}{\to}& f(c_2, d_2) } \end{displaymath} that respect daggering all morphisms, in the evident way. Keeping $d_1$ and $d_2$ fixed, respectively this makes $F(-,d_1), F(-,d_2) : C \to E$ \dag{}-functors. We see from the diagrams that $F(-,(d_1 \stackrel{g}{\to}) d_2)$ is a natural transformation between these \dag{}-functors, and the fact that $F$ intertwines the dagger operation of $D$ with that of $E$ means $F$ regarded as a functor $D \to [C,E]$ intertwines the \dag{}-structures of $D$ and $[D,E]_{dag}$, by the above definition. \end{proof} \hypertarget{terminology_and_wording}{}\subsection*{{Terminology and wording}}\label{terminology_and_wording} In Wikipedia \href{http://en.wikipedia.org/wiki/Dagger_category}{dagger category} is said to be the same as \emph{involutive category} or \emph{category with involution}, but \href{http://eom.springer.de/C/c020780.htm}{Springer's Encyclopedy} requires for a category with involution additional conditions namely a partial order on the set of morphisms and that the order is compatible with the composition of morphisms. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item The category [[Rel]] of sets and [[relations]] is a \dag{}-category, taking dagger as relational converse. \item More generally, let $C$ be a category with [[pullbacks]] and let $Span_1(C)$ be the 1-category of [[spans]] up to isomorphism: its morphisms are spans with one leg labeled as source, the other labeled as target. Then the functor $\dagger : Span_1(C)^{op} \to Span_1(C)$ which just exchanges this labeling is a \dag{}-structure on $Span_1(C)$. \item $\mathcal{R}(G)$, the category of unitary [[representation]]s of a (discrete) [[group]] $G$ and intertwining maps, is a \dag{}-category. For an intertwiner $\phi : R \rightarrow S$, let $\phi^\dagger : S \rightarrow R$ be the adjoint of $\phi$ in [[Hilb]]. \end{itemize} \hypertarget{variants}{}\subsection*{{Variants}}\label{variants} \hypertarget{ModelStructure}{}\subsubsection*{{Model structure on \dag{}-categories}}\label{ModelStructure} \begin{quote}% the following is based on a remark by [[Andre Joyal]], \href{http://permalink.gmane.org/gmane.science.mathematics.categories/5477}{posted} to the CategoryTheory mailing list on Jan 6, 2010 \end{quote} Consider \dag{}-categories from the point of view of [[homotopy theory]]. Recall that the category [[Cat]] of [[small category|small categories]] naturally admits the [[model category]] structure called the [[folk model structure on Cat]]. The category of small \dag{}-categories $DCat$ also admits a ``natural'' [[model category]] structure: \begin{itemize}% \item \dag{}-functor $f:A \to B$ is a weak equivalence iff it is \begin{itemize}% \item [[full and faithful functor|full and faithful]]; \item and unitary surjective, meaning that every object of $B$ is unitary isomorphic to an object in the image of the functor $f$; \end{itemize} \item the cofibrations and the trivial fibrations are as in [[folk model structure on Cat|Cat]]; \item fibrations are the unitary isofibration: maps having the [[right lifting property]] for unitary isomorphisms. \end{itemize} The [[forgetful functor]] $DCat \to Cat$ is a [[right adjoint]] but it is not a [[Quillen adjunction|right Quillen functor]] with respect to the natural model structures on these categories. Moreover, a forgetful functor $XStruc \to Cat$ should reflect weak equivalences in addition to preserving them. The forgetful functor $DCat\to Cat$ preserves weak equivalences but it does not reflect them. Because two objects in a \dag{}-category can be isomorphic without been unitary isomorphic. In other words the forgetful functor $DCat\to Cat$ is wrong. This may explains why a \dag{}-category cannot be regarded as a category equipped a homotopy invariant structure, as discussed in more detail in the example sections of the entry \emph{[[principle of equivalence]]}. But the notion of \dag{}-category is perfectly reasonable from an homotopy theoretic point of view. This is because the model category $DCat$ is a [[combinatorial model category]]. It follows, by a general result, that the notion of of \dag{}-category is homotopy essentially algebraic There a homotopy limit sketch whose category of models (in spaces) is [[Quillen equivalence|Quillen equivalent]] to the model category $DCat$. This is true also for the model category Cat. \hypertarget{simplicial_set}{}\subsubsection*{{$\dagger$-simplicial set}}\label{simplicial_set} \begin{quote}% the following is based on a remark by [[Andre Joyal]], posted to the CategoryTheory mailing list on Jan 6, 2010 \end{quote} A \dag{}-[[simplicial set]] can be defined to be a simplicial set $X$ equipped with an involutive [[isomorphism]] $\dagger :X\to X^{op}$ which is the identity on 0-cells. The category of \dag{}-simplicial sets (and dagger preserving maps) is the category of [[presheaf|presheaves]] on the category whose objects are the ordinals $[n]$, but where the maps $[m]\to [n]$ are order reversing or preserving. \hypertarget{graphs}{}\paragraph*{{$\dagger$-Graphs}}\label{graphs} \begin{itemize}% \item [[dagger-graph]] \end{itemize} \hypertarget{oo1Version}{}\subsubsection*{{$(\infty,1)$-\dag{}-categories}}\label{oo1Version} \begin{quote}% the following is based on a remark by [[Andre Joyal]], posted to the CategoryTheory mailing list on Jan 6, 2010 \end{quote} There should be a notion of \dag{}-[[quasi-category]] based on $\dagger$-simplicial sets as ordinary quasi-categories are based on ordinary simplicial sets. (\ldots{}) \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{relation_to_staralgebras}{}\subsubsection*{{Relation to star-algebras}}\label{relation_to_staralgebras} The [[category convolution algebra]] of a dagger category is naturally a [[star-algebra]]. The star-involution is given by [[pullback of functions]] along the $\dagger$-functor. \hypertarget{applications}{}\subsection*{{Applications}}\label{applications} \hypertarget{quantum_mechanics_in_terms_of_compact_categories}{}\subsubsection*{{Quantum mechanics in terms of $\dagger$-compact categories}}\label{quantum_mechanics_in_terms_of_compact_categories} Large parts of [[quantum mechanics]] and [[quantum computation]] are naturally formulated as the theory of $\dagger$-categories that are also [[compact closed categories]] in a compatible way -- [[dagger compact categories]]. For more on this see \begin{itemize}% \item [[quantum mechanics in terms of †-compact categories]]. \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[star-category]], [[C-star-category]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The concept of $\dagger$-category is discussed here: \begin{itemize}% \item [[Samson Abramsky|S. Abramsky]] and [[Bob Coecke|B. Coecke]], A categorical semantics of quantum protocols, \emph{Proceedings of the 19th IEEE conference on Logic in Computer Science (LiCS'04)}, IEEE Computer Science Press (2004). \href{http://arxiv.org/abs/quant-ph/0402130}{arXiv} \end{itemize} and further abstracted in: \begin{itemize}% \item [[Peter Selinger|P. Selinger]], Dagger compact closed categories and completely positive maps, \emph{Proceedings of the 3rd International Workshop on Quantum Programming Languages}, Chicago, June 30--July 1, 2005. \href{http://www.mscs.dal.ca/~selinger/papers.html#dagger}{web} \end{itemize} The importance of $\dagger$-categories in quantum theory is discussed here: \begin{itemize}% \item [[John Baez]], Quantum quandaries: a category-theoretic perspective, in \emph{Structural Foundations of Quantum Gravity}, eds. Steven French, Dean Rickles and Juha Saatsi, Oxford U. Press, 2006, pp. 240--265. See especially Section 3: The $\star$-category of Hilbert spaces. (\href{http://math.ucr.edu/home/baez/quantum/node3.html}{web}) \end{itemize} Note that in older literature, the term ``$\star$-category'' or ``star-category'' is sometimes used instead of $\dagger$-category. Certain specially nice $\dagger$-categories, such as $C^*$-categories and [[modular tensor category|modular tensor categories]], play an important role in topological quantum field theory and the theory of quantum groups: \begin{itemize}% \item [[Jürg Fröhlich]] and Thomas Kerler, \emph{Quantum Groups, Quantum Categories, and Quantum Field Theory}, Springer Lecture Notes in Mathematics 1542, Springer-Verlag, Berlin, 1991. \item Bojko Bakalov and Alexander Kirillov, Jr., \emph{Lectures on Tensor Categories and Modular Functors}, American Mathematical Society, Providence, Rhode Island, 2001. (\href{http://www.math.sunysb.edu/~kirillov/tensor/tensor.html}{web}) \end{itemize} [[!redirects dagger-category]] [[!redirects dagger-categories]] [[!redirects dagger categories]] [[!redirects †-category]] [[!redirects †-categories]] \end{document}