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characterizing free and invariant elements
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Expand Up @@ -792,11 +792,11 @@ \subsection{First examples}
\begin{remark}
\label{rem:whatAREabeliangroups}

Abelian groups have the amazing property that the classifying types are themselves identity types (of certain $2$-types).
Abelian groups have the amazing property that their classifying types are themselves identity types (of certain $2$-types).
This can be used to give a very important characterization of what it means to be abelian.
We will return to this point in \cref{sec:abelian-groups}.

Alternatively, the reference to $\isAb$ in the definition of abelian groups is avoidable using the ``one point union'' of pointed types $X\vee Y$ of \cref{def:wedge} below. (It is the sum of $X$ and $Y$ where the base points are identified.). \cref{xca:whatAREabeliangroups}
Alternatively, the reference to underlying symmetries in the definition of abelian groups is avoidable using the ``one point union'' of pointed types $X\vee Y$ of \cref{def:wedge} below. (It is the sum of $X$ and $Y$ where the base points are identified.). \cref{xca:whatAREabeliangroups}
\marginnote{%
\begin{tikzcd}[ampersand replacement=\&]
\BG\vee\BG\ar[r,"\text{fold}"]\ar[d,"\text{inclusion}"'] \& \BG \\
Expand Down Expand Up @@ -1457,19 +1457,16 @@ \section{Homomorphisms}

\begin{example}
\label{ex:Zinitial}
\cref{cha:circle} was all about the circle $S^1$ and its role as a
\cref{cha:circle} was all about the circle $\Sc$ and its role as a
``universal symmetry'' and how it related to the integers. In our
current language, $\ZZ\jdeq\mkgroup(\Sc,\base)$ and much\footnote{%
Not all: $\BG$ is a groupoid and not an arbitrary type,
cf.~\cref{sec:inftygps}.} of the
universality of $S^1$ is found in the following observation. If $G$ is a
universality of $\Sc$ is found in the following observation. If $G$ is a
group, then \cref{cor:circle-loopspace} yields an equivalence of sets
%the evaluation equivalence
%$\ev_{\BG_\div}:(S^1\to \BG_\div)\we \sum_{y:\BG_\div}(y=y)$ of
\[
\ev_{\BG}:\left((S^1,\base)\ptdto \BG\right)\equivto \USymG,
\quad
%f_\pt^{-1}\ap{f_\div}(\Sloop)f_\pt
\ev_{\BG}:\left((\Sc,\base)\ptdto \BG\right)\equivto \USymG,
\quad
\ev_{\BG}(f_\div,f_\pt)\defeq\loops(f_\div,f_\pt)(\Sloop).
\]
The domain of this equivalence is $\BHom(\ZZ,G)$.
Expand Down Expand Up @@ -2330,6 +2327,70 @@ \subsection{Transitive $G$-sets}
we only get injectivity, not an equivalence.
We'll study exactly when we get surjectivity in~\cref{sec:normal}
on ``normal'' subgroups.
\Cref{fig:not-normal} illustrates what can go wrong.
\begin{marginfigure}
\noindent\begin{tikzpicture}[scale=.1]
\node[dot,label=above:$x$] (two) at (0,10) {};
\node[dot] (one) at (0, 6) {};
\node[dot] (zero) at (0, 2) {};
\node[dot] (base) at (0,-5) {};

\pgfmathsetmacro\cc{.55228475}% = 4/3*tan(pi/8)
\pgfmathsetmacro\cy{2*\cc}%
\pgfmathsetmacro\cx{10*\cc}%
\pgfmathsetmacro\intx{3.5}%
\pgfmathsetmacro\inty{1.5}%
\pgfmathsetmacro\ay{.35165954}%

% right 3-cycle
\draw (zero.center) .. controls ++(0,-\cy+\ay) and ++(-\cx,-\ay)
.. (10,1) .. controls ++(\cx,+\ay) and ++(0,-\cy-\ay)
.. (20,4)
\foreach \y in {4,8} {
.. controls ++(0,\cy + \ay) and ++(\cx,-\ay)
.. (10,3 + \y) .. controls ++(-\cx,\ay) and ++(0,\cy-\ay)
.. (0,2 + \y) .. controls ++(0,-\cy+\ay) and ++(-\cx,-\ay)
.. (10,1 + \y) .. controls ++(\cx,\ay) and ++(0,-\cy-\ay)
.. (20,4 + \y) }
.. controls ++(0,+\cc) and ++(\cx,\ay)
.. (10+\intx,12 + \inty) .. controls ++(-\cx,-\ay) and ++(\cx,\ay)
.. (10-\intx,2 + \inty) .. controls ++(-\cx,-\ay) and ++(0,\cc)
.. (zero.center);

% left 2-cycle
\draw (one.center) .. controls ++(0,-\cy+\ay) and ++(\cx,-\ay)
.. (-10,5) .. controls ++(-\cx,+\ay) and ++(0,-\cy-\ay)
.. (-20,8) .. controls ++(0,\cy + \ay) and ++(-\cx,-\ay)
.. (-10,11) .. controls ++(+\cx,\ay) and ++(0,\cy-\ay)
.. (two.center) .. controls ++(0,-\cy+\ay) and ++(\cx,-\ay)
.. (-10,9) .. controls ++(-\cx,\ay) and ++(0,-\cy-\ay)
.. (-20,12) .. controls ++(0,+\cc) and ++(-\cx,\ay)
.. (-10-\intx,12 + \inty) .. controls ++(\cx,-\ay) and ++(-\cx,\ay)
.. (-10+\intx,6 + \inty) .. controls ++(\cx,-\ay) and ++(0,\cc)
.. (one.center);

% left 1-cycle
\draw (zero.center) .. controls ++(0,\cy) and ++(\cx,0)
.. (-10,4) .. controls ++(-\cx,0) and ++(0,\cy)
.. (-20,2) .. controls ++(0,-\cy) and ++(-\cx,0)
.. (-10,0) .. controls ++(\cx,0) and ++(0,-\cy)
.. (zero.center);

% base right
\draw (base.center) .. controls (0,-5+\cy) and ++(-\cx,0)
.. (10,-3) .. controls ++(\cx,0) and ++(0,\cy)
.. (20,-5) .. controls ++(0,-\cy) and ++(\cx,0)
.. (10,-7) .. controls ++(-\cx,0) and ++(0,-\cy) .. (base.center);
% base left
\draw (base.center) .. controls (0,-5 + \cy) and (-10+\cx,-3)
.. (-10,-3) .. controls (-10-\cx,-3) and (-20,-5 + \cy)
.. (-20,-5) .. controls (-20,-5 - \cy) and (-10-\cx,-7)
.. (-10,-7) .. controls (-10+\cx,-7) and (0,-5 - \cy)
.. (base.center);
\end{tikzpicture}
\caption{A $\mkgroup(\Sc\vee\Sc)$-set for which $\protect\ev_x$ is not surjective.}
\label{fig:not-normal}
\end{marginfigure}

\begin{lemma}
\label{lem:evisinjwhentransitive}
Expand Down Expand Up @@ -2409,7 +2470,7 @@ \subsection{Fixed points and orbits}
\label{sec:fixpts-orbits}
We now return to some important constructions involving $G$-sets for a group $G$.
However, since they make equally good sense for \emph{$G$-types} for \aninftygp
$G$, we'll work in that generality.
$G$, we'll mostly work in that generality.
\begin{definition}
\label{def:orbittype}
If $X\colon G\to\UU$, then the \emph{orbit type}\index{orbit type}
Expand All @@ -2429,18 +2490,34 @@ \subsection{Fixed points and orbits}
\[
X / G \defeq \Trunc{X_{hG}}_0.
\]
We say that the action is \emph{transitive} if $X / G$ is contractible.
We say that the action is \emph{transitive}\index{transitive}
if $X / G$ is contractible.
\end{definition}

\begin{xca}
Show that the above notion of transitive coincides with the one we introduced in \cref{def:transitiveGset} for a $G$-set $X$ for an ordinary group $G$:
that $X/G$ is contractible exactly encodes that there is just one ``orbit'':
there is some $x:X(\shape_G)$ so that for any $y:X(\shape_G)$
there is a $g:\USymG$ such that $x=g\cdot y$.
\end{xca}

We have seen many instances of orbit types before:
When $G$-sets are considered as \coverings $f : A \to \BG$,
they are the domains $A$.
Recall for example~\cref{fig:two-comp-S1-cover},
showing an action of $\ZZ$ on $\set{1,2,3,4,5}$ with no fixed points
and an orbit type equivalent to a sum of two circles.
In~\cref{fig:ZZ-set-orbits}, we show a similar $\ZZ$-set,
with underlying set $\set{0,1,2,3,4,5}$, three orbits,
and $5$ as a single fixed point.

\begin{marginfigure}
\begin{tikzpicture}[scale=.15]
\node (Sc) at (0,-5) {$\B\ZZ$};
\node[dot,label=left:$5$] (four) at (-10,22) {};
\node[dot,label=left:$4$] (three) at (-10,18) {};
\node[dot,label=left:$3$] (two) at (-10,10) {};
\node[dot,label=left:$2$] (one) at (-10, 6) {};
\node[dot,label=left:$1$] (zero) at (-10, 2) {};
\node[dot] (D) at (-10,-5) {};
\node[dot,label=left:$5$] (five) at (-10,30) {};
\node[dot,label=left:$4$] (four) at (-10,22) {};
\node[dot,label=left:$3$] (three) at (-10,18) {};
\node[dot] (base) at (-10,-5) {};
\node[label=left:$\Sloop$] (Sloop) at (10,-5) {};

\pgfmathsetmacro\cc{.55228475}% = 4/3*tan(pi/8)
Expand All @@ -2462,7 +2539,7 @@ \subsection{Fixed points and orbits}
and (-\intx + \cx,20 + \ay)
.. (-\intx,18 + \inty) .. controls (-\intx - \cx,18 + \inty - \ay)
and (-10,18 + \cc) .. (-10,18);
\draw (-10,2) .. controls (-10,2 - \cy + \ay) and (-\cx,1 - \ay)
\draw[casblue] (-10,2) .. controls (-10,2 - \cy + \ay) and (-\cx,1 - \ay)
.. (0,1) .. controls (\cx,1 + \ay) and (10,4 - \cy - \ay)
.. (10,4)
\foreach \y in {4,8} {
Expand All @@ -2476,56 +2553,174 @@ \subsection{Fixed points and orbits}
and (-\intx + \cx,4 + \ay)
.. (-\intx,2 + \inty) .. controls (-\intx - \cx,2 + \inty - \ay)
and (-10,2 + \cc) .. (-10,2);
\draw (10,-5) .. controls (10,-5 + \cy) and (\cx,-3)
.. (0,-3) .. controls (-\cx,-3) and (-10,-5 + \cy)
.. (-10,-5) .. controls (-10,-5 - \cy) and (-\cx,-7)
.. (0,-7) .. controls (\cx,-7) and (10,-5 - \cy) .. (10,-5);
\draw (10,-5) .. controls ++(0,\cy) and ++(\cx,0)
.. (0,-3) .. controls ++(-\cx,0) and ++(0,\cy)
.. (-10,-5) .. controls ++(0,-\cy) and ++(-\cx,0)
.. (0,-7) .. controls ++(\cx,0) and ++(0,-\cy) .. (10,-5);

\draw (10,30) .. controls ++(0,\cy) and ++(\cx,0)
.. (0,32) .. controls ++(-\cx,0) and ++(0,\cy)
.. (-10,30) .. controls ++(0,-\cy) and ++(-\cx,0)
.. (0,28) .. controls ++(\cx,0) and ++(0,-\cy) .. (10,30);

\node[dot,label=left:$2$,casred] (two) at (-10,10) {};
\node[dot,label=left:$1$,casred] (one) at (-10, 6) {};
\node[dot,label=left:$0$,casred] (zero) at (-10, 2) {};
\end{tikzpicture}
\caption{A $\ZZ$-set with three orbits and one fixed point.}
\label{fig:ZZ-set-orbits}
\end{marginfigure}
We have seen many instances of orbit types before:
When $G$-sets are considered a \coverings $f : A \to \BG$,
they are the domains $A$.
Recall for example~\cref{fig:two-comp-S1-cover}, reproduced in the margin,
showing an action of $\ZZ$ on $\bn 5$ with no fixed points
and an orbit type equivalent to $\Sc\amalg\Sc$.

\begin{xca}
Show that the above notion of transitive coincides with the one we introduced in \cref{def:transitiveGset} for a $G$-set $X$ for an ordinary group $G$:
that $X/G$ is contractible exactly encodes that there is just one ``orbit'':
there is some $x:X(\shape_G)$ so that for any $y:X(\shape_G)$
there is a $g:\USymG$ such that $x=g\cdot y$.
\end{xca}
In~\cref{fig:ZZ-set-orbits} we have highlighted a single component of
the orbit type in blue (\ie corresponding to an element of the set of orbits),
and we see that it contains a subset of the underlying set,
the three red elements $\set{0,1,2}$.
Such a set is what is traditionally called an orbit.
This connection is emphasized in the following result.

\begin{lemma}\label{lem:orbit-equiv}
The map
\[
[\blank] : X(\shape_G) \to X/G, \qquad
[x] \defeq \settrunc{(\shape_G,x)}
\]
is surjective, and $[x] = [y]$ is equivalent to
$\exists_{g:\USymG}(g\cdot x = y)$.
\end{lemma}
\begin{proof}
Surjectivity follows from the connectivity of $\BG$.
By~\cref{rem:set-trunc-as-quotient},
$X/G \jdeq \setTrunc{X_{hG}}$ is itself the
set quotient of $X_{hg} \jdeq \sum_{z:\BG}X(z)$ by
the relation $\sim$ defined by letting $(z,x)\sim(w,y)$
if and only if $\exists_{g:z\eqto w}(g\cdot x=y)$.
Thus,~\cref{thm:quotient-property} gives the
desired conclusion.
\end{proof}
Thus, both the underlying set $X(\shape_G)$ and the orbit type
$X_{hg}$ have equivalence relations with quotient set $X/G$.\footnote{%
This also justifies the notation $X/G$.
We have a diagram of surjective maps:
\[
\begin{tikzcd}[ampersand replacement=\&]
X(\shape_G) \ar[rr,"{x\mapsto(\shape_G,x)}"]\ar[dr,"{[\blank]}"']
\& \& X_{hG}\ar[dl,"{\settrunc\blank}"] \\
\& X/G \&
\end{tikzcd}
\]}
The equivalence classes of both are important:
\begin{definition}\label{def:orbit-stabilizer}
If $X : \BG \to \Set$ is a $G$-set, and $x : X(\shape_G)$ is an
element of the underlying set, then we let
\begin{enumerate}
\item $G_x \defeq \Aut_{X_{hg}}(\shape_G,x)$
be the \emph{stabilizer group}\index{stabilizer}%
\index{group!stabilizer} at $x$, and
\item $G\cdot x \defeq \setof{y : X(\shape_G)}{[x] =_{X/G} [y]}$
be the \emph{orbit}\index{orbit} of $x$.\qedhere
\end{enumerate}
\end{definition}
Note that the classifying type $\BG_x$ of $G_x$
is identified with the fiber of $\settrunc{\blank} : X_{hg} \to X/G$,
and $G\cdot x$ (pointed at $x$)
is identified with the fiber of $[\blank] : X(\shape_G) \to X/G$,
both taken at $[x]$, the orbit containing $x$.

Also, the base point of $\BG_x$ depends on the choice of $x$,
but the underlying type $(\BG_x)_\div$ only depends on $[x]:X/G$.
Thus, we can decompose our diagram by writing $X(\shape_G)$ and $X_{hG}$
as sums of the respective fibers.\footnote{%
Yielding the diagram
\[
\begin{tikzcd}[ampersand replacement=\&,column sep=tiny]
\displaystyle\sum_{b:X/G}G\cdot b \ar[rr]\ar[dr,"\fst"']
\& \& \displaystyle\sum_{b:X/G}(\BG_b)_\div\ar[dl,"\fst"] \\
\& X/G, \&
\end{tikzcd}
\]
where we use $b$ to denote a \emph{bane}/orbit. (Too cute?)}

The stabilizer group $G_x$ comes equipped with a homomorphism
$i_x : \Hom(G_x,G)$, classified by
the projection $\fst:X_{hG} \to \BG$.\footnote{%
Since the projection is still a \covering, $\iota_x$ is a monomorphism
(\cref{lem:eq-mono-cover}), so $G_x$ together with $i_x$
becomes a \emph{subgroup} of $G$. {\color{red}REORDER?}}
There are two possible extreme cases that are important:
\begin{definition}\label{def:invariant-free}
Let $X$ be a $G$-set and $x:X(\shape_G)$ an element of the underlying set.
We say that
\begin{enumerate}
\item $x$ is \emph{invariant}\index{invariant}
if $i_x$ is an isomorphism (so $G_x$ is all of $G$),
\item and $x$ is \emph{free}\index{free}\index{action!free}
is $G_x$ is trivial.
\end{enumerate}
We say that $X$ itself is \emph{free} if each $x:X(\shape_G)$ is free.
\end{definition}

\begin{lemma}\label{lem:invariant-char}
Given a $G$-set $X$, an element $x:(\shape_G)$ is
invariant if and only if the orbit $G\cdot x$ is contractible,
\ie $x = g\cdot x$ for all $g:\USymG$.
\end{lemma}
\begin{proof}
The orbit $G\cdot x$ is the fiber of $\Bi_x : \BG_x \ptdto \BG$
at $\shape_G$. Since $\BG$ is connected,
this is contractible if and only if all fiber of $\Bi$ are contractible,
\ie $\Bi_x$ is an equivalence, which in turn is equivalent to $i_x$
being an isomorphism.
\end{proof}

\begin{lemma}\label{lem:free-pt-char}
Given a $G$-set $X$, an element $x:(\shape_G)$ is\marginnote{%
Doesn't fit well here; move to where?}
free if and only if the (surjective) map
$\blank \cdot x : G \to G\cdot x$ is injective
(and hence a bijection).
\end{lemma}
\begin{proof}
Consider two elements of the orbit, say $g\cdot x,g'\cdot x$ for $g,g':\USymG$.
We have $g\cdot x=g' \cdot x$ if and only if $x = \inv{g} g'\cdot x$
if and only if $\inv{g} g'$ lies in $\USymG_x$.
\end{proof}

When $X : \BG \to \Set$ is a $G$-set for an ordinary group $G$,
there is another reasonable definition of the fixed points,
namely the subset
\[
\setof{x : X(\shape_G)}{\text{$g\cdot x = x$ for all $g:\USymG$}}
\setof{x : X(\shape_G)}{\text{$x$ is invariant}}
\]
of the underlying set consisting of all those elements $x$ that
are unmoved (\ie fixed) by all the symmetries in $G$.
consisting of the invariant elements.
If we evaluate a fixed point $f : \prod_{z:\BG}X(z)$ at $\shape_G$
we do indeed land in this subset. Letting $x\defeq f(\shape_G)$,
we do indeed land in this subset:
Letting $x\defeq f(\shape_G)$,
and taking the dependent action on paths,
$\apd{f}(g) : \pathover x X g x$,
we can use~\cref{def:pathover-trp} to conclude
$\trp[X] g(x)\jdeq g\cdot x = x$, for all $g:\USymG$.

\begin{lemma}\label{lem:fixpts-are-fixed}
For any $G$-set $X$, evaluation gives an equivalence
For any $G$-set $X$, evaluation at $\shape_G$ gives an equivalence
\[
X^{hG} \jdeq \prod_{z:\BG}X(z) \equivto
\setof{x : X(\shape_G)}{\text{$g\cdot x = x$ for all $g:\USymG$}}.
\setof{x : X(\shape_G)}{\text{$x$ is invariant}}.
\]
\end{lemma}
\begin{proof}
Fix $x : X(\shape_G)$ with $g\cdot x = x$ for all $g: \USymG$.
Fix an invariant $x : X(\shape_G)$,
so $g\cdot x = x$ for all $g: \USymG$.
We need to show that the type
\[
\sum_{f : \prod_{z:\BG}X(z)}f(\shape_G)=x
\]
is contractible. [TODO: move stabilizer group here
from~\cref{sec:orbit-stabilizer-theorem}
to make this more conspicuous?]
is contractible.
This is equivalent to the type of pointed sections
of the projection $\fst : (X_{hG},x) \ptdto \BG$.
Since $\BG$ is connected, this is in turn equivalent
to the type of pointed sections of $\Bi_x : \BG_x \ptdto \BG$,
\ie the type of sections of the inclusion homomorphism $i_x:\Hom(G_x,G)$.
This is a proposition, and it's true if and only if $i_x$ is an isomorphism.
\end{proof}

\section{The classifying type is the type of torsors}
Expand Down
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