
%\newpage
%
%{\bf \blue \'Etal\'e space of a sheaf}
%
%\definition
%Let ${\cal F}$ be a sheaf on $M$, and $E({\cal F}):= \bigcup_{x\in M}{\cal F}_x$
%the union of all stalks of ${\cal F}$. Given a section
%$f\in {\cal F}(U)$, let $U\arrow E({\cal F})$
%map each $x\in U$ to the germ of $f$ in $x$.
%The image of $f$ under this map is called $e(f)$.
%We equip $E({\cal F})$
%with a weakest topology for which each $e(f)$ is open.
%The space $E({\cal F})$ is called {\bf \blue \'etal\'e 
%space of a sheaf}.
%
%\remark
%The space $E({\cal F})$ is usually non-Hausdorff; however,
%the projection $E({\cal F})\arrow M$ is locally
%a diffeomorphism. The sheaf ${\cal F}$ can be reconstructed
%
%\exercise
%


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  \tiny M. Verbitsky }}
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\begin{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
{\Large\bf Geometry of manifolds \\[15mm]
\small Lecture 7: Categories and locally trivial fibrations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\\[14mm]

{\small Misha Verbitsky } 
\\[20mm]

{\tiny\bf Math in Moscow and HSE
\\[2mm]  April 1, 2013
}
\end{center}

\newpage

{\bf \blue Locally trivial fibrations}

\definition
A smooth map $f:\; X \arrow Y$ is called {\bf \blue a locally trivial
fibration} if each point $y\in Y$ has a neighbourhood
$U\ni y$ such that $f^{-1}(U)$ is diffeomorphic to
$U\times F$, and the map
$f:\; f^{-1}(U)= U\times F \arrow U$ is a projection.
In such situation, $F$ is called {\bf \blue the fiber}
of a locally trivial fibration.

\definition {\bf \blue A trivial fibration}
is a map $X \times Y \arrow Y$.

\example
The projection
$S^3\subset \C^2\backslash 0 \stackrel f\arrow \C P^1$
is called {\bf \blue the Hopf fibration}.
Given $U= \{x:1\}\subset \C P^1$, with $|x|\leq 1$, 
one has 
\[ f^{-1}(U)= \{z_1, z_2\in S^3 \ \ | \ \ |z_1|^2 +|z_2|^2=1, |z_1|\leq 1\} 
\]
(here $z_i$ are complex coordinates in $\C^2$).
Then \[ f^{-1}(U)= \left\{ 
(z_1, z_2) \ \ |\ \ z_2\in U(1)\cdot \sqrt{1-|z_1|^2} \right\},
\]
where $U(1)=\{z\in \C \ \ |\ \ |z|=1\}$.
Therefore, {\bf \red the Hopf fibration $f:\; S^3 \arrow S^2$ is
a locally trivial fibration}.

\remark Since $\pi_1(S^3)=0$ and $\pi_1(S^1 \times S^2)=\Z$,
{\bf \red the Hopf fibration is non-trivial.}

\newpage

{\bf \blue Vector bundles}

\definition
{\bf \blue A vector bundle} on $Y$ is a locally trivial fibration
$f:\; X\arrow Y$ with fiber $\R^n$, with each fiber 
$V:= f^{-1}(y)$ equipped with a structure of a vector
space, smoothly depending on $y\in Y$.

\remark This definition {\bf \green is not very precise or rigorous,}
because ``smoothly depending on $y\in Y$'' {\bf \red needs to be explained.}


\remark This definition is compatible with the one we used previously
(``a vector bundle is a locally free sheaf of $C^\infty M$-modules'').
This will be explained later.

For a more rigorous approach:

1. Define categories.

2. Define group objects and vector space objects

3. Formulate ``smoothly depending on $y\in Y$'' in these terms.


\newpage

{\bf \blue Categories: data}



\newcommand{\Ob}{\operatorname{{\cal O}b}}
\newcommand{\Mor}{\operatorname{{\cal M}or}}

\definition 
{\bf \blue A category} $\cac$ is a collection of data
called ``objects'' and ``morphisms between objects''
which satisfies the axioms below.

{\bf \green DATA.}\\
\phantom{XX} {\bf \red Objects:} The set $\Ob(\cac)$ of {\bf \blue objects}
of $\cac$. \\
\phantom{XX} {\bf \red Morphisms:} For each $X, Y\in \Ob(\cac)$,
one has a set $\Mor(X,Y)$ of {\bf \blue morphisms from $X$ to $Y$}.\\
\phantom{XX} {\bf \red Composition of morphisms:} For each
$\phi\in \Mor(X,Y), \psi \in \Mor(Y,Z)$ there exists
{\bf \blue the composition} $\phi\circ \psi \in \Mor(X, Z)$\\
\phantom{XX} {\bf \red Identity morphism:} For each 
$A\in \Ob(\cac)$ there exists a morphism $\Id_A \in \Mor(A,A)$.

\remark
In some versions of axiomatic set theory, one considers
not a set, but {\bf \purple a class} of objects, which could
be arbitrarily big, such as the class of all sets, or the class
of all linear spaces. The category with {\bf \red a set} 
of morphisms and objects is called {\bf \blue a small category},
and one with a class {\bf \blue a big category}.

In ZFC, one postulates existence of so-called 
{\bf \blue Grothendieck universe} (that is, {\bf \blue
a strongly inaccessible cardinal}). {\bf \blue Small sets}
are ones which belong to the Grothendieck universe,
the rest of the sets are {\bf \blue big}.

{\small Existence of a strongly inaccessible cardinal
{\bf \green implies consistency of ZFC} (Goedel).}


\newpage

{\bf \blue Categories: axioms}

{\bf \green AXIOMS.}\\
\phantom{XX} {\bf \red Associativity of composition:}
$\phi_1\circ(\phi_2\circ\phi_3)=(\phi_1\circ\phi_2)\circ\phi_3$.\\
\phantom{XX} {\bf \red Properties of identity morphism:}
For each $\phi\in \Mor(X,Y)$, one has
$\Id_x\circ \phi = \phi = \phi\circ \Id_Y$\\

\definition
Let $X, Y\in \Ob(\cac)$ -- objects of $\cac$.
A morphism $\phi\in \Mor(X,Y)$ is called {\bf \blue an isomorphism}
if there exists $\psi\in \Mor(Y,X)$ such that
$\phi \circ \psi = \Id_X$ and $\psi\circ\phi = \Id_Y$.
In this case, the objects $X$ and $Y$ are called
{\bf \blue isomorphic}.

{\bf \green Examples of categories:}\\
{\bf \purple Category of sets:} its morphisms are arbitrary maps.\\
{\bf \purple Category of vector spaces:} its morphisms are linear maps.\\
{\bf \purple Categories of rings, groups, fields:} morphisms are 
homomorphisms.\\
{\bf \purple Category of topological spaces:} morphisms are continuous maps.\\
{\bf \purple Category of smooth manifolds:} morphisms are smooth maps.


{\bf \red It is often convenient to express
morphisms by arrows, and call them ``maps''.}


\newpage

{\bf \blue Some categorical constructions}

\definition
{\bf \blue A product} $X_1\times X_2$ of $X_1, X_2 \in \Ob(cac)$
is an object of $\cac$ equipped with {\bf \blue projection maps}
$\pi_i:\; X_1\times X_2\arrow X_i$ such that {\bf \purple for any
pair of morphisms $\phi_i\in \Mor(Y, X_i)$ there is a unique morphism
$\phi\in \Mor(Y, X_1 \times X_2)$ such that $\phi\circ\pi_i =\phi_i$.}

\exercise Prove that {\bf \purple
a product is unique up to isomorphism,} if it exists.

\exercise Prove that the product is {\bf \red associative:}
$X\times(Y\times Z)\cong (X\times Y)\times Z$
and {\bf \red commutative:} $X\times Y\cong Y\times X$.

\exercise Find the product in the categories of
a. groups b. rings c. vector spaces d. sets e. topological spaces.

\definition
{\bf\blue An initial object} of a category is 
an object $I\in \Ob(\cac)$ such that
$\Mor(I,X)$ is always a set of one element.
{\bf \blue A terminal object} is $T\in \Ob(\cac)$ such that
$\Mor(X,T)$ is always a set of one element.

\exercise
Prove that {\bf \purple the initial and the terminal object is unique,}
up to isomorphism.

%\exercise Find the initial and the terminal object in the categories of
%a. groups b. rings c. vector spaces d. sets e. topological spaces,
%or show that it does nor exist.


\newpage

{\bf \blue Group objects in categories}

\exercise Let $T$ be a terminal object. {\bf \purple 
Prove that $X\times T\cong X$ for each $X\in \Ob(\cac)$.}

\definition
An object $G\in \Ob(\cac)$ is called {\bf \blue a group object}
if there exists a morphism $\mu\in \Mor(G\times G, G)$ ({\bf \blue
the product}), a morphism $e\in \Mor(T, G)$ from the terminal
object ({\bf \blue the unit}), and a morphism $i\in \Mor(G,G)$
({\bf \blue the inverse}),
satisfying the following axioms.

{\bf \red Associativity:} Consider the morphisms
$\mu_{12},\mu_{23}:\; G\times G\times G\arrow G\times G$,
the first map takes the product on the first two objects, and
acts as identity on the third, the second maps is a product
on last 2 and identity on the first. Then
$\mu_{12}\circ\mu=\mu_{23}\circ \mu:\; G\times G\times G\arrow G$.

{\bf \red Unit:} The compositions 
$G=G\times T \stackrel{\Id_G\times e}\arrow G\times G \stackrel \mu \arrow G$ 
and $G=G\times T \stackrel{e\times \Id_G}\arrow G\times G \stackrel \mu \arrow G$ 
are identities.

{\bf \red Inverse:} 
Let $\Delta:\; G \arrow G\times G$ be {\bf \blue the diagonal map,}
that is, a map $G \arrow G\times G$ obtained from
a pair of identity maps. Then 
the composition $G\stackrel\Delta\arrow  G\times
G\stackrel{\Id_G\times i}\arrow G\times G \stackrel \mu
\arrow G$ is equal to $G\arrow T 
\stackrel e\arrow G$.


\newpage

{\bf \blue Examples of group objects}

\example
{\bf \blue A topological group} is a group object 
in the category of topological spaces.

\example
{\bf \blue A Lie group} is a group object in the category
of smooth manifolds.

\definition
Let $\cac$ be a category.
{\bf \blue An opposite category} $\cac^\circ$ is a category
with the same sets of objects, $\Mor_{\cac^o}(X,Y)=\Mor_\cac(Y,X)$,
with the same compositions as in $\cac$ taken in inverse order.

\example
{\bf \red The category of finitely generated algebras withous
nilpotents over $\C$ is equivalent to $\cac^\circ$,}
where $\cac$ is a category with objects algebraic
subsets in $\C^n$ (common zeros of a system of polynomial
equations) and morphisms polynomial functions.
This statement is called {\bf \blue ``Hilbert's Nullstellensatz''.}

\example
{\bf \blue An algebraic group} is a group object in
the category $\cac^\circ$, where $\cac$ is a category of rings.

\example
{\bf \blue A formal group} is a group object in the category
$\cac^o$, where $\cac$ is a category of complete local rings
(over $\C$, these are local rings, obtained as quotients of the ring
of formal power series by an ideal).

\newpage

{\bf \blue Topological groups over a base}

\definition
Fix a topological space $M$, and let
$\cac_M$ be a category of pairs $(X, f:\; X\arrow M)$
with morphisms being continuous maps from $X_1$ to $X_2$
commuting with the projections to $M$. The product in
$\cac_M$ is called {\bf \blue fiber product:}
$X_1\times_M X_2:= \{(x_1,x_2)\in X_1\times
X_2\ \ |\ \ f_1(x_1)=f_2(x_2)\}$.
A group object in $\cac_M$ is called
{\bf \blue a topological group over $M$}.

\remark
{\bf \red
This definition is equivalent to the following.}

\definition
Let $B \stackrel \pi \arrow M$ be a continuous map, 
and $B\times_M B \stackrel \Psi \arrow M$ -
a morphism over $M$. This morphism is called {\bf\blue
associative multiplication} if it is associative on the fibers of
 $\pi$, that is, satisfies
$\Psi(a, \Psi(b,c))= \Psi(\Psi(a,b),c)$ for every
triple $a,b,c$ in the same fiber. \\
\phantom{X} A section $M \stackrel e \arrow B$ is called
{\bf\blue the unit} if the maps
$B \stackrel {\Id_B \times e}\arrow B\times_M B \stackrel \Psi \arrow B$
and $B \stackrel {e\times \Id_B}\arrow B\times_M B \stackrel \Psi \arrow B$
are equal to $Id_B$.  \\
\phantom{X} A morphism $\nu:\; B \arrow B$ over $M$ 
is called {\bf \blue a group inverse} if each of the maps
$B\stackrel \Delta \arrow B\times_M B \stackrel
{\Id_B\times \nu}\arrow B\times_M B \stackrel \Psi \arrow B$
and $B\stackrel \Delta \arrow B\times_M B \stackrel
{\nu\times \Id_B}\arrow B\times_M B \stackrel \Psi \arrow B$
 is a constant map, mapping $b$ to $e(\pi(b))$. 

A map $B \stackrel \pi \arrow M$
equipped with associative multiplication, unit and 
group inverse is called {\bf\blue a topological group over $M$}. 

\newpage

{\bf \blue Vector spaces over a base}

\remark
Let $\pi:\; G \arrow M$ be a topological group over $M$.
Then the fiber $\pi^{-1}(m)$ is a group for each $m\in M$.
{\bf \red This group structure depends on $m\in M$ continuously},
but {\bf \purple to state this dependency formaly, one needs to define
a topological group over $M$.}

\definition 
Let $k$ be a field.
{\bf\blue  A $k$-vector space object} in a category $\cac$
is a group object $V$ 
equipped with a set of morphisms $\lambda_x\in \Mor(V,V)$,
parametrized by $x\in k$, and satisfying the following conditions.\\
\phantom{X} {\bf \red Multiplicativity:} 
$\lambda_x \lambda_y=\lambda_{xy}$, \\
\phantom{X}
{\bf \red Zero:} $\lambda_1 =\Id_V$\\
\phantom{X}  {\bf \red Unit:} $\lambda_0:\; V \arrow V$
is a composition $V \arrow T \stackrel e \arrow V$.\\
\phantom{X}
{\bf \red Additivity:}  Let $\Delta$ be the diagonal map.
Then the composition $G\stackrel\Delta \arrow G\times G \arrow
\lambda_x\times\lambda_y\stackrel\mu\arrow G$ is equal to
$\lambda_{x+y}$.\\
\phantom{X}
{\bf \red Distributivity:} The composition
$G\times G \stackrel{\lambda_x\times \lambda_x}\arrow
G\times G \stackrel \mu\arrow G$ is equal to $\mu\circ \lambda_x$.

\definition
Let $k$ be a topological field (for instance, $\C$ or $\R$).
{\bf\blue A topological vector space $B$ over a base $M$}
is a vector space object in $\cac_M$, such that
the map $\lambda_x:\; k\times B \arrow B$ is continuous.

\newpage

{\bf \blue Vector spaces over a base (category-free definition)}

\definition
Let $G$ be an abelian group, and $k$ a field.
Suppose that for each non-zero $\lambda\in k$
there exists an automorphism $\phi_\lambda:\; G \arrow G$,
such that  $\phi_\lambda\circ\phi_{\lambda'}=\phi_{\lambda\lambda'}$,
and $\phi_{\lambda+\lambda'}(g)= \phi_{\lambda}(g)+\phi_{\lambda'}(g).$
Then $G$ is called {\bf \blue a vector space over $k$}.

\definition
Let $k=\R$ or $\C$.
An abelian topological group $B\stackrel \pi \arrow M$ over $M$
is called {\bf \blue a vector space over a base $M$},
or {\bf \blue a relative vector space over $M$}
if for each non-zero $\lambda\in k$ there exists a
continuous automorphism
$\phi_\lambda:\; B \arrow B$ of a group $B$ over $M$
satisfying assumptions of the above definition.

\remark
Let  $B\stackrel \pi \arrow M$ be a relative vector space
over $M$, $U\subset M$ an open subset,
and ${\cal B}(U)$ the space of sections of a map 
$\pi^{-1}(U) \stackrel \pi \arrow U$. Then
{\bf \red ${\cal B}(U)$ defines a sheaf of modules
over a sheaf $C^0(M)$ of continuous functions.}

\example
Let  $S\subset \R^n$ be a subset (not necessarily 
a smooth submanifold), $s\in S$ a point, and $v\in
T_s\R^n$ a vector. We sat that $v$ belongs to a {\bf\blue tangent cone} 
$C_sS$ if the distance from $S$ to a point $s+tv$
converges to 0 as $t\rightarrow 0$ faster than linearly:
$\lim\limits_{t \rightarrow 0}\frac{d(S, s+tv)}{t} = 0.$
{\bf \purple Then the set $CS$ of all pairs $(s, v), s\in S, v\in C_s S$
is a relative vector space over $S$}.

\newpage

{\bf \blue Total space of a vector bundle}


\definition
Let $B \arrow M$ be a smooth locally trivial
fibration with fiber $\R^n$.
Assume that  $B$ is equipped with a structure
of relative vector space over $M$, and all the maps
used in the definition of a relative vector space
are smooth. Then $B$
is called {\bf\blue a total space of a vector bundle.}

\remark
Let $\pi:\; B \arrow M$ be a total space of a vector bundle, 
$U\subset M$ open subset, and ${\cal B}(U)$ the space of all smooth sections
of $\pi^{-1}(U)\stackrel \pi \arrow U$. {\bf \purple Then
  ${\cal B}$ is a locally free sheaf of $C^\infty M$-modules}.

\theorem
{\bf \red Every locally free sheaf $C^\infty M$-modules is defined
from a total space of a vector bundle,} which
is determined uniquely by a sheaf.

The proof will be a couple of slides below.

\newpage

{\bf \blue Fiber of a locally free sheaf}


\definition
Let ${\cal B}$ be an $n$-dimensional locally free
sheaf of $C^\infty$-modules on $M$,
$x\in M$ a point,  ${\cal B}_x$ the space of germs of ${\cal B}$
in $x$, and ${\goth m}_x\subset C^\infty_x M$
the maximal ideal in the ring of germs $C^\infty_x M$
of smooth functions. Define {\bf\blue the fiber} of
${\cal B}$ in $x$ as a quotient
${\cal B}_x/{\goth m}_x{\cal B}_x$.
A fiber of ${\cal B}$ is denoted
${\cal B}\restrict x$.

\remark
{\bf \purple A  fiber of an $n$-dimensional bundle
is an $n$-dimensional vector space.}

\remark
Let  ${\cal B}= C^\infty M^n$, and  
$b\in {\cal B}\restrict x$ a point of a fiber, represented
by a germ $\phi \in {\cal B}_x=C^\infty_m M^n$,
$\phi=(f_1, ..., f_n)$. Consider a map $\Psi$ from the set
of all fibers ${\cal B}$ to $M \times \R^n$,
mapping $(x, \phi=(f_1, ..., f_n))$
to $(f_1(x), ..., f_n(x))$. {\bf \red Then $\Psi$ is bijective.}
Indeed, ${\cal B}\restrict x=\R^n$.


\newpage

{\bf \blue Total space of a vector bundle from its sheaf
  of sections}


\definition
Let ${\cal B}$ be an $n$-dimensional locally free sheaf
of $C^\infty$-modules. Denote the set of all
vectors in all fibers of  ${\cal B}$ over all points of $M$
by $\Tot {\cal B}$. Let  $U\subset M$ be an open subset of $M$,
with  ${\cal B}\restrict U$ a trivial bundle.
Using the local bijection $\Tot {\cal B}(U)=U \times \R^n$
we consider topology on $\Tot {\cal B}$
induced by open subsets in 
$\Tot {\cal B}(U)=U \times \R^n$ for all
open subsets $U\subset M$ and all trivializations of
${\cal B}\restrict U$. Then
$\Tot {\cal B}$ is called {\bf \blue a total space of a
  vector bundle ${\cal B}$.}

\claim 
The space $\Tot {\cal B}$ with this topology {\bf \purple is
a locally trivial fibration over $M$, with fiber $\R^n$.}
Moreover, it is a relative vector space over $M$, and
{\bf \purple the sheaf of smooth sections of $\Tot {\cal B}\arrow M$
is isomorphic to ${\cal B}$.}


\remark {\bf \red This gives an equivalence between locally free
sheaves of $\C^\infty$-modules and the 
total spaces of vector bundles,} defined abstractly in
terms of locally trivial fibrations.




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