\documentclass[11pt,reqno]{amsart}
\usepackage{amssymb,amsmath,mathrsfs,xspace,multicol}
\usepackage{amscd}
\usepackage[all]{xy}
%\usepackage[mathscr]{eucal}
\usepackage[breaklinks=true]{hyperref}%To get clickable links to papers


%%% SPRING 2007 SYMBOLS %%%

\newcommand{\hhxy}{\ensuremath{\hhom_{\calc}(x,y)}\xspace}
\newcommand{\cinf}{\ensuremath{\calc^\infty}\xspace}
\newcommand{\morcts}{\ensuremath{\morc {}_t\times_s \morc}\xspace}
\newcommand{\xfgy}{\ensuremath{X {}_f\times_g Y}\xspace}
\newcommand{\mfg}{\ensuremath{\mathfrak{g}}\xspace}
\newcommand{\cald}{\ensuremath{\mathcal{D}}\xspace}
\newcommand{\mapdef}[1]{\ensuremath{\overset{#1}{\longrightarrow}}\xspace}
\newcommand{\cech}{\v{C}ech }
\DeclareMathOperator{\trans}{\ensuremath{Trans}\xspace}
\DeclareMathOperator{\hol}{\ensuremath{hol}\xspace}
\DeclareMathOperator{\codisc}{\ensuremath{Codisc}\xspace}
\DeclareMathOperator{\disc}{\ensuremath{Disc}\xspace}

%%% WINTER 2007 SYMBOLS %%%

\newcommand{\hh}{\ensuremath{\widehat{H}}\xspace}
\newcommand{\zh}{\ensuremath{Z_{\hbar}}\xspace}
\newcommand{\D}{\ensuremath{\mathscr{D}}\xspace}
\newcommand{\scrp}{\ensuremath{\mathscr{P}}\xspace}
\newcommand{\calc}{\ensuremath{\mathcal{C}}\xspace}
\newcommand{\obc}{\ensuremath{{\rm Ob}(\mathcal{C})}\xspace}
\newcommand{\morc}{\ensuremath{{\rm Mor}(\mathcal{C})}\xspace}
\newcommand{\pq}{\ensuremath{P_{q_0 \rightarrow q_1}}\xspace}
\newcommand{\pt}{\ensuremath{P_{\bf t}}\xspace}
\newcommand{\ppx}{\ensuremath{P_{x_0 \rightarrow x_1}}\xspace}
\newcommand{\ltq}{\ensuremath{L^2(Q)}\xspace}
\newcommand{\rmin}{\ensuremath{\mathbb{R}^{min}}\xspace}
\newcommand{\grad}{\ensuremath{\overset{\longrightarrow}{\nabla}}\xspace}
\newcommand{\dd}{\ensuremath{\Delta}\xspace}
\newcommand{\tphipsi}{\ensuremath{\langle \phi, \psi \rangle}\xspace}
\newcommand{\twopi}{\ensuremath{\frac{1}{\sqrt{2 \pi}}}\xspace}
\newcommand{\schr}{{Schr\"odinger}\xspace}
\newcommand{\eish}[1]{\ensuremath{e^{i S(#1) / \hbar}}\xspace}
\newcommand{\diff}[1]{\ensuremath{\frac{\Delta x_{#1}}{\Delta t_{#1}}}\xspace}
\newcommand{\tangle}[1]{\ensuremath{\langle #1 \rangle}\xspace}
\DeclareMathOperator{\im}{\ensuremath{im}\xspace}
\DeclareMathOperator{\hhom}{\ensuremath{Hom}\xspace}
\DeclareMathOperator{\fun}{\ensuremath{Fun}\xspace}
\DeclareMathOperator{\Tor}{\ensuremath{Tor}\xspace}
\DeclareMathOperator{\Ob}{\ensuremath{Ob}\xspace}
\DeclareMathOperator{\Mor}{\ensuremath{Mor}\xspace}
\DeclareMathOperator{\vectc}{\ensuremath{Vect_{\mathbb{C}}}\xspace}

%%% FALL 2006 SYMBOLS %%%

\newcommand{\aaa}{\ensuremath{\alpha}\xspace}
\newcommand{\aaat}{\ensuremath{\widetilde{\alpha}}\xspace}
\newcommand{\la}{\ensuremath{\lambda}\xspace}
\newcommand{\s}{\ensuremath{\Sigma}\xspace}
\newcommand{\g}{\ensuremath{\gamma}\xspace}
\newcommand{\gt}{\ensuremath{\widetilde{\gamma}}\xspace}
\newcommand{\gd}{\ensuremath{\dot{\gamma}}\xspace}
\newcommand{\qd}{\ensuremath{\dot{q}}\xspace}
\newcommand{\omt}{\ensuremath{\widetilde{\omega}}\xspace}
\newcommand{\px}{\ensuremath{P_{x_0 \rightarrow x_1} X}\xspace}
\newcommand{\ppm}{\ensuremath{P_{m_0 \rightarrow m_1} M}\xspace}
\newcommand{\N}{\ensuremath{\mathbb{N}}\xspace}
\newcommand{\Z}{\ensuremath{\mathbb{Z}}\xspace}
\newcommand{\R}{\ensuremath{\mathbb{R}}\xspace}
\newcommand{\C}{\ensuremath{\mathbb{C}}\xspace}
\newcommand{\ap}{\ensuremath{A_\wp}\xspace}
\newcommand{\To}{\ensuremath{\Rightarrow}\xspace}
\newcommand{\Iff}{\ensuremath{\Leftrightarrow}\xspace}
\newcommand{\ddt}{\ensuremath{\frac{d}{dt}}\xspace}
\newcommand{\intt}{\ensuremath{\int_{t_0}^{t_1}}\xspace}
\newcommand{\wpt}{\ensuremath{\bigwedge{}^p}\xspace}
\newcommand{\ega}{\ensuremath{e \int_{\gamma} A}\xspace}
\newcommand{\del}[1]{\ensuremath{\frac{\partial}{\partial #1}}\xspace}
\newcommand{\der}[1]{\ensuremath{\frac{d #1}{ds}}\xspace}
\newcommand{\dell}[1]{\ensuremath{\frac{\partial L}{\partial #1}}\xspace}
\newcommand{\delh}[1]{\ensuremath{\frac{\partial H}{\partial #1}}\xspace}
\newcommand{\delg}[1]{\ensuremath{\frac{\partial \gamma(x)}{\partial
s_{#1}}}\xspace}
\newcommand{\ph}[1]{\ensuremath{e^{i #1}}\xspace}
\newcommand{\smiley}{\ensuremath{\ddot{\smile} }}


\DeclareMathOperator{\vol}{\ensuremath{vol}\xspace}
\DeclareMathOperator{\Vol}{\ensuremath{Vol}\xspace}
\DeclareMathOperator{\Ph}{\ensuremath{Phase}\xspace}
\DeclareMathOperator{\area}{\ensuremath{area}\xspace}

\theoremstyle{plain}
\newtheorem{theoremeq}{Theorem}[section]
\newtheorem{prop}[theoremeq]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theoremeq]{Definition}
%\newtheorem{stand}[theoremeq]{Standing Assumption}
\newtheorem{remark}[theoremeq]{Remark}

\begin{document}
\title[Course notes on Quantization and Cohomology]{Course notes on
Quantization and Cohomology, Spring 2007}

\author[John Baez]{John Baez*\\ Notes by Apoorva Khare*}
\footnotetext{$^*$Department of Mathematics, University of California at
Riverside, USA}
\maketitle

\tableofcontents
\newpage

\addtocontents{toc}{\protect\vspace{2ex}}
\section{Preface}

These are lecture notes taken at U.C.\ Riverside, in the Tuesday lectures of
John Baez's Quantum Gravity Seminar, Spring 2007.
Apoorva Khare typed notes and
Christine Dantas prepared figures based on handwritten notes by Derek Wise.
This is the third and final quarter of a year-long course.
You can find the latest version of notes for all three quarters here:

\begin{center}
\href{http://math.ucr.edu/home/baez/qg-spring2007/index.html#quantization}{\tt
http://math.ucr.edu/home/baez/qg-spring2007/}
\end{center}

\noindent
If you see typos or other problems with any of these notes, please let
John Baez know ({\tt baez@math.ucr.edu}).

\newpage

\addtocontents{toc}{\protect\vspace{2ex}}
\section{Apr 3, 2007: Smooth categories}

We've been starting with a category $\calc$ equipped with an ``action"
functor
\[ S : \calc \to (\R, +) \]

\noindent and doing classical and quantum mechanics using this.

To do quantum mechanics, we formed
\[ e^{iS} : \calc \to (U(1), \cdot) \]

\noindent and did path integrals - integrals over objects and/or
morphisms of $\calc$ - so we needed something like a measure space (or
perhaps a generalized measure space) of objects and a (generalized)
measure space of morphisms.

In classical mechanics, instead of integrating, we {\it minimize} the
action, or find critical points of the action

\begin{remark}
Note that minimization requires {\it no} extra structure on $\calc$,
whereas finding critical points requires something like a ``smooth
structure" on the sets $\obc$ of objects and (especially) $\morc$ of
morphisms. For example, we might want $\obc, \morc$ or each $\hhxy
\subset \morc$ to be a smooth manifold. If $\hhxy$ were a manifold, we
could demand that $S : \hhxy \to \R$ be smooth, and look for $\g \in
\hhxy$ with
\[ dS(\g) = 0 \]

\noindent Alas, in examples, $\hhxy$ is usually a more general ``smooth
space".\medskip
\end{remark}


\subsection{Example: Path groupoids}

The path groupoid $PM$ of a manifold $M$ has
\begin{itemize}
\item points of $M$ as objects: Ob$(PM) = M$.

\item thin homotopy classes of smooth paths $\g : [0,1] \to M$ which are
constant near 0 and 1, as morphisms.
\end{itemize}

Here, a {\it thin homotopy} between paths $\g_0, \g_1 : [0,1] \to M$ is a
smooth map:
\[ H : [0,1] \times [0,1] \to M \]

\noindent such that
\begin{enumerate}
\item $H(t,0) \equiv \g_0(t)$. 

\item $H(t,1) \equiv \g_1(t)$. (These two together explain the homotopy.)

\item $H(0,s)$ is independent of $s$.

\item $H(1,s)$ is independent of $s$.\medskip

\noindent and

\item $H$ is {\it thin}, i.e. rank$(dH) \leq 1$ (i.e. the homotopy
``sweeps out no area"), or equivalently, $\det(dH) = 0\ \forall (t,s)$.
\end{enumerate}

Thin homotopies can do this:

\vspace{4ex}
\noindent [figure: point isomorphically maps to two points connected by
an edge ``traveling" both ways]
\vspace{4ex}

\noindent Also, any reparametrization is a thin homotopy, so the obvious
composition of morphisms in $PM$ is associative and has identities and
inverses (trace ``backwards" along any path, staying near the original
point for some time at the beginning and at the end, to get the ``thin"
inverse path, or a composite path that is thin homotopic to the identity
map - one that stays at the original point throughout.)

Thus, $PM$ is a groupoid.\medskip

If $\aaa$ is a 1-form on $M$, we get a functor
\[ S : PM \to (\R, +) \]

\noindent sending every object of $PM$ to the one object of $(\R, +)$,
and every morphism $[\g]$ to
\[ S([\g]) := \int_{\g} \aaa \in \R \]

\noindent Note that if $\g_0, \g_1$ are thinly homotopic,
then
\[ \int_{\g_0} \aaa - \int_{\g_1} \aaa = \int_H d \aaa = 0 \]

\noindent since $\det(dH) = 0$. Thus $S$ is well-defined.\medskip


\subsection{Introduction to smooth categories}

This kind of example arises all over in classical mechanics, especially
when $M$ is a cotangent bundle and $\aaa$ is the canonical 1-form.

We would like to say that $PM$ is a ``smooth category", $S : PM \to (\R,
+)$ is a ``smooth functor", and $\hhom_{PM}(x,y) \subset$ Mor$(PM)$ is a
``smooth space", so that we can find critical points.\medskip

Suppose we knew what ``smooth spaces" and ``smooth maps between smooth
spaces" were. They'd better form a category, $\cinf$, say. Then, what's a
``smooth category" and a ``smooth functor"?\medskip

A smooth category $\calc$ should have
\begin{enumerate}
\item a smooth space of objects: $\obc \in \cinf$.

\item a smooth space of morphisms: $\morc \in \cinf$.

\item a smooth identity-assigning map $i : \obc \to \morc$, i.e. $i$is a
morphism in $\cinf$.

\item smooth source and target maps $s,t : \morc \to \obc$.

\item a smooth ``composition" map:
\[ \circ : \morcts \to \morc\]

\noindent where $\morcts$ is the smooth space of composable pairs of
morphisms in $\calc$ (i.e. $\{ (f,g) : t(f) = s(g) \}$ - so that $g \circ
f =\ \overset{f}{\to}\ \overset{g}{\to}$ is defined), i.e. the pullback
of the diagram
\[ \begin{CD}
\; @. \morc\\
@.    @VsVV\\
\morc @>t>> \obc
\end{CD} \]

\noindent i.e. the universal object making
\[ \begin{CD}
\morcts @>\pi_1>> \morc\\
@V\pi_2VV    @VsVV\\
\morc @>t>> \obc
\end{CD} \]

\noindent commute. Here, $\pi_1, \pi_2$ denote the projection to the
first/second component respectively. (So $\cinf$ should have pullbacks or
at least {\it this} pullback. The category of smooth manifolds and smooth
maps doesn't have pullbacks!)\medskip

\item the associative law, written as a commutative diagram in $\cinf$.
In other words, the following diagram commutes:
\[ \begin{CD}
\morcts {}_t\times_s \morc @>1 \times \circ>> \morcts\\
@V\circ \times 1 VV    @V\circ VV\\
\morcts @>\circ >> \morc
\end{CD} \]

\item the left and right unit laws.

\item the source and target of a composite morphism, to be what they {\it
should} be.
\end{enumerate}
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{Apr 10, 2007: Smooth spaces}

Last time we posed a question: what's a ``smooth functor" $S : \calc \to
\R$?

\noindent For this, we need $\calc$ to be a ``smooth category", which we
succeeded in defining, {\it given} a category $\cinf$ of ``smooth spaces"
and ``smooth maps". More generally, given a category $K$, we can define a
{\it category in} $K$ (say $\calc$) to consist of
\begin{itemize}
\item $\obc \in K$
\item $\morc \in K$
\item $s,t : \morc \to \obc$ in $K$, i.e. $s,t \in \hom_K(\morc, \obc)$
\item $i : \obc \to \morc$ in $K$
\item $\circ : \morcts \to \morc$ in $K$
\end{itemize}\medskip

\noindent satisfying the usual category theory axioms.

This trick - taking a definition and replacing sets and functions with
objects and morphisms of $K$ - is called {\it internalization} (in $K$).
The result is often called an {\it internal category} (in $K$) or simply
a {\it category in $K$}.

This works best if $K$ has pullbacks, so we can write ``$\morcts$" and
know it's defined.\medskip

(We now look at examples.) The category $K =$ {\tt Set} has pullbacks:
\[ \begin{CD}
\xfgy @>\pi_1>> X\\
@V\pi_2VV    @VfVV\\
Y @>g>> Z
\end{CD} \]

\noindent where in this case,
\[ \xfgy :=  \{ (x,y) \in X \times Y : f(x) = g(y) \} \]

The category {\tt Diff} of smooth manifolds and smooth maps doesn't have
pullbacks. For a rough idea why, consider
\[ \begin{CD}
\{ (x,y) : f(x) = g(y) \} @>\pi_1>> X=\R\\
@V\pi_2VV    @VfVV\\
Y=\R @>g>> Z
\end{CD} \]

\noindent where $g(x) \equiv 0$, and
\[ f(x) =
  \begin{cases}
    e^{-1/x}, 	&\text{if}\ x > 0;\\
    0,		&\text{if}\ x \leq 0.\\
  \end{cases}
\]

\vspace{5ex}
[figure: the graph of $f(x)$]
\vspace{5ex}

\noindent Here, $\{ (x,y) : f(x) = g(y) \} = \{ (x,y) : x \leq 0 \}$ is a
manifold with boundary - {\it not} a smooth manifold.\medskip

\noindent {\it A more sophisticated example}. $f(x) = 0$ only on the
Cantor set, $g(x) \equiv 0$.

\vspace{5ex}
[figure: graph of such an $f$]
\vspace{5ex}

So, let's introduce a bigger category of smooth spaces that {\it does}
have pullbacks, following Grothendieck's dictum: {\it a nice category
with some bad objects is better than a bad category with only nice
objects}. He used this to invent ``schemes", generalizing ``algebraic
varieties", and revolutionizing algebraic geometry.

Let's do the same for differential geometry. So, what's a smooth space?
Following Chen's ideas:\medskip

\begin{definition}\hfill
\begin{enumerate}
\item A {\it convex set} is a convex subset of $\R^n$. For example,

\vspace{8ex}
[mini-figures: $\R^n$, a half-space, a quarter-space, etc.]
\vspace{8ex}

\noindent (Note: Convex sets need not be open.)\medskip


\item A function $f : C \to C'$ between convex sets is {\it smooth} if it
has continuous $n$th derivatives for all $n \geq 0$, defined in the usual
way.\medskip


\item A {\it smooth space} is a set $X$ equipped with, for each convex
set $C$, a set of {\it plots}
\[ \varphi : C \to X \]

\noindent (which we think of as ``smooth"), so that
\begin{enumerate}
\item Given a plot $\varphi : C \to X$ and a smooth map $f : C' \to C$
between convex sets, $\varphi \circ f : C' \to X$ is a plot.

\item Suppose we are given inclusions $i_\aaa : C_\aaa \to C$ such that
$\{ C_\aaa \}$ is an open cover of $C$, and a (set) map $\varphi : C \to
X$. If $\varphi \circ i_\aaa : C_\aaa \to X$ is a plot for each $\aaa$,
then $\varphi : C \to X$ is a plot.

\item Every map from a point (in $\R^n$) to $X$ is a plot.
\end{enumerate}\medskip


\item Given smooth spaces $X,Y$, a function $f : X \to Y$ is a {\it
smooth map} if, for every plot $\varphi : C \to X$, the composite $f
\circ \varphi : C \to Y$ is a plot in $Y$.\medskip

\item The category of smooth spaces and smooth maps above, is denoted by
$\cinf$. (Note that we need to verify that smooth maps compose to give
smooth maps - but this is easy.)
\end{enumerate}\medskip
\end{definition}
\newpage

\noindent {\bf Examples:}
\begin{enumerate}
\item Any convex set $C$ becomes a smooth space, where the plots $\varphi
: C' \to C$ are just the smooth maps (as defined earlier).

\item Any set $X$ has a {\it discrete} smooth structure such that the
plots $\varphi : C \to X$ are just the constant functions.

\item Any set $X$ has an {\it indiscrete} smooth structure where every
function $\varphi : C \to X$ is a plot.

\item Any smooth manifold $X$ becomes a smooth space where $\varphi : C
\to X$ is a plot if and only if $\varphi$ is smooth in the usual sense of
smooth manifolds. Moreover, if $X,Y$ are manifolds, $f : X \to Y$ is
smooth according to our new definition if and only if it's smooth in the
usual sense.
\end{enumerate}\medskip


We now prove some functoriality results in $\cinf$.

\begin{theoremeq}
$\cinf$ has products and coproducts, and is closed under taking subsets.
\end{theoremeq}

\begin{proof}
The product $X \times Y$ of smooth spaces becomes a smooth space where a
function $\varphi : C \to X \times Y$ is a plot if and only if the
composites $C \mapdef{\varphi} X \times Y \mapdef{p_1} X$ and $C
\mapdef{\varphi} X \times Y \mapdef{p_2} Y$ are plots. Thus $\cinf$ has
products (note that the maps $p_1, p_2 : X \times Y \to X,Y$
respectively, are indeed smooth maps, and hence in $\cinf$).\medskip


Similarly, the disjoint union $X+Y := X \coprod Y$ of smooth spaces,
becomes a smooth space where a function $\varphi : C \to X+Y$ is a plot
iff either

\vspace{5ex}
[commutative diagram : $C \mapdef{\varphi_1} X \mapdef{i_1} X+Y$ equals
$\varphi : C \to X+Y$]
\vspace{5ex}

\noindent commutes for some plot $\varphi_1 : C \to X$, or

\vspace{5ex}
[commutative diagram : $C \mapdef{\varphi_2} Y \mapdef{i_2} X+Y$ equals
$\varphi : C \to X+Y$]
\vspace{5ex}

\noindent commutes for some plot $\varphi_2 : C \to Y$. Thus $\cinf$ has
coproducts. (Once again, the maps $: X,Y \to X+Y$ are smooth by
definition.)\medskip


Finally, any subset $Y \subset X$ of a smooth space $X$ becomes a smooth
space (and the inclusion a smooth map), where a plot $\varphi : C \to Y$
is any function such that
\[ C \mapdef{\varphi} Y \hookrightarrow X \]

\noindent is a plot in $X$. (So for example, the Cantor set is a smooth
space.)
\end{proof}\medskip


\noindent {\bf Homework.} Show the following result.

\begin{theoremeq}
$\cinf$ has pullbacks.
\end{theoremeq}

\begin{proof}
Given a pair of morphisms in $\cinf$
\[ \begin{CD}
\; @.    X\\
@.       @VfVV\\
Y  @>g>> Z
\end{CD} \]

\noindent define $\xfgy$ to be
\[ \xfgy := \{ (x,y) \in X \times Y : f(x) = g(y) \} \subset X \times Y
\]

\noindent By the above theorem, this is a smooth space. We now show that
$\xfgy \mapdef{p_1} X$ is a smooth map (the projection to $Y$ is also
smooth for similar reasons). Clearly, this map is a composite of the
inclusion map $: \xfgy \hookrightarrow X \times Y$ and $p_1 : X \times Y
\to X$ - but both of these maps are smooth (as shown above).

To complete the proof, we show that $\xfgy$ satisfies the universal
property of pullbacks: given $W \in \cinf$ so that the diagram
\[ \begin{CD}
W @>r_X>>    X\\
@Vr_YVV       @VfVV\\
Y  @>g>> Z
\end{CD} \]

\noindent commutes in $\cinf$, we form $W \mapdef{r_X \oplus r_Y} X
\oplus Y = X \times Y$. It can be seen that $r_X, r_Y, r_X \oplus r_Y$
are all smooth maps, and (set-theoretically, for instance, that) $\im(r_X
\oplus r_Y) \subset \xfgy$, so that the obvious diagram for universality,
now commutes.
\end{proof}
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{Apr 17, 2007: The path groupoid of a smooth space}

We've defined a notion of ``smooth space" - a set $X$ equipped with a
collection of plots $\varphi : C \to X$ for convex sets $C$, satisfying
some properties. We've also defined a notion of ``smooth map" - a
function $f : X \to Y$ (with $X,Y$ smooth spaces) such that
\[ \varphi : C \to X \mbox{ is a plot} \]

\noindent implies that
\[ f \circ \varphi : C \to Y \mbox{ is a plot.} \]

We claimed that there's a category of smooth spaces and smooth maps. We
listed a bunch of nice properties of $\cinf$; here are some more.
\begin{enumerate}
\item (This is dual to the notion of subsets.) Any {\it quotient} $Y = X
/ \sim$, where $X$ is a smooth space and $\sim$ is an equivalence
relation on $X$, becomes a smooth space, where a plot in $Y$ is a
composite
\[ C \mapdef{\varphi} X \mapdef{p} Y = X /\sim \]

\noindent where $\varphi$ is any plot in $X$.
\begin{itemize}
\item This definition guarantees that the quotient map $p : X \to Y$ is
smooth.

\item This definition also lets us see that $\cinf$ has {\it pushouts}:
\[ \begin{CD}
Z @>f>>    X\\
@VgVV      @Vi_1VV\\
Y  @>i_2>> X \coprod_\circ Y
\end{CD} \]

\noindent where the {\it pushout} $X \coprod_\circ Y$ is defined to be
$(X \coprod Y)/ \{ i_1(f(z)) = i_2(g(z)) : z \in Z \}$.

The pushout is a quotient of a disjoint union $X + Y = X \coprod Y$ (i.e.
the coproduct).
\end{itemize}\medskip


\item $\cinf$ is {\it Cartesian-closed}: that is, given smooth spaces $X$
and $Y$, the set
\[ \cinf(X,Y) := \{ f : X \to Y : f \mbox{ is smooth } \} \]

\noindent can be made into a smooth space such that there's a natural
isomorphism
\[ \cinf(X \times Y, Z) \cong \cinf(X, \cinf(Y,Z)) \]

\noindent under $f \mapsto \widetilde{f}$, where in this case,
\[ \widetilde{f}(x)(y) = f(x,y). \]

\noindent To do this, we define a plot in $\cinf(Y,Z)$, say
\[ \varphi : C \to \cinf(Y,Z) \]

\noindent to be any map of the form $\widetilde{f}$, where
\[ f : C \times Y \to Z \]

\noindent is smooth.%using currying in $Set$ ??

(Note that one must check that $\cinf(Y,Z)$ really is a smooth space and
that then we get a one-to-one and onto map
\[ \cinf(X \times Y, Z) \cong \cinf(X, \cinf(Y,Z)) \]

\noindent which turns out to be smooth.)
\end{enumerate}\medskip

Using these properties above, we can make a smooth category $PX$, the
{\it path groupoid} of any smooth space $X$. This has
\begin{itemize}
\item a smooth space of objects $\Ob(PX) = X$.

\item a smooth space of morphisms $\Mor(PX) = \cinf([0,1],X)/\sim$, where
$[0,1]$ is smooth since it's a convex set, $\cinf([0,1], X)$ is smooth by
Cartesian closedness, and modding out by ``thin homotopy" gives a smooth
space $\cinf([0,1],X)/\sim$. We also need to check that the source,
target, identity-assigning, and composition maps are smooth.
\end{itemize}\medskip

Next, we want to understand smooth functors
\[ S : PX \to (\R, +) \]

\noindent since such a functor describes the ``action" for some classical
system.

\begin{definition}
Given smooth categories $\calc$ and $\cald$, a {\it smooth functor} is
\begin{itemize}
\item a smooth map $F : \obc \to \Ob(\cald)$
\item a smooth map $F : \morc \to \Mor(\cald)$
\end{itemize}

\noindent such that the usual properties of a functor hold, written as
commutative diagrams:
\begin{enumerate}
\item $F$ preserves the source for morphisms:
\[ \begin{CD}
\morc @>F>> \Mor(\cald)\\
@Vs_CVV     @Vs_DVV\\
\obc  @>F>> \Ob(\cald)
\end{CD} \]

\noindent and similarly,
\item $F$ preserves targets.

\item $F$ preserves the identity-assigning map.

\item $F$ preserves composition.
\end{enumerate}
\end{definition}

\noindent (Just as you can define ``categories in $K$" for any category
$K$ with pullbacks, you can define ``functors in $K$".)\medskip


\begin{theoremeq}
For any smooth space $M$, there's a one-to-one correspondence between
smooth functors
\[ S : PM \to (\R, +) \]

\noindent and 1-forms $A$ on $M$ (defined below), given by
\[ S([\g]) = \int_{\g} A. \]
\end{theoremeq}

\noindent where we have

\begin{definition}
A $p$-{\it form} $\aaa$ on a smooth space $X$ assigns to each plot
$\varphi : C \to X$ a $p$-form $\aaa_\varphi$ on $C$, such that given a
smooth map $f : C'\to C$ between convex sets,
\[ f^* \aaa_\varphi = \aaa_{\varphi \circ f} \]

\noindent where $f^*$ is the operation of pulling back $p$-forms from $C$
to $C'$ along $f : C' \to C$.
\end{definition}\medskip

\noindent (For a proof of the theorem, see {\it Higher Gauge Theory - II}
by Baez and Schreiber.)
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{May 1, 2007: Smooth functors and beyond}

We've seen that if $M$ is a smooth space (e.g. a manifold), then there is
a smooth category $PM$, where
\begin{itemize}
\item objects are points $x \in M$;
\item morphisms $\g : x \to y$ are thin homotopy classes of smooth maps
$f : [0,1] \to M$ with $f(0) = x, f(1) = y$, and $f$ constant near $0,1$.
\end{itemize}\medskip

We think of $PM$ as a category of ``configurations" and ``processes" for
some physical system. So to formulate the Lagrangian approach to the
physics of this system, we need a smooth functor
\[ S : PM \to \R \]

\noindent describing the ``action" of any process. Last time, we saw a
Theorem that gave an explicit one-to-one correspondence between smooth
functors $S : PM \to \R$ and 1-forms $A$ on $M$, via
\[ S([\g]) = \int_{\g} A \]

\noindent where we pick up a representative (path) for $\g$ to define the
integral.\medskip

Alas, this isn't general enough... as we'll soon see.\medskip


\subsection{Quantum physics and circle-bundles}

To do quantum physics, what matters is not
\[ S : PM \to \R \]

\noindent but the phase
\[ e^{iS} : PM \to U(1) \]

\noindent which has less information, since $\exp : \R \to U(1)$ is
many-to-one. In fact, $e^{iS}$ is also sufficient to do classical
physics!

\vspace{2ex}
\noindent [figure: path from $x$ to $y$ with ``perturbations" in the
middle, and accompanied by the equation: $\qquad \displaystyle
\frac{d}{ds} S(\g_s) = 0$]
\vspace{2ex}

If we seek {\it critical points} of the action (instead of minima), we
can work with $e^{iS}$ instead of $S$:
\[ \frac{d}{ds} e^{i S(\g_s)} = 0 \]

\noindent (for all smooth homotopies $\g_s$ of $\g$, that hold the
endpoints fixed). This is because the critical points of $e^{iS}$ are the
same as those of $S$.

So this doesn't seem like a big deal.\medskip

\begin{theoremeq}
There's a one-to-one correspondence between smooth functors
\[ P : PM \to U(1) \]

\noindent and 1-forms $A$ on $M$, given by
\[ P([\g]) = e^{i \int_\g A} \]
\end{theoremeq}

\noindent Here, $P$ stands for ``phase". So far the picture looks like:

\vspace{8ex}
\noindent [figure: cylinder along $x \to y$ on the $X$-axis, which plots
points of $M$, and there's a path from one circular face of the cylinder
(at $x$) to the other circular face (at $y$)]
\vspace{8ex}

For each point $x \in M$, we have a circle of possible phases for the
system in configuration $x$, so we have a ``trivial principal
$U(1)$-bundle":
\[
 \begin{CD}
  M \times U(1) \qquad @.\ni\qquad (x,\aaa)\\
  @V\pi_1 VV    \qquad\             @V\pi_1 VV\\
  M             @.\ni \qquad        x\\
 \end{CD}
\]

\noindent Sitting over $x \in M$, we have a {\it fiber}
\[ \pi_1^{-1}(x) \subset M \times U(1) \]

\noindent which is a circle - the set of possible phases our system could
have at $x$.

This example is called ``trivial" because each fiber {\it is} $U(1)$ - or
is canonically isomorphic to $U(1)$:
\[ \pi_1^{-1}(x) = \{ (x, \aaa) : \aaa \in U(1) \} \mapdef{\sim} U(1) \]

\noindent where the isomorphism sends $(x, \aaa)$ to $\aaa$.\medskip

More interesting are the {\it nontivial} principal $U(1)$-bundles:

\vspace{8ex}
[figure: sphere, $M = S^2 = \C P^1$, with fiber described below]
\vspace{8ex}

\noindent For example, let the fiber over $x$ be the set of points in
$S^2$ that are perpendicular to $x$. Then we can't smoothly identify all
the fibers with $U(1)$, since that would produce a nowhere vanishing
smooth vector field on $S^2$!

\vspace{8ex}
[figure: the figure of a possible(?!) sphere with such a vector field]
\vspace{8ex}


\subsection{Torsors and principal bundles}

Now let's get a bit more formal. What's the difference between {\it a}
circle and {\it the} circle? The circle is $U(1) \subset \C$. {\it A}
circle is a ``$U(1)$-torsor" - a copy of $U(1)$ that's forgotten what the
element 1 is. (We now recall lots of stuff from the end of last quarter.)

\begin{definition}
For any group $G$, a $G$-{\it torsor} is a set $X$ equipped with an
action (a right action) of $G$:
\begin{eqnarray*}
\aaa : X \times G & \to & X\\
(x, g) & \mapsto & xg
\end{eqnarray*}

\noindent so that $x1 = x$, and $(xg)h = x(gh)$, and such that $X$ is
isomorphic to $G$ as a space with a right $G$-action.
\end{definition}

\noindent In other words, there's a bijection
\[ \beta : X \to G \]

\noindent so that
\[ \beta(xg) = \beta(x) g \]

If $G = U(1)$, the difference between right and left action is
inessential, since $U(1)$ is abelian. More importantly, any circle
equipped with the ability to rotate it by any phase $g \in U(1)$, is a
$U(1)$-torsor.

\vspace{8ex}
\noindent [figure: circle with a fixed point $x$, and a varying point $y$
on the circle orthogonal to it; so $P \mapdef{\pi} M$ with $(x,y) \mapsto
x$]
\vspace{8ex}

\noindent If a point in $P$ is a point $x \in S^2$ together with a point
in the circle perpendicular to $x$, and $\pi : P \to M$ is the obvious
map, then $\pi^{-1}(x)$ is a $U(1)$-torsor.

More precisely, $\pi^{-1}(x)$ {\it becomes} a $U(1)$-torsor after we pick
a ``right- or left- hand rule" for rotating $y \in \pi^{-1}(x)$ by a
phase $g \in U(1)$.\medskip

Getting back to principal $U(1)$-bundles, they are (among other things)
smooth spaces $P$ with a smooth map
\[ \begin{CD} P\\ @V\pi VV\\M\\ \end{CD} \]

\noindent such that each fiber $\pi^{-1}(x)$ (where $x \in M$) is
equipped with the structure of being a $U(1)$-torsor.\medskip

Next time we'll {\it really} define a ``principal $U(1)$-bundle", and
include a clause saying that the $U(1)$-torsor structure on
$\pi^{-1}(x)$ varies smoothly with $x$.
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{May 8, 2007: Smooth functors vs.~connections on principal
bundles}

We're generalizing the idea of a smooth functor
\[ e^{iS} : PM \to U(1) \]

\noindent to the concept of ``parallel concept in a principal
$U(1)$-bundle".\medskip


\subsection{Principal bundles - the formal definition}

We first define principal $U(1)$-bundles, as promised last time; we give
the definition in greater generality.

\begin{definition}
For any Lie group $G$, a {\it principal $G$-bundle over the smooth space
$M$} is a smooth space $P$ equipped with a (smooth) right action of $G$
\[ \aaa : G \times P \to P \]

\noindent and a map
\[ \begin{CD}
P @. \qquad \mbox{[figure: tubular cylinder and circular cross
section]}\\
@V\pi VV\\
M @. \mbox{[figure: the base space } M \mbox{ and a point } x \mbox{ in
it]}\\
\end{CD} \]

\noindent such that
\begin{enumerate}
\item the action $\aaa$ preserves the {\it fibers} $\pi^{-1}(x)\ \forall
x \in M$ (i.e. for all $p \in \pi^{-1}(x),\ g \in G$, we have $\aaa(p,g)
\in \pi^{-1}(x)$); and
\item $\pi : P \to M$ is {\it locally trivial}: $\forall x \in M,$ there
exists open $U \ni x$ and an isomorphism of right $G$-spaces $\g :
\pi^{-1}(U) \to U \times G$ such that
\[
\xymatrix{
  \pi^{-1}(U)
     \ar[rr]^{\g}
     \ar[dr]_{\pi}
  && U \times G
     \ar[dl]^{\pi_1} \\
 & U
}
\]

\noindent commutes.
\end{enumerate}
\end{definition}\medskip


\begin{remark}
This implies that every fiber $P_x := \pi^{-1}(x)$ is a right $G$-space,
but in fact we have an isomorphism of right $G$-spaces
\[ \g :  P_x \to \{ x \} \times G \cong G \]

\noindent so $P_x$ is a $G$-torsor!
\end{remark}\medskip


\subsection{Parallel transport and connections}

Now let's talk about parallel transport - a new way. Given a principal
$G$-bundle $\pi : P \to M$, we can form the {\it transport groupoid}
$\trans(P)$, where:
\begin{itemize}
\item objects are points $x \in M$ (or if you prefer, fibers $P_x$);
\item morphisms $f : x \to y$ are maps (hence isomorphisms - since a
$G$-set map between $G$-torsors is an isomorphism) between the torsors
$P_x$ and $P_y$.
\end{itemize}\medskip

\noindent {\bf Example.} When $G = U(1)$, we think of $P$ as a ``circle
bundle" over $M$, giving one circle of possible phases $P_x$ for each $x
\in M$. Then $\trans(P)$ has these circles as objects, and ``rotations"
from one circle to another, as morphisms.\medskip

\begin{definition}
A {\it connection} on $P$ is a smooth functor
\[ \hol : PM \to \trans(P) \]

\noindent where $PM$ is the smooth category (groupoid!) of equivalence
classes of paths in $M$, such that
\[ \hol(x) = P_x\ \forall x \in M \]
\end{definition}

\vspace{8ex}
\noindent [figure: base of $M$, with $\g : x \to y$ in it below; and
tubular space above, with circular cross sections $\hol(x) = P_x, \hol(y)
= P_y$ above $x,y$ respectively; $\hol(\g) : P_x \to P_y$ by
functoriality]
\vspace{8ex}


\noindent We can think of any smooth space $M$ as a smooth category with
\begin{itemize}
\item points of $M$ as objects: $\Ob(M) = M$; and
\item only identity morphisms: $\Mor(M) = M$.
\end{itemize}

Then the condition $\hol(x) = P_x$ says:
\[
\xymatrix{
  PM
     \ar[rr]^{\hol}
  && \trans(P)\\
 & M
     \ar[ul]
     \ar[ur]
}
\]

\noindent commutes.\medskip


\subsection{Getting back smooth circle-valued functors}

Let's examine this locally: for any $x \in M$, there's an open set $U \ni
x$ and an isomorphism $\g$ such that
\[
\xymatrix{
  P|_U := \pi^{-1}(U)
     \ar[rr]^{\g}_\sim
     \ar[dr]_{\pi}
  && U \times G
     \ar[dl]^{\pi_1} \\
 & U
}
\]

\noindent commutes.\medskip

The inclusion $U \hookrightarrow M$ gives a smooth functor
\[ PU \to PM \]

\noindent and a smooth functor
\[ \trans(P|_U) \to \trans(P) \]

In fact, we get a commuting prism
\[
\xymatrix{
  &&& PM
     \ar[rr]^{\hol}
  && \trans(P)\\
  P|_U := \pi^{-1}(U)
     \ar[rr]^{\hol|_U}
     \ar[rrru]
  && \trans(P|_U)
     \ar[rrru]
  && M
     \ar[lu]
     \ar[ru]\\
  & U
     \ar[lu]
     \ar[ru]
     \ar[rrru]
}
\]

\noindent for a unique $\hol|_U : PU \to \trans(P|_U)$. So we can
``restrict parallel transport to $U$". But we have (as above)
\[
\xymatrix{
  P|_U
     \ar[rr]^{\g}_\sim
     \ar[dr]_{\pi}
  && U \times G
     \ar[dl]^{\pi_1} \\
 & U
}
\]

\noindent so $\trans(P|_U) \cong \trans(U \times G)$, where $U \times G$
is the trivial principal $G$-bundle over $U$. But $\trans(U \times G)$ is
easy to understand:

\vspace{5ex}
\noindent [figure: cylinder called $U \times G$ over $U$, and $\g : x \to
y$ gives some rotation by $g$]
\vspace{5ex}

\begin{eqnarray*}
\Ob(\trans(U \times G)) & \cong & U\\
\Mor(\trans(U \times G)) & \cong & U^2 \times G
\end{eqnarray*}

\noindent and in fact, $\trans(U \times G)$ is a product of categories:
\[ \trans(U \times G) \cong \codisc(U) \times G \]

\noindent where $\codisc(U)$ has elements $x \in U$ as objects, and
1-morphisms from any object to any other - and $G$ is regarded as a
1-object category. So we get
\[ PU \mapdef{\hol|_U} \trans(P|_U) \cong \trans(U \times G) \cong
\codisc(U) \times G \mapdef{\pi_2} G \]

\noindent which is a smooth functor from $PU$ to $G$ which contains all
the information in $\hol|_U$. But...

\begin{theoremeq}
Smooth functors $\hol : PU \to G$ are in one-to-one correspondence with
$\mfg$-valued 1-forms, where $\mfg$ is the Lie algebra of $G$.
\end{theoremeq}

\noindent So connections on principal $G$-bundles are locally described
by $\mfg$-valued 1-forms. For $G = U(1)$, these are just 1-forms.
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{May 15, 2007: Three approaches to connections - and smooth
anafunctors}

Last time we described a connection on a principal $G$-bundle $P \to M$
as a smooth functor:
\[ \hol : PM \to \trans(P) \]

\noindent such that $\hol(x) = P_x$. Last time, we expressed this last
clause by saying that 
\[
\xymatrix{
 & \disc(M)
   \ar[dl]
   \ar[dr]\\
 PM
   \ar[rr]^{\hol}
 && \trans(P)\\
}
\]

\noindent commutes, where $\disc(M)$ is the {\it discrete category on
$M$}. In other words, we're looking at smooth categories and functors
{\it under} $\disc(M)$.

A better way to express this clause is to say that
\[
\xymatrix{
  PM
     \ar[rr]^{\hol}
     \ar[dr]
  && \trans(P)
     \ar[dl]\\
 & \codisc(M)
}
\]

\noindent commutes, where $\codisc(M)$ is the codiscrete category on
$M$.\medskip

Okay$\dots$ now for a big chart$\dots$\newpage

\noindent{\small
\begin{tabular}{|p{0.8in}|p{2in}|p{2in}|}
\hline
$G$-bundles & Connections & Gauge Transformations\\
\hline
\hline
\vspace{0.3ex}
Trivial $G$-bundles $M \times G \to M$
&
\vspace{0.3ex}
smooth functors\hfill\break
$\hol : PM \to G$,\hfill\break
or equivalently, $\mfg$-valued 1-forms on $M$
&
\vspace{0.3ex}
smooth natural transformations\hfill\break
\hspace*{5ex}$\xymatrix{
 PM
   \ar@/^2pc/[rr]^{\hol_1}_{}="0"
   \ar@/_2pc/[rr]_{\hol_2}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g}
&& G
}$\hfill\break
\noindent or equivalently, smooth functions\hfill\break
$g : M \to G$

\noindent with $\hol_2(\g) = g_y \hol_1(\g) g_x^{-1}$ for $\g : x \to y$
in $PM$.\\
\hline
\vspace{0.3ex}
Any fixed $G$-bundle\hfill\break
$P \to M$
&
\vspace{0.3ex}
smooth functors\hfill\break
$\xymatrix{
  PM
     \ar[r]^{\hol}
     \ar[dr]
  & \trans(P)
     \ar[d]\\
 & \codisc(M)
}$
&
\vspace{0.3ex}
smooth natural transformations\hfill\break
$\xymatrix{
 PM
   \ar@/^2pc/[rr]^{\hol_1}_{}="0"
   \ar@/_2pc/[rr]_{\hol_2}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g}
    \ar[ddr]
&& G
   \ar[ddl]\\
&&\\
& \codisc(M)}$\\
\hline
\vspace{0.3ex}
A variable $G$-bundle over $M$
&
\vspace{0.3ex}
smooth anafunctors\hfill\break $\hol : PM \to G$
&
\vspace{0.3ex}
smooth ananatural transformations\hfill\break
\hspace*{5ex}$\xymatrix{
 PM
   \ar@/^2pc/[rr]^{\hol_1}_{}="0"
   \ar@/_2pc/[rr]_{\hol_2}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g}
&& G
}$\\
\hline
\end{tabular}
}


\begin{remark}\hfill
\begin{enumerate}
\item Connections mod gauge transformations are classified by various
forms of cohomology. For example,
\[ \{ U(1)\mbox{-bundles with connections over } M \} / \tangle{
\mbox{gauge transformations}} \]

\noindent is an example of Deligne cohomology.\medskip


\item To get a smooth functor, ``parallel transport", or
\[ \hol : PM \to G \]

\noindent from a $\mfg$-valued 1-form $A$ on $M$, we set
\[ \hol(\g) = Pe^{\int_{\g} A} \]

\noindent If $G = U(1)$, then $\mfg = \mathfrak{u}(1) = i \R \cong \R$,
so a $\mfg$-valued 1-form amounts to a 1-form $A$ and then
\[ \hol(\g) = e^{i \int_{\g} A} \]

\noindent (i.e., path-ordered exponentiation reduces to orginary
exponentiation, since $U(1)$ is abelian).\medskip


\item Given smooth functors
\[ F_1, F_2 : \calc \to \cald \]

\noindent a {\it smooth natural transformation} is a smooth map
\[ \aaa : \Ob(\calc) \to \Mor(\cald) \]

\noindent such that for all morphisms $f : x \to y$ in $\calc$, the
following square exists
\[ \begin{CD}
F_1(x) @>F_1(f)>> F_1(y)\\
@V\aaa_xVV    @V\aaa_yVV\\
F_2(x) @>F_2(f)>> F_2(y)
\end{CD} \]

\noindent and commutes.

So a gauge transformation
\[ \xymatrix{
 PM
   \ar@/^2pc/[rr]^{\hol_1}_{}="0"
   \ar@/_2pc/[rr]_{\hol_2}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g}
&& G
}
\]

\noindent is a smooth natural transformation from $\hol_1$ to $\hol_2$,
i.e. a smooth map
\[ g : \Ob(PM) \to \Mor(G) \]

\noindent i.e.
\[ g : M \to G \]

\noindent such that given a path $\g : x \to y$ in $M$, this square
commutes:
\[ \begin{CD}
* @>\hol_1(\g)>> *\\
@Vg_xVV    @Vg_yVV\\
* @>\hol_2(\g)>> * 
\end{CD} \]

\noindent where $*$ is the one object of our group $G$. This says that
\[ \hol_2(\g) = g_y \hol_1(\g) g_x^{-1} \]


\item Given any $G$-bundle $P \to M$ and two connections
\[ \hol_1, \hol_2 : PM \to \trans(P), \]

\noindent a smooth natural transformation
\[ \xymatrix{
 PM
   \ar@/^2pc/[rr]^{\hol_1}_{}="0"
   \ar@/_2pc/[rr]_{\hol_2}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g}
&& \trans(P)
}
\]

\noindent is a smooth map
\[ g : \Ob(PM) \to \Mor(\trans(P)) \]

\noindent (here, $\Ob(PM) = M$) such that the following square commutes:
\[ \begin{CD}
P_x @>\hol_1(\g)>> P_y\\
@Vg_xVV    @Vg_yVV\\
P_x @>\hol_2(\g)>> P_y
\end{CD}
\qquad [\mbox{figure: cylinder above } \g : x \to y \mbox{ with cross
sections } P_x, P_y \mbox{ above } x,y] \]

\noindent where $g_x : P_x \to P_x$ is a $G$-torsor morphism
\[ g_x(ph) = g_x(p)h\ \forall p \in P_x, h \in G \]

\noindent and similarly for $g_y$.\medskip


\item What's a smooth anafunctor? Given smooth categories $\calc$ and
$\cald$, a smooth anafunctor is the {\it right} kind of thing going from
$\calc$ to $\cald$, generalizing a smooth functor. A smooth anafunctor
``looks locally like a smooth functor", so it can be thought of as a
functor which is locally isomorphic (via natural isomorphisms) to a
smooth one. More precisely:

\begin{definition}
Let $\calc,\cald$ be smooth categories. A {\it smooth anafunctor} $F :
\calc \to \cald$ consists of
\begin{enumerate}
\item an open cover $\{ U_\aaa \}$ of $\Ob(\calc)$, where $\Ob(\calc)$ is
a topological space with the finest topology such that every plot in
$\Ob(\calc)$ is continuous.

\item smooth functors $F_\aaa : \calc_\aaa \to \cald$, where $\calc_\aaa$
has
\begin{itemize}
\item objects in $U_\aaa$ as objects
\item morphisms between these as morphisms.
\end{itemize}

\item smooth natural isomorphisms\[
\xy
 (0,15)*+{\calc_\aaa}="T";
 (-15,0)*+{\calc_{\aaa \beta}}="L";
 (15,0)*+{\cald}="R";
 (0,-15)*+{\calc_\beta}="B";
 (0,7)*{}="MT";
 (0,-7)*{}="MB";
     {\ar "L";"T"};
     {\ar "L";"B"};
     {\ar^{F_\aaa} "T";"R"};
     {\ar_{F_\beta} "B";"R"};
         {\ar@{=>}^{g_{\aaa \beta}} "MT";"MB"};
\endxy
\]

\noindent where $\calc_{\aaa \beta} \hookrightarrow \calc_\aaa,
\calc_\beta$ has
\begin{itemize}
\item objects in $U_{\aaa \beta} := U_\aaa \cap U_\beta$ as objects;
\item morphisms between these as morphisms.
\end{itemize}

\item the ``cocycle conditions" that are finally requred:
\[ g_{\aaa \beta} g_{\beta \g} = g_{\aaa \g} \]

\noindent on objects in $U_{\aaa \beta \g} := U_\aaa \cap U_\beta \cap
U_\g$.
\end{enumerate}
\end{definition}
\end{enumerate}
\end{remark}
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{May 22, 2007: Bundles, connections, cohomology, and anafunctors}

Last time JB claimed that if $M$ is a smooth space (e.g. a manifold), and
$G$ a Lie group, then principal $G$-bundles with connection over $M$
correspond to smooth anafunctors
\[ \hol : PM \to G \]

\noindent JB also claimed that isomorphisms between principal $G$-bundles
with connection, correspond to smooth ananatural transformations
\[ \xymatrix{
 PM
   \ar@/^2pc/[rr]^{\hol}_{}="0"
   \ar@/_2pc/[rr]_{\hol'}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g}
&& G
}
\]


\subsection{The picture without connection}

If we leave out the connection, we get a simpler version of this story.
Principal $G$-bundles over $M$ correspond to smooth anafunctors
\[ \hol : \disc(M) \to G \]

\noindent and isomorphisms between principal $G$-bundles correspond to
smooth ananatural transformations
\[ \xymatrix{
 \disc(M)
   \ar@/^2pc/[rr]^{\hol}_{}="0"
   \ar@/_2pc/[rr]_{\hol'}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g}
&& G
}
\]

Let's see why this simpler version works. Our ultimate goal is to
show
\[ \frac{\{ \mbox{Principal } G\mbox{-bundles over } M \}}{\{
\mbox{isomorphisms}\}} \cong \frac{\tangle{\mbox{smooth anafunctors} :
\disc(M) \to G }}{\tangle{\mbox{ananatural transformations}}} \cong
\check{H}^1(M,G) \]

\noindent where $\check{H}^1(M,G)$ is the first \cech cohomology of $M$
with coefficients in $G$. A famous example:
\[ \check{H}^1(M,U(1)) \cong H^2(M,\Z) \]

\noindent where $H^2(M,\Z)$ is the second cohomology of $M$ with
coefficients in $\Z$.

For example, the sphere

\vspace{4ex}
[figure: sphere $M = S^2$]
\vspace{4ex}

\noindent has $H^2(M,\Z) \cong \Z$ ($S^2$ has ``one 2-dimensional hole").
So, (isomorphism classes of) principal $U(1)$-bundles over $S^2$ are
classified by an integer, the ``first Chern class" $c_1$.\medskip


\subsection{A first look at \cech cohomology}

Suppose $P \to M$ is a principal $G$-bundle. It's locally trivial: for
each point $x \in M$ there's an open neighbourhood $U \ni x$ for which we
have an isomorphism of $G$-spaces:
\[
\xymatrix{
  P|_U := \pi^{-1}(U)
     \ar[rr]^{\g}_\sim
     \ar[dr]_{\pi}
  && U \times G
     \ar[dl]^{\pi_1} \\
 & U
}
\]

Let's go ahead and pick an open cover $\{ U_i : i \in I \}$ of $M$, and
isomorphisms
\[
\xymatrix{
  P|_{U_i} := \pi^{-1}(U_i)
     \ar[rr]^{\g_i}_\sim
     \ar[dr]_{\pi}
  && U_i \times G
     \ar[dl]^{\pi_1} \\
 & U_i
}
\]

\noindent Let $U_{ij} = U_i \cap U_j$; we get ``transition functions"
\[ g_{ij} := \g_i \circ \g_j^{-1} : U_{ij} \times G \mapdef{\sim} U_{ij}
\times G \]

\noindent (here, $\g_i$ means $\g_i$ restricted to $P|_{U_{ij}}$;
similarly for $\g_j$). This is an isomorphism of right $G$-spaces,
necessarily given by left multiplication by some $G$-valued function on
$U_{ij}$.

By abuse of notation, we also call this function
\[ g_{ij} : U_{ij} \to G \]

\noindent This is a ``\cech 1-cochain": in general a \cech $n$-cochain
would be a bunch of maps
\[ g_{i_0, \dots, i_n} : U_{i_0, \dots, i_n} = \bigcap_{j=0}^n U_{i_j}
\to G \]

\noindent So: any principal $G$-bundle over $M$ gives a \cech 1-cochain.
In fact, $g_{ij}$ satisfies an equation:
\[ g_{ij} \cdot g_{jk} = \g_i \g_j^{-1} \circ \g_j \g_k^{-1} = \g_i
\g_k^{-1} = g_{ik} \]

\noindent on $U_{ijk} := U_i \cap U_j \cap U_k$. This is called the
cocycle condition, and we say $g_{ij}$ is a {\it \cech 1-cocycle}. The
cocycle condition really says that this 2-cochain
\[ g_{ij} g_{jk} g_{ik}^{-1} : U_{ijk} \to G \]

\noindent is trivial, i.e. it equals 1.

Conversely, given a \cech 1-cocycle, you can build a principal $G$-bundle
over $M$. So:

{\it principal $G$-bundles correspond to \cech 1-cocycles}.\medskip

\noindent What do isomorphisms of principal $G$-bundles correspond
to?\medskip


\subsection{Quotienting out the isomorphisms}

Suppose $P,P'$ are two principal $G$-bundles over $M$, and we have an
isomorphism
\[
\xymatrix{
  P
     \ar[rr]^{f}_\sim
     \ar[dr]
  && P'
     \ar[dl] \\
 & M
}
\]

\noindent (i.e. $f$ is a smooth map of right $G$-spaces making the
triangle commute, where we can think of $M$ as a trivial $G$-space - so
this diagram lives in the category of smooth $G$-spaces). We can find an
open cover $\{ U_i : i \in I \}$ of $M$ such that $P|_{U_i}$ and
$P'|_{U_i}$ are trivializable:
\[ U_i \times G \mapdef{\g_i^{-1}} P|_{U_i} \mapdef{f} P'|_{U_i}
\mapdef{\g'_i} U_i \times G \]

\noindent (Over here, all maps are isomorphisms, and $\g_i, \g'_i$ are
trivializations - isomorophisms of right $G$-spaces over $U_i$.) Let
\[ f_i := \g'_i \circ f \circ \g_i^{-1} \]

\noindent for $i \in I$ be the composite; these $f_i$'s describe our
bundle isomorphism $f$ locally, i.e. we can reconstruct $f: P|_{U_i} \to
P'|_{U_i}$ from $f_i, \g_i, \g'_i$. Since $f_i : U_i \times G \to U_i
\times G$ is an isomorphism of $G$-bundles, it's determined by a function
from $U_i$ to $G$, which we also call $f_i$: we can write\
\[ f_i(x,g) = (x, f_i(x,g)) \]

\noindent for some $f_i : U_i \to G$.\medskip

Recall that a bunch of maps $\{ f_i : U_i \to G : i \in I \}$ is called a
{\it \cech 0-cochain}. The bundle $P$ is described by a \cech 1-cochain:
\[ g_{ij} := \g_i \g_j^{-1} \]

\noindent and similarly for $P': g'_{ij} = \g'_i (\g'_j)^{-1}$, where
$\g_{ij}, \g'_{ij} : U_{ij} \to G$. In fact, $g_{ij}$ and $g'_{ij}$ are
related by the $f_i$'s as follows:
\[ f_i g_{ij} = g'_{ij} f_j \]

\noindent Let's \v{c}eck this! We indeed have
\[ \g'_i f \g_i^{-1} \circ \g_i \g^{-1}_j = \g'_i (\g'_j)^{-1} \circ
\g'_j f \g^{-1}_j \]

\noindent because both sides equal $\g'_i f \g^{-1}_j$.\medskip

In short, the ``difference" of the two \cech 1-cochains (actually
1-cocycles) $g_{ij}$ and $g'_{ij}$ is given by the \cech 0-cochain $f_i$

\begin{remark}
But if $G$ is nonabelian, we {\it cannot} actually write $g_{ij}
(g'_{ij})^{-1} = f_j f^{-1}_i$. This works only if $G$ is abelian. We're
studying ``nonabelian \cech cohomology"!
\end{remark}\medskip

\noindent So:
\[ \frac{ \{ G\mbox{-bundles } P \to M \mbox{ such that } P|_{U_i} \mbox{
are trivial} \} }{ \tangle{ \mbox{isomorphisms}} } \cong \frac{ \{
\mbox{\cech 1-cocycles } g_{ij} : U_{ij} \to G \} }{\tangle{g_{ij} \sim
g'_{ij} \mbox{ iff } f_i g_{ij} = g'_{ij} f_j}} \]

\noindent Taking the ``limit" (really the colimit!) as the cover $\{ U_i
\}$ gets finer, we get
\[ \frac{ \{ G\mbox{-bundles } P \to M \} }{\tangle{\mbox{isomorphisms}}}
= \lim_{\overset{\longrightarrow}{U}} \frac{\{ \mbox{\cech 1-cocycles }
g_{ij} : U_{ij} \to G \} }{\sim} \]

\noindent and the right-hand side is defined to be $\check{H}^1(M,G)$,
the first \cech cohomology of $M$ with coefficients in $G$!\medskip

\begin{remark}
There's another description of this (due to Toby Bartels):
\[ \check{H}^1(M,G) = \frac{\{ \mbox{smooth anafunctors } F : \disc(M)
\to G \} }{\tangle{ \mbox{ananatural isomorphisms} } } \]
\end{remark}
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{May 29, 2007: \cech cohomology and ananatural category theory}


\subsection{Definitions - old and new}

Recall: given smooth categories $\calc,\cald$, a smooth anafunctor $F :
\calc \to \cald$ consists of
\begin{enumerate}
\item an open cover $\{ U_i : i \in I \}$ of $\Ob(\calc)$

\item smooth functors $F_i : \calc|_{U_i} \to \cald$

\item smooth natural isomorphisms
\[ \xymatrix{
 \calc|_{U_{ij}}
   \ar@/^2pc/[rr]^{F_j}_{}="0"
   \ar@/_2pc/[rr]_{F_i}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
  \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g_{ij}}
&& \cald
}
\]

\noindent such that

\item the cocycle condition
\[ g_{ij} g_{jk} = g_{ik} \]

\noindent holds on $\calc|_{U_{ijk}}$.
\end{enumerate}\medskip

So - what's a smooth anafunctor
\[ F : \disc(M) \to G ? \]

Well, it must consist of
\begin{enumerate}
\item an open cover $\{ U_i : i \in I \}$ of $\Ob(\disc(M)) = M$

\item smooth functors $F_i : \disc(U_i) \to G$ - but there's only one,
since $\disc(U_i)$ is discrete, and $G$ has only one object.

\item smooth natural isomorphisms
\[ \xymatrix{
 \disc(U_{ij})
   \ar@/^2pc/[rr]^{F_j}_{}="0"
   \ar@/_2pc/[rr]_{F_i}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{g_{ij}}
&& G
}
\]

\noindent For each object in $\disc(U_{ij})$, i.e. $x \in U_{ij}$, this
gives an isomorphism $g_{ij}(x) : F_j(x) = * \to * = F_i(x)$, where $*
\in G$ is unique.\medskip

In short, $g_{ij} \in G$. Why is $g_{ij}$ natural? Well,
\[ \begin{CD}
F_j(x) @>F_j(1_x)>> F_j(x)\\
@Vg_{ij}(x)VV    @Vg_{ij}(x)VV\\
F_i(x) @>F_i(1_x)>> F_i(x)\\
\end{CD} \]

\noindent must commute, and obviously does!

\item And we require $g_{ij} g_{jk} = g_{ik}$.
\end{enumerate}\medskip

All this is just a \cech 1-cocycle! So:
\[ \{ \mbox{smooth anafunctors } F : \disc(M) \to G \} \cong \{
\mbox{\cech 1-cocycles} \} \]

\noindent and we can cook up a definition of ``ananatural
transformation", such that
\[ \frac{\{ \mbox{smooth anafunctors } F : \disc(M) \to G
\}}{\tangle{\mbox{ananatural transformations}}} \cong \check{H}^1(M,G) \]

Here is the definition in general.

\begin{definition}
Given smooth anafunctors
\[ \xymatrix{
 \calc
   \ar@/^2pc/[rr]^{F}_{}="0"
   \ar@/_2pc/[rr]_{F'}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
&& \cald
}, \]

\noindent a {\it smooth ananatural transformation}
\[ \xymatrix{
 \calc
   \ar@/^2pc/[rr]^{F}_{}="0"
   \ar@/_2pc/[rr]_{F'}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{f}
&& \cald
} \]

\noindent is a collection of smooth natural transformations
\[ \xymatrix{
 \calc|_{U_i}
   \ar@/^2pc/[rr]^{F_i}_{}="0"
   \ar@/_2pc/[rr]_{F'_i}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8" ^{f_i}
&& \cald
}, \]

\noindent such that
\[ f_i g_{ij} = g'_{ij} f_j\ \forall i,j \]

\noindent where $g_{ij}, g'_{ij}$ come from the anafunctors $F,F'$, in
the way we described (above).
\end{definition}\medskip


\subsection{One possible generalization}

In summary:
\[ \frac{ \{ G\mbox{-bundles over } M \}}{\tangle{\mbox{isomorphisms}}}
\cong \check{H}^1(M,G) \cong \frac{\{ \mbox{smooth anafunctors } F :
\disc(M) \to G \}}{\tangle{\mbox{ananatural isomorphisms}}} \]

\noindent but in fact:
\[ \frac{ \{ G\mbox{-bundles with connection over } M
\}}{\tangle{\mbox{isomorphisms}}} \cong \frac{\{ \mbox{smooth anafunctors
} \hol : PM \to G \}}{\tangle{\mbox{ananatural isomorphisms}}} \]

\noindent and this deserves to be called ``$\check{H}^1(PM,G)$" - now
we're really doing \cech cohomology of a smooth category! More generally,
given two smooth 1-categories $\calc,\cald$, let's define
\[ \check{H}^1(\calc,\cald) := \frac{\{ \mbox{smooth anafunctors } F :
\calc \to \cald \}}{\tangle{\mbox{ananatural isomorphisms}}} \]

\noindent Then replace 1 by ``$n$"!
\newpage


\addtocontents{toc}{\protect\vspace{2ex}}
\section{June 5, 2007: Review and prospectus}


\subsection{Phases and functors}
We've been trying to understand classical and quantum physics in a
unified way, starting with a category $\calc$ of ``configurations" (ways
things can be) and ``processes" (ways things can become), with a functor
\[ S : \calc \to \R \]

\noindent or
\[ \Phi : \calc \to U(1) \]

\noindent assigning to any process $\g : x \to y$ its ``action" $S(\g)
\in \R$ or ``phase" $\Phi(\g) \in U(1)$. These are often related by
\[ \Phi(\g) = e^{iS(\g)} \]

\noindent but $\Phi$ need not be of this form.
\vspace{2ex}


\noindent{\small
\begin{tabular}{p{2.15in}|p{2.5in}}
\hline
Classical & Quantum\\
\hline
\hline
\vspace{0.3ex}
$\calc$ is a smooth category.
&
\vspace{0.3ex}
$\calc$ is a measurable category.\\
\hline
\vspace{0.3ex}
$\Phi : \calc \to U(1)$ is a smooth functor.
&
\vspace{0.3ex}
$\Phi : \calc \to U(1)$ is a measurable functor.\\
\hline
\vspace{0.3ex}
Given $x,y \in \calc$, find $\g : x \to y$ with\hfill\break
$\delta \Phi(\g) = 0$.
&
\vspace{0.3ex}
Given $x,y \in \calc$, find\hfill\break
$\displaystyle \tangle{y,x} = \int_{\g : x \to y} \Phi(\g)\ \D \g \in
\C$.\\
\hline
\end{tabular}
}
\vspace{2ex}


We can try to unify these using {\it rigs}, but in both cases there are
subtleties that complicate the picture above. In the quantum case, we
need to stretch the theory of integration beyond traditional measure
theory, in order to do the ``path integral" above. In the classical case,
smooth functors aren't enough - we need smooth anafunctors. So, there's a
lot more to be understood, both classically and quantum-ly.\medskip


Look at the classical side of the prospectus. In classical mechanis, we
often start with a phase space $(X, \omega)$ - a symplectic manifold.
From this, we build a smooth category
\[ \calc = PX, \]

\noindent the path groupoid. Then we seek a smooth anafunctor
\[ \Phi : PX \to U(1) \]

\noindent such that if $\g$ is a loop that bounds a surface $\Sigma$
(i.e. $\partial \Sigma = \g$)

\vspace{8ex}
[figure: surface $\Sigma$, possibly with handles, bounded by (oriented)
$\g$]
\vspace{8ex}

\noindent then we have
\[ \Phi(\g) = e^{i\int_{\Sigma} \omega} \]


There could be many $\Phi$'s, or no $\Phi$'s, that do this. For such a
$\Phi$ to exist, we need:
\[ \partial \Sigma = \emptyset \To \int_\Sigma \omega \in 2 \pi \Z \]

\vspace{5ex}
[figure: $\Sigma$ with a point boundary $x$ - so $\g = 1_x$]
\vspace{8ex}

\noindent since we need $\Phi(1_x) = 1$. We say a closed 2-form is {\it
integral} if $\displaystyle \int_\Sigma \omega \in 2 \pi \Z$ for all
closed $\Sigma \subset X$.
Conversely, if $\omega$ is integral, we can indeed find $\Phi$ as
desired. If $\omega$ is integral,
\[ [\omega] \in H^2(X,\Z) \cong \check{H}^1(X, U(1)) \]

\noindent and we've seen that $\check{H}^1(X,U(1))$ classifies principal
$U(1)$-bundles over $X$. In fact, our smooth anafunctor
\[ \Phi : PX \to U(1) \]

\noindent arises as follows: pick a $U(1)$-bundle $P \to X$ corresponding
to $[\omega]$, pick a connection $A$ on it, and let
\[ \Phi(\g) = \hol(\g) \]

\noindent where holonomy is defined using $A$.\medskip


To go further, we should use geometric quantization to get a Hilbert
space from $(X,\omega,P \to X, A)$ - this requires extra structure on
$X$, e.g. a K\"ahler structure. It would be great to show that these
Hilbert spaces match those given by our path integral procedure!\medskip


\subsection{From particles to strings: categorification}

Also, in this course we {\it hinted} at how to categorify all this stuff,
to go from particle physics to string physics. Naively, we could
categorify our previous chart:
\vspace{2ex}


\noindent{\small
\begin{tabular}{p{2.15in}|p{2.5in}}
\hline
Classical & Quantum\\
\hline
\hline
\vspace{0.3ex}
$\calc$ is a smooth 2-category.
&
\vspace{0.3ex}
$\calc$ is a measurable 2-category.\\
\hline
\vspace{0.3ex}
$\Phi : \calc \to U(1)-\Tor \cong U(1)[1]$ is a smooth functor.
&
\vspace{0.3ex}
$\Phi : \calc \to U(1)[1]$ is a measurable functor.\\
\hline
\vspace{0.3ex}
Given
$\xymatrix{
 x
   \ar@/^2pc/[rr]^{\g_1}_{}="0"
   \ar@/_2pc/[rr]_{\g_2}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
&& y
}$,
find $\Sigma : \g_1 \To \g_2$ such that
$\delta \Phi(\Sigma) = 0$.
&
\vspace{0.3ex}
Given
$\xymatrix{
 x
   \ar@/^2pc/[rr]^{\g_1}_{}="0"
   \ar@/_2pc/[rr]_{\g_2}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
&& y
}$,
we can compute an amplitude\hfill\break
$\displaystyle \tangle{\g_2,\g_1} = \int_{\Sigma : \g_1 \To \g_2}
\Phi(\Sigma)\ \D \Sigma$.
\vspace{0.3ex}\\
\hline
\end{tabular}
}
\vspace{2ex}

The path integrals become a lot harder now, but are still manageable.
Over on the classical side, we really need 2-anafunctors.\medskip

Urs Schreiber and JB showed:\medskip

Given a smooth space $X$, there's a smooth 2-groupoid $P_2 X$, where:

\noindent{\small
\begin{tabular}{p{2.in}p{2.5in}}
\vspace{0.3ex}
$\bullet$
&
\vspace{0.3ex}
objects are points of $X$;
\vspace{4ex}\\
$\xymatrix{
 \bullet
   \ar@/^2pc/[rr]="0"
&& \bullet
}$
&
morphisms are smooth paths in $X$; and\\
\vspace{2ex}
$\xymatrix{
 \bullet 
   \ar@/^2pc/[rr]^{}_{}="0"
   \ar@/_2pc/[rr]^{}_{}="2"
   \ar@{}"0";"2"|(.2){\,}="7"
   \ar@{}"0";"2"|(.8){\,}="8"
   \ar@{=>}"7";"8"
&& \bullet
}$
&
\vspace{5ex}
2-morphisms are thin homotopy classes of paths-of-paths in $X$
\end{tabular}
}
\vspace{2ex}


Given a ``smooth 2-group" $G$ (e.g. $U(1)[1]$), 2-connections on
principal $G$-2-bundles over $X$ correspond to smooth 2-anafunctors
\[ \hol : P_2 X \to G \]

\noindent So - everything we said about bundles and connections
categorifies!\medskip


In string theory, there are nice examples of smooth 2-groups. For
example, given any compact simple (simply connected) Lie group $G$,
there's a 2-group ``String$_H$". When $H =$ Spin$(n)$ is the double cover
of $SO(n)$, ``String$_H$" is the symmetry 2-group involved in studying
``spinning strings", and basic in elliptic cohomology, as studied by
Stolz and Teichner.
\end{document}
