我正在使用 cleveref 和 hyperref。我将When using the hyperref package, cleveref automatically makes all cross references into hyperlinks to the corresponding reference.其解释为我可以使用在 上定义的目标\label，但\hyperlink{def:ModTop}{Model topology}在 上定义 def:ModTop 的命令\label生成了指向错误页面的链接。是我读错了文档还是有其他问题？
下面的代码会生成一堆指向错误页面的超链接：
\documentclass{article}
\usepackage{amsmath}
%\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{bm}
\usepackage{enumitem}
\usepackage{ifthen}
%\usepackage{mathrsfs}
\usepackage{mathtools}
\usepackage{scalerel}
\usepackage{stix2}[notext,not1] %does not reqire XeTeX or luaTeX
\usepackage{thmtools}
\usepackage{tikz-cd}
\usepackage{xparse} % loads expl3
%See interface3.pdf
\usepackage{xstring}
\usepackage[colorlinks,hidelinks,draft=false]{hyperref}
\usepackage{cleveref}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\def\lemmaautorefname{Lemma} % Needed for \autoref
\newtheorem{corollary}[theorem]{Corollary}
\def\corollaryautorefname{Corollary} % Needed for \autoref
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\def\definitionutorefname{Definition} % Needed for \autoref
\newtheorem{example}[theorem]{Example}
\newtheorem{xca}[theorem]{Exercise}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\hypersetup {
bookmarksnumbered=true,
colorlinks,
pdfinfo={
Author={Shmuel (Seymour J.) Metz},
Keywords={fiber bundles,manifolds},
Subject={Topology},
Title={Local Coordinate Spaces: a proposed unification of manifolds with fiber bundles, and associated machinery}
}
}
\usepackage[draft]{showlabels}
\showlabels{cite}
\showlabels{cref}
\showlabels{crefrange}
\begin{document}
\begin{abstract}
Manifolds and fiber bundles, while superficially different, have strong
parallels; in particular, they are both defined in terms of equivalence
classes of atlases or in terms of maximal atlases, with the atlases
treated as mere adjuncts. This paper introduces the notions of an
m-atlas and of a local coordinate space, and shows that special cases
are equivalent to fiber bundles and manifolds. It defines categories of,
e.g., atlases, and constructs functors among them.
\end{abstract}
\part {Introduction}
\label{part:intro}
Historically, manifolds were formalized in terms of atlases based on
Euclidean spaces and fiber bundles were formalized in terms of atlases
based on product spaces, using either equivalence classes of atlases or
maximal atlases. The concept of pseudo-groups allowed unifying
manifolds and manifolds with boundary. The definitions of fiber bundles
and manifolds have strong parallels, and can be unified in a similar
fashion; there are several ways to do so. The central part of this
paper, LCS, defines an approach using categories and
commutative diagrams that is designed for easy exposition at the
possible expense of abstractness and generality. In particular, I have
chosen to assume the Axiom of Choice (AOC).
This paper treats atlases as objects of interest in their own right,
although it does not give them primacy. It introduces notions that are
convenient for use here and others that, while not used here, may be
useful for future work. \Cref{sec:new} below lists the more imortant of
these.
Although this paper incidentally defines partial equivalents to
manifolds and fiber bundles using model spaces and model atlases, it
proposes the more general
\hyperlink{def:LCS}{Local Coordinate Space (LCS)} in order to explicitly
reflect the role of the group in fiber bundles. \cite{Taxonomy} further
develops some of the notions used here.
A local coordinate space (LCS) is a space (total space) with some
additional structure, including a coordinate model space and an atlas
whose transition functions are restricted to morphisms of the coordinate
model space; one can impose, e.g., differentiability restrictions, by
appropriate choice of the coordinate category. There is an equivalent
paradigm that avoids explicit mention of the total space by imposing a
cocycle (compatibility) condition on the transition functions, but that
approach is beyond the scope of this paper.
This paper defines functors among categories of atlases, categories of
model spaces, categories of local coordinate spaces, categories of
manifolds and categories of fiber bundles; it constructs more machinery
than is customary in order to facilitate the presentation of those
categories and functors.
\section{New concepts and notation}
\label{sec:new}
This paper introduces a significant number of new concepts and some
modifications of the definitions for some conventional concepts. It also
introduces some notation of lesser importance. The following are the
most important.
\begin{enumerate}
\item \hyperlink{def:NCD}{Nearly commutative diagram (NCD)}, NCD at a point,
locally NCD and special cases with related nomenclature.
These are cases where a diagram can be modified to make it commutative.
\item \hyperlink{def:model}{Model space} and related concepts.
A \hyperlink{def:model}{model space}\footnote{
The phrase has been used before, but with a different meaning.
}
is a topological space with a category of permissible open sets and
mappings satisfying specified conditions. It is similar to a
pseudo-group, with some important differences.
\item \hyperlink{def:ModTop}{Model topology} and
\hyperref[def:M-para]{M-paracompactness}
\item \hyperlink{sec:sig}{Signature},
\hyperlink{def:Sigmacomm}{$\Sigma$-commutation} and related concepts.
A signature is a means of characterising the domains and range of a
multi-variable function.
$\Sigma$-commutation is a natuality condition for a sequence of functions.
\item
An \hyperref[def:m-atlas]{m-atlas} is a generalization of atlases in
manifolds.
\item
An \hyperref[def:M-ATLmorph]{m-atlas morphism} is a generalization of a
$Ck$ map between differentiable manifolds. It differs in using both a
function between the manifolds and a function between the coordinate
spaces.
\item
\hyperlink{def:LCS}{Local Coordinate Space (LCS)} and related concepts
\item
\hyperlink{def:lin}{Linear space and related concepts}
A linear space is a subspace of a Banach space or of a Fr\'echet space
that is suitable for defining coordinate patches in a manifold.
\item
\hyperlink{def:trivck}{Trivial $Ck$ linear model space} and related
concepts.
A trivial $Ck$ linear model space is a maximal model space
with morphisms $Ck$ maps between open subsets of a linear space.
\item \hyperlink{def:BunAtl}{$G$-$\rho$ bundle atlases}%
\footnote{Similar to coordinate bundles}
and related concepts
\end{enumerate}
%newpage
\part {Conventions}
\label{part:conv}
An arrow with an Equal-Tilde ($A {\,\to/{>}->>/^{iso}}_\phi B$) represents an
isomorphism. One with a hook ($A \underset{i}{\hookrightarrow} B$)
represents an inclusion map. One with a tail represents
a monomorphism. One with a double head represents a
surjection.
\part {General notions}
\label{part:notions}
This part of the paper describes nomenclature used throughout the paper.
In some cases this reflects new nomenclature or notions, in others it
simply makes a choice from among the various conventions in the
literature.
%newpage
\part{Nearly commutative diagrams}
\label{part:ncd}
The notion of commutative diagrams is very useful in, e.g., Algebraic
Topology. Often one encounters commutative diagrams in which two
outgoing terminal nodes can be connected by a bridging function such
that the resulting diagram is still commutative. This paper uses the
term nearly commutative to describe a restricted class of such diagrams.
\begin{definition}[Nearly commutative diagrams in category $C$]
\label{def:NCD}
$D$ is left (right) nearly commutative in category $C$ iff the
two final nodes are in $C$ and there is a morphism
$\hat{f} maps U_m \to V_n arin C$
($\hat{f'} maps V_n \to U_m arin C$)
making the graph a commutative diagram, as shown in redacted.
$D$ is nearly commutative in category $C$ iff it is eiher
left nearly commutative or right nearly commutative.
\end{definition}
%newpage
\part{Model spaces and allied notions}
\label{part:modeldef}
Let $S$ be a topological space. We need to formalize the notions of an
open cover by sets that are ``well behaved'' in some sense, e.g.,
convex, sufficiently small, and of "well behaved" functions among those
sets, e.g., preserving fibers, smooth. We do this by associating a
category of acceptable sets and functions.
\section{Model spaces}
\begin{definition}[Model spaces]
\label{def:model}
Let $S$ be a topological space and $catname{S}$ a small category whose
objects are open subsets of $S$ and whose morphisms are continuous
functions. $seqname{S} = (S, catname{S})$ is a model space for
$S$ iff
\end{definition}
\section{M-paracompact model spaces}
\label{sec:m-para}
Paracompactness is an important property for topological spaces because
of partitions of unity. There is an analogous property for model
spaces.
\begin{definition}[Model topology]
\label{def:ModTop}
Let $seqname{M} defeq (S, catname{S})$ be a model space for $S$.
Then the model topology $topname{M}^*$ for $M$ is the topology
generated by $Ob(catname{S})$.
\begin{remark}
$topname{M}^*$ is not guarantied to be T0 even if $S$ is T4. However,
$topname{M}^*$ may be normal or regular even if $S$ is not.
\end{remark}
\end{definition}
\begin{definition}[m-paracompactness]
\label{def:M-para}
A model space $seqname{M} defeq (S, catname{S})$ is \\
m-paracompact iff $topname{M}^*$ is regular and every cover of $S$ by
model neighborhoods has a locally finite refinement by model neighborhoods.
\begin{remark}
This is a stronger condition than merely requiring $topname{M}^*$ to
be paracompact.
\end{remark}
\end{definition}
%newpage
\part{Signatures}
\label{part:sig}
Given a sequence of spaces, we need a way to characterise a function
that takes arguments in those spaces and has a value in them. For this
purpose we use a sequence of ordinals where the last ordinal in the
sequence identifies the space containing the function's value.
%newpage
\part{Prestructures}
\label{part:pre}
Prestructures and prestructure morphisms are generalizations of
$Omega$-algebras and $\Omega$-homomorphisms, and are notational
conveniences to simplify imposing commutation relations on multiple
unrelated functions of multiple variables.
%newpage
\part{M-charts and m-atlases}
\label{part:M-charts}
The literature defines fiber bundles and manifolds using the language of
charts, atlases and transition functions; it has multiple equivalent
definitions. Some authors start with topological spaces and define
atlases over them. Some start with abstract sets and define atlases over
them, deriving the topology from the atlas. Some start with an indexed
set of open patches in a coordinate space and transition functions among
them satisfying a cocycle (compatibility) constraint, and then derive
the total space as a quotient space of the disjoin union of the patches.
Some authors use maximal atlases while others use equivalence classes of
atlases.
\section{M-atlases}
\label{sec:M-atlas}
A set of charts can be atlases for different coordinate model spaces
even if it is for the same total model space. In order to aggregate
atlases into categories, there must be a way to distinguish them.
Including the two\footnote{
The spaces are redundant, but convenient.}
spaces in the definitions of the categories serves the purpose.
\begin{definition}[M-atlases]
\label{def:M-atlas}
Let $seqname{A}$ be a set of mutually m-compatible m-charts of
$seqname{E}=(E, catname{E})$ in the coordinate space
$seqname{C}=(C,catname{C})$. Then $seqname{A}$ is an m-atlas of
$seqname{E}$ in the coordinate space $seqname{C}$, abbreviated
$isAtl_Ob(seqname{A}, seqname{E}, seqname{C})$, iff $seqname{A}$
covers $E$. $seqname{A}$ is a full atlas of $seqname{E}$ in the
coordinate space $seqname{C}$, abbreviated
$full{isAtl_Ob}(seqname{A}, E, seqname{C})$, iff $seqname{A}$
covers $E$ and $\pi_2[seqname{A}]$ covers $C$. The triple
$(seqname{A}, seqname{E}, seqname{C})$ refers to $seqname{A}$
considered as an m-atlas of $seqname{E}$ in the coordinate space
$seqname{C}$.
$seqname{E}$ is the total model space or total space for the atlas and
$seqname{C}$ is the coordinate model space or coordinate space for the
atlas.
\end{definition}
\section{M-atlas morphisms and functors}
\label{sec:M-ATLmorph}
This section introduces a taxonomy for morphisms between m-atlases,
defines categories of m-atlases, defines functors amomg them and proves
some basic reasults.
\subsection{M-atlas morphisms}
\label{sub:M-ATLmorph}
This subsection introduces the notions of m-atlas morphisms, semi-strict
m-atlas morphisms and strict m-atlas morphisms.
\subsubsection{Definitions of m-atlas morphisms}
\begin{definition}[M-atlas morphisms for model spaces]
\label{def:M-ATLmorph}
Let $catname{E}^i$, $catname{C}^i$, $i=1,2$, be model categories,
$seqname{E}^i objin catname{E}^i$,
$seqname{C}^i objin catname{C}^i$, $seqname{A}^i$ be an m-atlas of
$seqname{E}^i$ in the coordinate space $seqname{C}^i$ and
$
funcseqname{f} defeq
(
funcname{f}_0 maps seqname{E}^1 to seqname{E}^2,
funcname{f}_1 maps seqname{C}^1 to seqname{C}^2
)
$
a pair of model functions.
\end{definition}
%newpage
\part{Local Coordinate Spaces}
\label{part:lcs}
This part of the paper defines local coordinate spaces, morphisms among
them and categories of them. The definitions given here are based on
m-atlas morphisms, but they could as well have been based on classic
m-atlas morphisms.
\section{Local M-Sigma and M-Sigma coordinate spaces}
\label{sec:lcs}
\begin{definition}[Local M-Sigma and M-Sigma coordinate spaces]
\label{def:LCS}
Let $catseqname{M} defeq (M_\alpha, \alpha \prec Alpha)$ be
a sequence of categories,
$seqname{M} defeq (M_\alpha, \alpha \prec Alpha)$ be a
sequence of spaces and
$funcseqname{F} defeq (funcname{F}_\gamma, \gamma \prec Gamma)$ a
sequence of functions.
$
seqname{L} defeq
(catseqname{M}, seqname{M}, seqname{A}, funcseqname{F}, \Sigma)
$
is a local $seqname{M}-Sigma$ coordinate space, abbreviated
$isLCS_Ob seqname{L}$, a local $catseqname{M}-\Sigma$ coordinate
space, abbreviated $isLCS_Ob seqname{L}$, and a local
$Alpha-\Sigma$ coordinate space, abbreviated
$isLCS_Ob(seqname{L},Alpha)$, iff
\end{definition}
\part{Equivalence of manifolds}
\label{part:man}
For a manifold\footnote{
The literature has several different definitions of a manifold.
This paper uses one chosen for ease of exposition.
}
the coordinate category is open subsets of a Banach space or more
generally a Fr\'echet space, with an appropriate choice of morphisms.
Choosing a separating hyperplane and half space, with open sets in the
chosen half space, allows manifolds with boundary. Similarly, choosing a
ball with tails allows a manifold with tails.
\section {Linear spaces and linear model spaces}
{
\showlabelsinline
\label{sec:lin}
}
This subsection defines spaces and related model spaces suitable for use
as the coordinate spaces of generalized $Ck$ manifolds.
\begin{definition}[Linear spaces]
{
\showlabelsinline
\label{def:lin}
}
Let $C$ be a a localrcwise connected topological subspace of a real
(complex) Banach or Fr\'echet space. Then $C$ is a real (complex) linear
space. If $C$ has a non-void interior, i.e., contains a ball, then it
is non-degenerate.
\end{definition}
\begin{definition}[Trivial \ensuremath{Ck} linear model spaces]
\label{def:trivck}
Let $C$ be a real (complex) linear space and $catname{C}$ the category
of all $Ck$ functions between open sets of $C$. Then
$Triv[Ck-]{C} defeq (C, catname{C})$ is the trivial $Ck$ linear
model space of $C$ and $Triv[Ck-]{C}$ is a real (complex) trivial
$Ck$ linear model space.
\end{definition}
\part{Equivalence of fiber bundles}
\label{part:bun}
For fiber bundles\footnote{
The literature has several definitions of fiber bundle. This paper
uses one chosen for clarity of exposition. It differs from
cite[p.~8]{TopFib} in that, e.g., it uses the machinery of maximal
atlases rather than equivalence classes of coordinate bundles, the
nomenclature differs in several minor regards.
},
the adjunct spaces are the base space $seqname{X} = (X,catname{X})$,
the fiber $seqname{Y} = (Y, catname{Y})$
and the group $G$;
the category of the coordinate space is the category of Cartesian
products $\{U \times Y \mid U objin catname{X}\}$ of model
neighborhoods in the base space with the entire fiber, with morphisms
$funcname{t} maps U \times Y \to U \times Y$ that preserve the
fibers, i.e., $\pi_1 \circ funcname{t} = \pi_1$, and are
generated by the group action on the fiber
\section{Bundle atlases}
\label{sec:bunatlases}
A set of charts can be atlases for different fiber bundles even if it is
for the same total model space, base space and fiber. In order to
aggregate atlases into categories, there must be a way to distinguish
them. Including the spaces\footnote{
The spaces are redundant, but convenient.},
group and group action in the definitions of the categories serves the
purpose.
\begin{definition}[Bundle atlases]
\label{def:BunAtl}
Let $seqname{B} defeq (E, X, Y, \pi, G, \rho)$, be a protobundle. Then
$seqname{A}$ is a bundle atlas of $B$, abbreviated
$isAtl[Bun]_Ob(seqname{A}, seqname{B})$ and $seqname{A}$ is a
$\pi$-$G$-$\rho$-bundle atlas of $E$ in the coordinate space
$X \times Y$, abbreviated
$isAtl[Bun]_Ob(seqname{A}, E, X, Y, \pi, G, \rho)$, iff it consists
of a set of mutually $G$-$\rho$-compatible $Y$-$\pi$-bundle charts of
$E$ in the coordinate space $X \times Y$ that covers $E$\footnote{
There is no need to introduce the concept of a full
$\pi$-$G$-$\rho$-bundle atlas because a $\pi$-$G$-$\rho$-bundle
atlas is automatically full.
}.
\end{definition}
\end{document}
答案1
而hyperlink应该使用hyperref[]{}。或者将标签定义为\hypertarget{def:ModTop}{}然后就可以使用\hyperlink。不过，我会\hyperref在这里使用 。