\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*{cobordism hypothesis} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{manifolds_and_cobordisms}{}\paragraph*{{Manifolds and cobordisms}}\label{manifolds_and_cobordisms} [[!include manifolds and cobordisms - contents]] \hypertarget{cobordism_theory}{}\paragraph*{{Cobordism theory}}\label{cobordism_theory} [[!include cobordism theory -- contents]] \hypertarget{quantum_field_theory}{}\paragraph*{{Quantum field theory}}\label{quantum_field_theory} [[!include functorial quantum field theory - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{formalization}{Formalization}\dotfill \pageref*{formalization} \linebreak \noindent\hyperlink{ForFramed}{For framed cobordisms}\dotfill \pageref*{ForFramed} \linebreak \noindent\hyperlink{statement}{Statement}\dotfill \pageref*{statement} \linebreak \noindent\hyperlink{TheCanonicalOnAction}{Implications -- The canonical $O(n)$-∞-action}\dotfill \pageref*{TheCanonicalOnAction} \linebreak \noindent\hyperlink{CobsWithExtraTopStructure}{For cobordisms with extra topological structure}\dotfill \pageref*{CobsWithExtraTopStructure} \linebreak \noindent\hyperlink{ForCobordismsWithXXiStructure}{For cobordisms with any structure (``$(X,\xi)$-structure'')}\dotfill \pageref*{ForCobordismsWithXXiStructure} \linebreak \noindent\hyperlink{StatementForCobordismsInAManifold}{For framed cobordisms in a topological space}\dotfill \pageref*{StatementForCobordismsInAManifold} \linebreak \noindent\hyperlink{CobordismsWithGStructure}{For cobordisms with $G$-structure}\dotfill \pageref*{CobordismsWithGStructure} \linebreak \noindent\hyperlink{ForUnorientedCobordisms}{For (un-)oriented cobordisms}\dotfill \pageref*{ForUnorientedCobordisms} \linebreak \noindent\hyperlink{ForHQFTs}{For HQFTs}\dotfill \pageref*{ForHQFTs} \linebreak \noindent\hyperlink{ForCobordismsWithSingularities}{For cobordisms with singularities (boundaries/branes and defects/domain walls)}\dotfill \pageref*{ForCobordismsWithSingularities} \linebreak \noindent\hyperlink{ForNoncompactCobordisms}{For noncompact cobordisms}\dotfill \pageref*{ForNoncompactCobordisms} \linebreak \noindent\hyperlink{for_cobordisms_with_geometric_structure}{For cobordisms with geometric structure}\dotfill \pageref*{for_cobordisms_with_geometric_structure} \linebreak \noindent\hyperlink{remarks}{Remarks}\dotfill \pageref*{remarks} \linebreak \noindent\hyperlink{morphisms_of_tqfts}{Morphisms of TQFTs}\dotfill \pageref*{morphisms_of_tqfts} \linebreak \noindent\hyperlink{invariants_determined_from_the_point}{Invariants determined from the point}\dotfill \pageref*{invariants_determined_from_the_point} \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 \textbf{Cobordism Hypothesis} states, roughly, that the [[(∞,n)-category of cobordisms]] $Bord_n^{fr}$ is the [[free functor|free]] [[symmetric monoidal (∞,n)-category]] [[(∞,n)-category with duals|with duals]] on a single object. Since a fully [[extended topological quantum field theory]] may be identified with a [[monoidal (∞,n)-functor|monoidal]] [[(∞,n)-functor]] $Z : Bord_n \to C$, this implies that all these [[TQFT]]s are entirely determined by their value on the point: ``the [[n-vector space]] of [[state]]s'' of the theory. As motivation, notice that by \href{cobordism+category#GMWT}{Galatius-Madsen-Tillmann-Weiss 09} we have that the [[loop space]] of the [[geometric realization]] of the [[framed manifold|framed]] [[cobordism category]] is equivalent to the [[sphere spectrum]] \begin{displaymath} \Omega \Vert Cob_d^{fr} \Vert \simeq \lim_{\to_{n \to \infty}} Maps_*(S^n, S^n) \simeq \Omega^\infty S^\infty \end{displaymath} which can be understood as the free [[infinite loop space]] on the point. \hypertarget{formalization}{}\subsection*{{Formalization}}\label{formalization} In (\hyperlink{Lurie}{Lurie}) a formalization and proof of the cobordism hypothesis is described. \hypertarget{ForFramed}{}\subsubsection*{{For framed cobordisms}}\label{ForFramed} \hypertarget{statement}{}\paragraph*{{Statement}}\label{statement} For $\mathcal{C}$ a [[symmetric monoidal (∞,n)-category with duals]] write $Core(\mathcal{C})$ for its [[core]] (the maximal [[∞-groupoid]] in $\mathcal{C}$). For $\mathcal{C}$, $\mathcal{D}$ two [[symmetric monoidal (∞,n)-categories]], write $Fun^\otimes(\mathcal{D}, \mathcal{C} )$ for the [[(∞,n)-category]] of symmetric monoidal [[(∞,n)-functors]] between them. \begin{defn} \label{FramedCobordismsNCat}\hypertarget{FramedCobordismsNCat}{} Write $Bord_n^{fr}$ be the [[symmetric monoidal (∞,n)-category|symmetric monoidal]] [[(∞,n)-category of cobordisms]] with [[framed manifold|n-framing]]. \end{defn} \begin{theorem} \label{CobordismHypothesisFramedVersion}\hypertarget{CobordismHypothesisFramedVersion}{} Evaluation of any such functor $F$ on the [[point]] ${*}$ \begin{displaymath} F \mapsto F({*}) \end{displaymath} induces an [[(∞,n)-functor]] \begin{displaymath} pt^* : Fun^\otimes(Bord_n^{fr} , \mathcal{C} ) \to \mathcal{C} . \end{displaymath} such that \begin{itemize}% \item this factors through the [[core]] of $\mathcal{C}$; \item the map \begin{displaymath} pt^* \;\colon\; Fun^\otimes(Bord_n^{fr} , \mathcal{C} ) \to Core(\mathcal{C}) \end{displaymath} is an [[equivalence in an (∞,1)-category|equivalence]] of [[(∞,n)-categories]]. \end{itemize} \end{theorem} This is (\hyperlink{Lurie}{Lurie, theorem 2.4.6}). \begin{proof} The proof is based on \begin{enumerate}% \item the [[Galatius-Madsen-Tillmann-Weiss theorem]], which characterizes the [[geometric realization]] $|Bord_n^{or}|$ in terms of the [[suspension]] of the [[Thom spectrum]]; \item Igusa's connectivity result which he uses to show that putting ``framed [[Morse function]]s'' on cobordisms doesn't change their [[homotopy type]] (theorem 3.4.7, page 73) \end{enumerate} In fact, the Galatius-Madsen-Weiss theorem is now supposed to be a corollary of Lurie's result. \end{proof} \hypertarget{TheCanonicalOnAction}{}\paragraph*{{Implications -- The canonical $O(n)$-∞-action}}\label{TheCanonicalOnAction} One of the striking consequences of theorem \ref{CobordismHypothesisFramedVersion} is that it implies that \begin{cor} \label{CanonicalOnAction}\hypertarget{CanonicalOnAction}{} Every [[∞-groupoid]] \begin{displaymath} \mathcal{C}^{fd} \hookrightarrow \mathcal{C} \end{displaymath} of [[fully dualizable objects]] in a [[symmetric monoidal (∞,n)-category]] $\mathcal{C}$ carries a canonical [[∞-action]] of (the [[∞-group]] structure on the [[homotopy type]] of) the [[orthogonal group]] $O(n)$, induced by the action of $O(n)$ on the [[framed manifold|n-framing]] of the point in $Bord_n^{fr}$. \end{cor} (\hyperlink{Lurie}{Lurie, corollary 2.4.10}) \begin{example} \label{ExamplesOfTheCanonicalOnActions}\hypertarget{ExamplesOfTheCanonicalOnActions}{} The action in corollary \ref{CanonicalOnAction} is \begin{itemize}% \item for $n = 1$: the $O(1) = \mathbb{Z}/2\mathbb{Z}$ action action given by passing to [[dual objects]]; \item for $n = 2$ the $O(2)$-action the \emph{[[Serre automorphism]]}. \item for $n = \infty$ the $O(n)$-action on [[n-fold loop spaces]] (see e.g. \hyperlink{GaudensMenichi07}{Gaudens-Menichi 07, section 5}) (see also at \emph{[[orthogonal spectra]]}). \end{itemize} \end{example} (\hyperlink{Lurie}{Lurie, examples 2.4.12, 2.4.14. 2.4.15}) \begin{prop} \label{CanonicalSOActionOnBnZ}\hypertarget{CanonicalSOActionOnBnZ}{} For all $n \in \mathbb{N}$, the canonical $SO$-[[∞-action]] on \begin{displaymath} B^n \mathbb{Z} \in Ab_\infty(\infty Grpd) \hookrightarrow (\infty,n)CatWithDuals \end{displaymath} is trivial. \end{prop} \begin{proof} The action on a [[connective spectrum]] $\Omega^\infty X$ factors through the [[J-homomorphism]] \begin{displaymath} SO \times \Omega^\infty X \stackrel{(J,id)}{\longrightarrow} \Omega^\infty S^\infty \times \Omega^\infty X \stackrel{precomp}{\longrightarrow} \Omega^\infty X \,. \end{displaymath} But on [[homotopy groups]] the [[image of J]] is pure [[torsion]] which means that for $\Omega^\infty X = B^n \mathbb{Z}$ the induced actions on homotopy groups are all trivial. From this and using the [[long exact sequence of homotopy groups]] it follows that the $\infty$-action itself is trivial. \end{proof} \hypertarget{CobsWithExtraTopStructure}{}\subsubsection*{{For cobordisms with extra topological structure}}\label{CobsWithExtraTopStructure} We discuss the cobordism hypothesis for cobordisms that are equipped with the extra [[structure]] of maps into some [[topological space]] equipped with a [[vector bundle]]. This is the case for which an [[extended TQFT]] is (the local refinement of) what has also been called an \emph{[[HQFT]]}. \hypertarget{ForCobordismsWithXXiStructure}{}\paragraph*{{For cobordisms with any structure (``$(X,\xi)$-structure'')}}\label{ForCobordismsWithXXiStructure} \begin{defn} \label{XXiStructure}\hypertarget{XXiStructure}{} Let $X$ be a [[topological space]] and $\xi \to X$ a [[real number|real]] [[vector bundle]] on $X$ of [[rank]] $n$. Let $N$ be a [[smooth manifold]] of [[dimension]] $m \leq n$. An \textbf{$(X,\xi)$-structure} on $N$ consists of the following data \begin{itemize}% \item A [[continuous function]] $f : N \to X$; \item An [[isomorphism]] of [[vector bundle]]s \begin{displaymath} T N \oplus \mathbb{R}^{n-m} \simeq f^* \xi \end{displaymath} between the [[fiber]]wise [[direct sum]] of the [[tangent bundle]] $T N$ with the trivial rank $(n-m)$ bundle and the [[pullback]] of $\xi$ along $f$. \end{itemize} \end{defn} This is (\hyperlink{Lurie}{Lurie, notation 2.4.16}). The two extreme cases of def. \ref{XXiStructure} are the following \begin{example} \label{FramingAsXXiStructur}\hypertarget{FramingAsXXiStructur}{} For $X = \ast$ the point and $\xi = \mathbb{R}^n$, then an $(X,\xi)$-structure is the same as an $n$-[[framing]], hence \begin{displaymath} (Bord_n^{(\ast, \mathbb{R}^n)}) \simeq Bord_n^{fr} \end{displaymath} reproduces the $(\infty,n)$-category of framed cobordisms of def. \ref{FramedCobordismsNCat}. \end{example} \begin{example} \label{UnorientedAsXXiStructur}\hypertarget{UnorientedAsXXiStructur}{} For $X = B O(n)$ the [[classifying space]] of real [[vector bundles]] of [[rank]] $n$ (the [[delooping]] of the [[∞-group]] $O(n)$ underlying the [[orthogonal group]]) and for $\xi = E O(n) \underset{O(n)}{\times} \mathbb{R}^n$ the vector bundle [[associated bundle|associated]] to the $O(n)$-[[universal bundle]], then $(X,\xi)$-structure on $n$-dimensional manifolds is essentially \emph{no}-structure (the [[maximal compact subgroup]]-inclusion $O(n)\to GL(n)$ is a [[weak homotopy equivalence]]). Cobordisms with this structure will also be called \emph{unoriented cobordisms} \begin{displaymath} Bord_n^{un} \coloneqq Bord_n^{(B O(n), E O(n)\underset{O(n)}{\times} \mathbb{R}^n)} \,. \end{displaymath} Accordingly, for $X = B SO(n)$ the [[delooping]] of the [[special orthogonal group]], the corresponding $(X,\xi)$-structure makes \emph{oriented manifolds} \begin{displaymath} Bord_n^{or} \coloneqq Bord_n^{(B SO(n), E SO(n)\underset{SO(n)}{\times} \mathbb{R}^n)} \,. \end{displaymath} \end{example} Generally: \begin{example} \label{GStructureAsXXiStructure}\hypertarget{GStructureAsXXiStructure}{} For $\chi \colon G \to O(n)$ a [[topological group]] mapping via a [[homomorphism]] to $O(n)$, then $X = B G$ and $\xi = \chi^\ast (E O(n)\underset{O(n)}{\times} \mathbb{R}^n)$, the $(X,\xi)$-structure is [[G-structure]]. This we get to \hyperlink{CobordismsWithGStructure}{below}. \end{example} \begin{defn} \label{CatOfCobordismsWithXIStructure}\hypertarget{CatOfCobordismsWithXIStructure}{} Let $X$ be a [[topological space]] and $\xi \to X$ an $n$-[[dimensional]] [[vector bundle]]. The [[(∞,n)-category]] $Bord_n^{(X, \xi)}$ is defined analogously to $Bord_n$ but with all manifolds equipped with $(X,\xi)$-structure, def. \ref{XXiStructure}. \end{defn} This is (\hyperlink{Lurie}{Lurie, def. 2.4.17}). \begin{theorem} \label{CobHypTheoremForTargetSpaceAndVectorBundle}\hypertarget{CobHypTheoremForTargetSpaceAndVectorBundle}{} Let $\mathcal{C}$ be a [[symmetric monoidal (∞,n)-category]] with duals, let $X$ be a [[CW-complex]], let $\xi \to X$ be an $n$-[[dimensional]] [[vector bundle]] over $X$ equipped with an inner product, and let $\tilde X \to X$ be the [[associated bundle|associated]] [[orthogonal group|O(n)]]-[[principal bundle]] of orthonormal [[frames]] in $\xi$. There is an [[equivalence in an (∞,1)-category|equivalence]] in [[∞Grpd]] \begin{displaymath} Fun^\otimes(Bord_n^{(X,\xi)}, \mathcal{C}) \simeq Top_{O(n)}(\tilde X, \tilde \mathcal{C}) \,, \end{displaymath} where on the right we regard $\tilde C$ as a [[topological space]] carrying the canonical $O(n)$-[[action]] discussed \hyperlink{TheCanonicalOnAction}{above}. \end{theorem} This is (\hyperlink{Lurie}{Lurie, theorem. 2.4.18}). The following is some aspects of the idea of the proof in (\hyperlink{Lurie}{Lurie, p. 57}). \begin{remark} \label{ActionHomomorphismsAsInfinityActionHomomorphisms}\hypertarget{ActionHomomorphismsAsInfinityActionHomomorphisms}{} In the language of [[∞-actions]] (as discussed there), the space $Top_{O(n)}(\tilde X, \tilde \mathcal{C})$ is that of horizontal maps fitting into \begin{displaymath} \itexarray{ X && \longrightarrow && \tilde X//O(n) \\ & \searrow && \swarrow \\ && B SO(n) } \end{displaymath} where the left map is the classifying map for $\xi$ and the right one is the canonical one out of the [[homotopy quotient]]. \end{remark} \begin{proof} Notice that for each point $x \colon \ast \to X$ there is an induced inclusion \begin{displaymath} Bord_n^{fr} \stackrel{x}{\longrightarrow} Bord_n^{(X,\xi)} \end{displaymath} of the framed cobordisms, def. \ref{FramedCobordismsNCat}, into those of $(X,\xi)$-structure, def. \ref{CatOfCobordismsWithXIStructure}, including those cobordisms whose map to $X$ is constant on $X$, and observing that for these an $(X,\xi)$-structure is equivalently an $n$-[[framing]]. Moreover, by corollary \ref{CanonicalOnAction} the induced point evaluation is $O(n)$-equivariant, hence yielding a morphism of [[∞-groupoids]] \begin{displaymath} \alpha \;\colon\; Func^\otimes(Bord_n^{(X,\xi)}, \mathcal{C}) \longrightarrow Maps_{O(n)}(\tilde X, \tilde \mathcal{C}) \,, \end{displaymath} where $\tilde X$ denotes the $O(n)$-[[principal bundle]] to which $\xi$ is [[associated bundle|associated]]. More generally, this is true for the pullback structure of $\xi$ along along any map $Y \to X$, yielding \begin{displaymath} \alpha_Y \;\colon\; Func^\otimes(Bord_n^{(Y,\xi|Y)}, \mathcal{C}) \longrightarrow Maps_{O(n)}(\tilde X\underset{X}{\times} Y, \tilde \mathcal{C}) \,. \end{displaymath} By the previous comment, observe that $\alpha_Y$ is an equivalence for $Y = \ast$. Now the [[codomain]] of this [[natural transformation]] sends [[(∞,1)-colimits]] in $Y$ over $X$ to [[(∞,1)-limits]]. (\hyperlink{Lurie}{Lurie, theorem 3.1.8}) shows that the same is true for the domain. Hence $\alpha_Y$ is an equivalence for all $Y$ that appear as [[(∞,1)-colimits]] of the point. But this is the case for all [[∞-groupoids]] $Y$, by \href{limit+in+a+quasi-category#EveryInfinityGroupoidIsHomotopyColimitOfConstantFunctorOverItself}{this proposition}. \end{proof} We consider now some special cases of the general definition of local structure-topological field theory \hypertarget{StatementForCobordismsInAManifold}{}\paragraph*{{For framed cobordisms in a topological space}}\label{StatementForCobordismsInAManifold} We discuss the special case of the cobordism hypothesis for $(X,\xi)$-cobordisms (def. \ref{CatOfCobordismsWithXIStructure}) for the case that the vector bundle $\xi$ is the trivial vector bundle $\xi = \mathbb{R}^n \otimes X$. In this case $\tilde X = O(n) \times X$. Write \begin{displaymath} Bord_n^{fr}(X) := Bord_n^{(X,X \times \mathbb{R}^n)} \,. \end{displaymath} Write $\Pi(X) \in$ [[∞Grpd]] for the [[fundamental ∞-groupoid]] of $X$. \begin{prop} \label{}\hypertarget{}{} There is an [[equivalence in an (∞,1)-category|equivalence]] in [[∞Grpd]] \begin{displaymath} Fun^\otimes(Bord^{fr}_n(X), C) \simeq (\infty,n)Cat(\Pi(X), \tilde C) \simeq \infty Grpd(\Pi(X), Core(\tilde C)) \,, \end{displaymath} \end{prop} This is a special case of the \hyperlink{CobHypTheoremForTargetSpaceAndVectorBundle}{above theorem}. Notice that one can read this as saying that $Cob_n(X)$ is roughly like the [[free functor|free]] [[symmetric monoidal (∞,n)-category]] on the [[fundamental ∞-groupoid]] of $X$ (relative to $\infty$-categories of fully dualizable objects at least). \hypertarget{CobordismsWithGStructure}{}\paragraph*{{For cobordisms with $G$-structure}}\label{CobordismsWithGStructure} We discuss the special case of the cobordism hypothesis for $(X,\xi)$-bundles (def. \ref{CatOfCobordismsWithXIStructure}) for the special case of [[G-structure]] (example \ref{GStructureAsXXiStructure}), hence for the case that $X$ is the [[classifying space]] of a [[topological group]]. Let $G$ be a [[topological group]] equipped with a [[homomorphism]] $\chi : G \to O(n)$ to the [[orthogonal group]]. Notice that via the canonical linear [[representation]] $\mathbf{B}O(n) \to$ [[Vect]] of $O(n)$ on $\mathbb{R}^n$, this induces accordingly a representation of $G$ on $\mathbb{R}^n$.. Let then \begin{itemize}% \item $X := B G$ be the [[classifying space]] for $G$; \item $\xi_\chi := \mathbb{R}^n \times_G E G$ be the corresponding [[associated bundle|associated vector bundle]] to the [[universal principal bundle]] $E G \to B G$. \end{itemize} \begin{defn} \label{}\hypertarget{}{} We say \begin{displaymath} Bord^G_n := Bord_n^{(B G, \xi_\chi)} \,. \end{displaymath} is the $(\infty,n)$-category of \textbf{cobordisms with $G$-structure}. \end{defn} See (\hyperlink{Lurie}{Lurie, notation 2.4.21}) \begin{defn} \label{}\hypertarget{}{} We have \begin{itemize}% \item For $G = 1$ the trivial group, a $G$-structure is just a framing and so \begin{displaymath} Bord_n^{(1,\xi)} \simeq Bord_n^{fr} \end{displaymath} reproduces the $(\infty,n)$-category of [[framed cobordisms]], def. \ref{FramedCobordismsNCat}. \item For $G = SO(n)$ the [[special orthogonal group]] equipped with the canonical embedding $\chi : SO(n) \to O(n)$ a $G$-structure is an [[orientation]] \begin{displaymath} Bord_n^{(SO(n))} \simeq Bord_n^{or} \,. \end{displaymath} \item For $G = O(n)$ the [[orthogonal group]] itself equipped with the identity map $\chi : O(n) \to O(n)$ a $G$-structure is no structure at all, \begin{displaymath} Bord_n^{O(n)} \simeq Bord_n \,. \end{displaymath} \end{itemize} \end{defn} See (\hyperlink{Lurie}{Lurie, example 2.4.22}). Then we have the following version of the cobordism hypothesis for manifolds with $G$-structure. \begin{cor} \label{GEquivariantVersion}\hypertarget{GEquivariantVersion}{} For $G$ an [[∞-group]] equipped with a homomorphism $G \to O(n)$ to the [[orthogonal group]] (regarded as an [[∞-group]] in [[∞Grpd]]), then evaluation on the point induces an equivalence \begin{displaymath} Fun^\otimes( Bord_n^{G}, \mathcal{C} ) \simeq (\tilde {\mathcal{C}})^{G} \end{displaymath} between extended TQFTs on $n$-dimensional manifolds with [[G-structure]] and the [[∞-groupoid]] \href{invariant#ForInfinityGroupActions}{homotopy invariants} of the [[infinity-action]] of $G$ on $\tilde \mathcal{C}$ (which is induced by the evaluation on the point). \end{cor} This is (\hyperlink{Lurie}{Lurie, theorem 2.4.26}). \begin{proof} Theorem \ref{CobHypTheoremForTargetSpaceAndVectorBundle} asserts that \begin{displaymath} Fun^\otimes( Bord_n^{G}, \mathcal{C} ) \simeq Maps_{G}(E G , \tilde C) \,. \end{displaymath} Hence it remains to see that the right hand side are equivalently the [[homotopy invariants]] of the $G$-[[∞-action]]. This follows for instance with the discussion at [[∞-action]], by which \begin{displaymath} Maps_G(V,W)\simeq \infty Grpd_{/B G}(V/\!/G, W/\!/G) \,. \end{displaymath} This yields \begin{displaymath} Maps_{G}(E G , \tilde C) \simeq \infty Grpd_{/ B G}( B G, \tilde C /\!/B G ) \,. \end{displaymath} By the discussion at [[dependent product]] \begin{displaymath} \infty Grpd_{/ B G}( B G, \tilde C /\!/B G ) \simeq \underset{B G}{\prod} (\tilde C /\!/B G) \end{displaymath} which are the [[homotopy invariants]]. \end{proof} \hypertarget{ForUnorientedCobordisms}{}\paragraph*{{For (un-)oriented cobordisms}}\label{ForUnorientedCobordisms} The case that $\chi \colon G \longrightarrow O(n)$ is the identity is at the other extreme of the framed case, and turns out to be similarly fundamental. For $\mathbf{H}$ an [[(∞,1)-topos]], write $Corr_n(\mathbf{H})^\otimes$ for the [[(∞,n)-category of correspondences]] in $\mathbf{H}$. For $Phases \in DCat_n(\mathbf{H})$ an [[(∞,n)-category with duals]] [[internal (∞,n)-category|internal]] to $\mathbf{H}$, write $Corr_n(\mathbf{H}_/{Phases})^{\otimes_{phases}}$ for the [[(∞,n)-category of correspondences]] over $Phases$ and equipped with the [[phased tensor product]]. There is the [[forgetful functor|forgetful]] [[monoidal (∞,n)-functor]] \begin{displaymath} Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \longrightarrow Corr_n(\mathbf{H})^\otimes \end{displaymath} By the discussion at \emph{[[(∞,n)-category of correspondences]]} these are [[(∞,n)-categories with duals]] and the canonical $O(n)$-[[∞-action]] on them, corollary \ref{CanonicalOnAction}, is trivial for $Corr_n(\mathbf{H})$. This means that an $O(n)$-[[homotopy fixed point]] in $Corr_n(\mathbf{H})$ is just an object of $\mathbf{H}$ equipped in turn with an $O(n)$-[[∞-action]]. Therefore \begin{prop} \label{UnorientedBulkFieldTheories}\hypertarget{UnorientedBulkFieldTheories}{} Local unoriented-topological field theory \begin{displaymath} Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H})^\otimes \end{displaymath} are equivalent to objects $X \in \mathbf{H}$ equipped with an $O(n)$-[[∞-action]]. At least for $\mathbf{H} =$ [[∞Grpd]], then given such, the corresponding field theory $Z_{X/\!/O(n)}$ sends a cobordism $\Sigma$ to the space of maps \begin{displaymath} \itexarray{ \Pi(\Sigma) && \longrightarrow && X//O(n) \\ & {}_{\mathllap{T \Sigma \oplus \mathbb{R}^{n-dim(\Sigma)}}}\searrow && \swarrow \\ && B O(n) } \end{displaymath} hence \begin{displaymath} Z_{X//O(n)} \colon \Sigma \mapsto [\Pi(\Sigma),X]^{O(n)} \,. \end{displaymath} In particular this means that the assignment to the point is again $X$ itself. \end{prop} This is a slight rephrasing of the paragraph pp 58-59 in (\hyperlink{LurieTFT}{Lurie}). \begin{prop} \label{ExchangingFieldsForStructures}\hypertarget{ExchangingFieldsForStructures}{} At least for $\mathbf{H} =$ [[∞Grpd]], with $X \in \mathbf{H}$ an object equipped with an $O(n)$-[[∞-action]], then horizontal lifts in \begin{displaymath} \itexarray{ && Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \\ & \nearrow & \downarrow \\ Bord_n^\sqcup &\underset{X//O(n)}{\longrightarrow}& Corr_n(\mathbf{H})^\otimes } \end{displaymath} are equivalent to \begin{displaymath} (Bord_n^{(X//O(n), X \underset{O(n)}{\times}\mathbb{R}^n)})^{\sqcup} \longrightarrow Phases^\otimes \,. \end{displaymath} \end{prop} This is (\hyperlink{Lurie}{Lurie, prop. 3.2.8}). \begin{remark} \label{}\hypertarget{}{} Via the interpretation of local field theories with coefficients in $Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}}$ as [[schreiber:Local prequantum field theory]], the statement of prop. \ref{ExchangingFieldsForStructures} translates in quantum field theory jargon to the statement that ``All background structures are fields.'' This is essentially the slogan of [[general covariance]]. \end{remark} \begin{cor} \label{UnorientedLocalPrequantumFieldTheory}\hypertarget{UnorientedLocalPrequantumFieldTheory}{} Let $Phases^\otimes \in Ab_\infty(\mathbf{H})$ be an [[abelian ∞-group]] object, regarded as a [[(∞,n)-category with duals]] [[internal (∞,n)-category|internal]] to $\mathbf{H}$. At least if $\mathbf{H} =$ [[∞Grpd]], then local unoriented-topological field theories of the form \begin{displaymath} Bord_n^\sqcup \longrightarrow Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \end{displaymath} are equivalent to a choice \begin{enumerate}% \item of $X \in \mathbf{H}$ equipped with an $O(n)$-[[∞-action]] \item a homomorphism of $O(n)$-[[∞-actions]] $L \colon X \to Phases$ (where $Phases^\otimes$ is equipped with the canonical $\infty$-action induced from the framed cobordism hypothesis), hence (by the discussion at \emph{[[∞-action]]}) to a horizontal morphism in $\mathbf{H}$ fitting into the diagram \end{enumerate} \begin{displaymath} \itexarray{ X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n) \\ & \searrow &\swArrow_\simeq& \swarrow \\ && B O(n) } \,. \end{displaymath} \end{cor} \begin{proof} By prop. \ref{UnorientedBulkFieldTheories} the co-restriction \begin{displaymath} Bord_n^\sqcup \stackrel{Z}{\longrightarrow} Corr_n(\mathbf{H}_{/Phases})^{\otimes_{phased}} \longrightarrow Corr_n(\mathbf{H})^\otimes \end{displaymath} is equivalent to an [[∞-action]] \begin{displaymath} \itexarray{ X &\longrightarrow& X//O(n) \\ && \downarrow \\ && B O(n) } \end{displaymath} Therefore by prop. \ref{ExchangingFieldsForStructures} $Z$ is equivalent to \begin{displaymath} (Bord_n^{(X//O(n), X \underset{O(n)}{\times} \mathbb{R}^n) })^\sqcup \longrightarrow Phases^\otimes \,. \end{displaymath} Finally, by theorem \ref{CobHypTheoremForTargetSpaceAndVectorBundle} and in view of remark \ref{ActionHomomorphismsAsInfinityActionHomomorphisms}, this is equivalent to maps of the form \begin{displaymath} \itexarray{ X//O(n) && \stackrel{L//O(n)}{\longrightarrow} && Phases//O(n) \\ & \searrow && \swarrow \\ && B O(n) } \,. \end{displaymath} \end{proof} By the discussion at \emph{[[schreiber:Local prequantum field theory]]}, these statements hold also for fields with moduli spaces in more general $(\infty,1)$-toposes $\mathbf{H}$ (one sufficient condition is that $\mathbf{H}$ has an [[(infinity,1)-site]] of definition all whose objects are [[etale homotopy theory|etale contractible]]). Some examples are discussed at \emph{[[prequantum field theory]]} in the section \emph{\href{prequantum+field+theory#HigherChern-SimonsLocalPrequantumFieldTheoryLevels}{Higher Chern-Simons field theory -- Levels}}. \hypertarget{ForHQFTs}{}\paragraph*{{For HQFTs}}\label{ForHQFTs} If in def. \ref{CatOfCobordismsWithXIStructure} one chooses $X = B SO(n) \times Y$ for any topological space $Y$, and $\xi$ the [[pullback]] of the canonical vector bundle bundle on $B SO$ to $B SO \times Y$, then an $(\infty,n)$-functor $Bord^{X}_n \to C$ is similar to what Turaev calls an [[HQFT]] over $Y$. \hypertarget{ForCobordismsWithSingularities}{}\subsubsection*{{For cobordisms with singularities (boundaries/branes and defects/domain walls)}}\label{ForCobordismsWithSingularities} There is a vast generalization of the plain $(\infty,n)$-category of cobordisms (with topological structure) considered above given by allowing the [[cobordisms]] to be equipped with various types of [[singularities]] (\hyperlink{Lurie}{Lurie 09, Definition Sketch 4.3.2}). Each type of singularity in dimension $k$ now corresponds to a new generator [[k-morphisms]], and the (framed) $(\infty,n)$-category of cobordisms with singularities is now no longer the free symmetric monoidal $(\infty,n)$-category freely generated from just a point (a 0-morphisms), but freely generated from these chosen generators. This general version is (\hyperlink{Lurie}{Lurie 09, Theorem 4.3.11}). For instance if the generator on top of the point $\ast$ is a [[1-morphism]] of the form $\emptyset \to \ast$, then this defines a type of [[codimension]] $(n-1)$-[[boundary]]; and hence extended TQFTs with such boundary data and with coefficients in some symmetric monoidal $(\infty,n)$-category $\mathcal{C}$ with all dual are equivalent to choices of morphisms $1 \to A$, where $A \in \mathcal{C}$ is the fully dualizable object assigned to the point, as before, and now equipped with a morphism from the tensor unit into it. Indeed, this is the usual datum that describes [[branes]] in QFT (see for instance at [[FRS formalism]]). For more on this see at \emph{[[QFT with defects]]}. \hypertarget{ForNoncompactCobordisms}{}\subsubsection*{{For noncompact cobordisms}}\label{ForNoncompactCobordisms} One important variant of the category of cobordisms is obtained by discarding all those morphisms which have non-empty incoming (say, dually one could use outgoing) boundary component. Then a representation of this category imposes on its values ``cups but no caps'', hence only half of the data of a [[dualizable object]] in the given degree. Accordingly, in this case the cobordism hypothesis says that such a functor is given not quite by a [[fully dualizable object]], but by a weaker structure called a \emph{[[Calabi-Yau object]]} (see there for more). (\hyperlink{Lurie}{Lurie, section 4.2}) 2-dimensional TQFT of this form is known as \emph{[[TCFT]]}, see there for more \hypertarget{for_cobordisms_with_geometric_structure}{}\subsubsection*{{For cobordisms with geometric structure}}\label{for_cobordisms_with_geometric_structure} A non-topological [[quantum field theory]] is a representation of a [[cobordism category]] for cobordisms equipped with extra [[stuff, structure, property]] that is ``not just topological'', meaning roughly not of the form of def. \ref{CatOfCobordismsWithXIStructure}. The theory for this more general case is not as far developed yet. \begin{itemize}% \item steps towards classification of quantum field theories with \emph{super-Euclidean structure} are discussed at \begin{itemize}% \item [[(1,1)-dimensional Euclidean field theories and K-theory]] \item [[(2,1)-dimensional Euclidean field theories and tmf]] \end{itemize} \item a general definition of a [[cobordism category]] of cobordisms equipped with \textbf{geometric structure} given by a morphism into, roughly, a [[smooth infinity-groupoid]] of structure is discussed in (\hyperlink{Ayala}{Ayala}). \end{itemize} \hypertarget{remarks}{}\subsection*{{Remarks}}\label{remarks} \hypertarget{morphisms_of_tqfts}{}\subsubsection*{{Morphisms of TQFTs}}\label{morphisms_of_tqfts} In particular this means that $Fun^\otimes(Bord_n^{fr} , C )$ is itself an $(\infinity,0)$-category, i.e. an [[∞-groupoid]]. When interpreting symmetric monoidal functors from bordisms to $C$ as [[TQFT]]s this means that TQFTs with given codomain $C$ form a [[topological space|space]], an [[∞-groupoid]]. In particular, any two of them are either equivalent or have no morphism between them. According to [[Chris Schommer-Pries]] interesting morphisms of [[TQFT]]s arise when looking at transformations only on sub-categories on all of $Bord_n$. This is described at [[QFT with defects]] . \hypertarget{invariants_determined_from_the_point}{}\subsubsection*{{Invariants determined from the point}}\label{invariants_determined_from_the_point} The theorem does say that the invariant attached by an extended TQFT to the point determines all the higher invariants -- however it is important to notice that there are strong constraints on what is assigned to the point. For an $n$-dimensional \emph{framed} theory one needs to assign a [[fully dualizable object]], and the meaning of the term ``fully dualizable'' depends on $n$, and gets increasingly hard to satisfy as n grows.. For an $n$-dimensional unoriented theory, the object assigned to the point has to be a fixed point for the $O(n)$- action on fully dualizable objects that is obtained from the framed case of the theorem. In the 1d case, this $O(1)$ action on dualizable objects takes every object to its dual, and an $O(1)$ fixed point is indeed a vector space with a nondegenerate symmetric inner product. For an \emph{oriented} theory $n$-dimensional theory need an $SO(n)$-fixed point, which for $n=1$ is nothing but for $n=2$ ends up meaning a [[Calabi-Yau category]] (in the case the target 2-category is that of categories). In fact something more general is true: if one wants a theory that takes values on manifolds equipped with a $G$-structure, for $G$ any group mapping to $O(n)$ (such as for instance [[orientation]] already discussed or its higher versions [[Spin structure]] or [[String structure]] or [[Fivebrane structure]] or \ldots{}) one needs to assign to the point a $G$-fixed point in dualizable objects in your category (with $G$ acting through $O(n)$). This beautifully includes all the above plus for example [[manifold]]s with maps (up to [[homotopy]]) to some auxiliary (connected) space $X$ -- here we take $G$ to be the [[loop space]] $\Omega X$ of $X$ (mapping trivially to $O(n)$), so that a reduction of the structure group of the manifold to $G$ involves a map to the [[delooping]] $\mathcal{B}G \simeq X$. Such theories are classified by $X$-families of fully dualizable objects. Notice that there is an important subtlety of Lurie's theorem in the case of manifolds with $G$-structure which is easy to confuse. The general version of the theorem about TFTs does not say that they are the $G$-fixed points for the $G$-action on fully dualizable objects, but rather they are the \emph{homotopy} fixed points. This is very important because a homotopy fixed point is not just a property. It is additional structure. Depending on $G$, this additional structure is often encoded in the higher dimensional portion of the field theory. One can see this in the 1 dimensional case: there is no property of vector spaces which automatically endows them with an inner product, but it is [[stuff, structure, property|extra structure]]. \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[generalized tangle hypothesis]] \item [[conformal cobordism category]] \item [[opetopic type theory]] \end{itemize} [[!include Isbell duality - table]] \hypertarget{references}{}\subsection*{{References}}\label{references} The original hypothesis is formulated in \begin{itemize}% \item [[John Baez]], [[James Dolan]], \emph{Higher dimensional algebra and Topological Quantum Field Theory} J.Math.Phys. 36 (1995) 6073-6105 (\href{http://arxiv.org/abs/q-alg/9503002}{arXiv:q-alg/9503002}) \end{itemize} The formalization and proof is described in \begin{itemize}% \item [[Jacob Lurie]], \emph{[[On the Classification of Topological Field Theories]]}, Current Developments in Mathematics Volume 2008 (2009), 129-280 (\href{http://arxiv.org/abs/0905.0465}{arXiv:0905.0465}) \end{itemize} This is almost complete, except for one step that is not discussed in detail. But a new (unpublished) result by [[Søren Galatius]] bridges that step in particular and drastically simplifies the whole proof in general. The comparatively simple case of $n = 1$ is spelled out in detail in \begin{itemize}% \item [[Yonatan Harpaz]], \emph{The Cobordism Hypothesis in Dimension 1} (\href{http://arxiv.org/abs/1210.0229}{arXiv:1210.0229}) \end{itemize} and aspects of the case $n = 2$ (see also at \emph{[[2d TQFT]]}) are discussed in \begin{itemize}% \item [[Chris Schommer-Pries]], \emph{The Classification of Two-Dimensional Extended Topological Field Theories} (\href{http://arxiv.org/abs/1112.1000}{arXiv:1112.1000}) \end{itemize} Lecture notes and reviews on the topic of the cobordisms hypothesis include \begin{itemize}% \item [[Jacob Lurie]], \emph{TQFT and the Cobordism Hypothesis} (\href{http://www.ma.utexas.edu/video/dafr/lurie/}{video}, \href{http://www.ma.utexas.edu/users/plowrey/dev/rtg/notes/perspectives_TQFT_notes.html}{notes}) \item [[Julie Bergner]], \emph{[[UC Riverside Seminar on Cobordism and Topological Field Theories]]} (2009). \item [[Julie Bergner]], \emph{Models for $(\infty,n)$-Categories and the Cobordism Hypothesis} , in [[Hisham Sati]], [[Urs Schreiber]] (eds.) \emph{[[schreiber:Mathematical Foundations of Quantum Field and Perturbative String Theory]]}, AMS 2011 \item [[Daniel Freed]], \emph{The cobordism hypothesis}, , Bulletin of the American Mathematical Society 50 (2013), pp. 57-92, (\href{http://arxiv.org/abs/1210.5100}{arXiv:1210.5100}) \item [[Chris Schommer-Pries]], \emph{Dualizability in Low-Dimensional Higher Category Theory} (\href{http://arxiv.org/abs/1308.3574}{arXiv:1308.3574}) \item [[Constantin Teleman]], \emph{Five lectures on topological field theory}, 2014 (\href{http://math.berkeley.edu/~teleman/math/barclect.pdf}{pdf}) \end{itemize} An approach to the proof of the cobordism hypothesis via [[factorization homology]] is in \begin{itemize}% \item [[David Ayala]], [[John Francis]], \emph{The cobordism hypothesis}, (\href{https://arxiv.org/abs/1705.02240}{arXiv:1705.02240}) \end{itemize} Discussion of the canonical $O(n)$-action on [[n-fold loop spaces]] (which may be thought of as a special case of the cobordism hypothesis) includes \begin{itemize}% \item Gerald Gaudens, [[Luc Menichi]], section 5 of \emph{Batalin-Vilkovisky algebras and the $J$-homomorphism}, Topology and its Applications Volume 156, Issue 2, 1 December 2008, Pages 365--374 (\href{http://arxiv.org/abs/0707.3103}{arXiv:0707.3103}) \end{itemize} Cobordisms with geometric structure are discussed in \begin{itemize}% \item [[David Ayala]], \emph{Geometric cobordism categories} (\href{https://arxiv.org/abs/0811.2280}{arXiv:0811.2280}) \end{itemize} [[!redirects cobordism theorem]] \end{document}