polish 3.6.1-3

circle.tex

@@ -1598,9 +1598,7 @@ \section{Connected \coverings over the circle}
 By \cref{cor:fib-vs-path}\ref{set-fib-vs-path}, $f$ is a \covering
 over $\Sc$ if and only if each map induced by $f$ on identity types is injective.
 Assume that $f:A\to \Sc$ is a \covering with $A$ connected.
-Let $(a_0,p)$ be an element of $f^{-1}(\base)$.
-(THURSDAY: DISCUSS REMOVING $p$)
-By \cref{xca:component-connected}
+Let $a_0$ be an element of $A$. By \cref{xca:component-connected}
 the condition that \emph{each} $\ap{f}$ is injective can be relaxed 
 to $\ap{f}: (a_0\eqto a_0)\to(f(a_0)\eqto f(a_0))$ being injective.
 Now look at the following diagram, with $\wdg$ the winding number
@@ -1620,10 +1618,13 @@ \section{Connected \coverings over the circle}
 depend on the choice of $a_0$, so the subset only depends on $f$.}
 Clearly, $g_f$ is an injection, so that its fibers are propositions,
 and the image is the subset $\sum_{n:\zet} \inv g_f(n)$.
-%OLD Consider $\setof{q: (\base\eqto\base)}{\ap{f}^{-1}(pqp^{-1})}$ etc.
 Obviously, a classification of connected \coverings over the circle
 also classifies certain subsets of $\zet$, or, equivalently,
-certain subsets of symmetries of $\base$.
+certain subsets of symmetries of $\base$.\marginnote{%
+  In the more general situation in \cref{ch:subgroups}, one needs
+  in addition to $a_0 :A$ a path $p:\base \eqto f(a_0)$ and
+  $\ap{f}$ is composed with conjugation, $\inv p \blank\, p$,
+  See also \cref{def:loops-map}.}
 Such subsets of $\zet$ are closed under addition and negation,
 and those of $(\base\eqto\base)$ are closed under concatenation and inverses,
 since $\ap{f}$, $\wdg$ and ${\Sloop}^{-}$ are compatible with these operations.
@@ -1770,7 +1771,7 @@ \section{Connected \coverings over the circle}
 is equal to the exponential \covering, which in turn corresponds 
 to the infinite cycle $(\zet,\zs)$
 consisting of the set of integers $\zet$ with the successor permutation.
-Another example is the left one of the two examples given
+Another example is the left one of the two examples given in
 \cref{fig:covering}.
 
 We now introduce the remaining \coverings over the circle, first as functions to the circle, then as families of sets.