Are there more intuitive and/or illustrative reason for why covering maps do not compose besides the...
$begingroup$
In an introductory topology class, one learns about covering maps $p : E to B$. A natural first question to ask is if the composition of two covering maps is also a covering map, and the answer is no. The classical counterexample is the covering map involving a "countable series of hawaiian earrings" (see here), and in fact the only counterexample I've ever heard of.
However, to me it seems that the fact that the covering maps don't form a category should be a more substantial fact. I feel like there should be a better / more high-level explanation than "look at this crazy space!" So I suppose my questions are:
Is there some property of the counter-example in question that is key in the failing of the composability, and can this point us in the direction of other counter-examples?
Along the lines of the above, is there some kind of restriction we can make on the property of the spaces (or the maps) which does indeed make covering spaces into a category?
general-topology soft-question covering-spaces
asked Jan 21 at 17:33
MCTMCT
14.5k42668
$endgroup$
add a comment |
$begingroup$
In an introductory topology class, one learns about covering maps $p : E to B$. A natural first question to ask is if the composition of two covering maps is also a covering map, and the answer is no. The classical counterexample is the covering map involving a "countable series of hawaiian earrings" (see here), and in fact the only counterexample I've ever heard of.
However, to me it seems that the fact that the covering maps don't form a category should be a more substantial fact. I feel like there should be a better / more high-level explanation than "look at this crazy space!" So I suppose my questions are:
Is there some property of the counter-example in question that is key in the failing of the composability, and can this point us in the direction of other counter-examples?
Along the lines of the above, is there some kind of restriction we can make on the property of the spaces (or the maps) which does indeed make covering spaces into a category?
general-topology soft-question covering-spaces
asked Jan 21 at 17:33
MCTMCT
14.5k42668
$endgroup$
$begingroup$
See Rob Arthan's anwer to math.stackexchange.com/q/146976.
$endgroup$
– Paul Frost
Jan 22 at 16:11
add a comment |
$begingroup$
In an introductory topology class, one learns about covering maps $p : E to B$. A natural first question to ask is if the composition of two covering maps is also a covering map, and the answer is no. The classical counterexample is the covering map involving a "countable series of hawaiian earrings" (see here), and in fact the only counterexample I've ever heard of.
However, to me it seems that the fact that the covering maps don't form a category should be a more substantial fact. I feel like there should be a better / more high-level explanation than "look at this crazy space!" So I suppose my questions are:
Is there some property of the counter-example in question that is key in the failing of the composability, and can this point us in the direction of other counter-examples?
Along the lines of the above, is there some kind of restriction we can make on the property of the spaces (or the maps) which does indeed make covering spaces into a category?
general-topology soft-question covering-spaces
asked Jan 21 at 17:33
MCTMCT
14.5k42668
$endgroup$
In an introductory topology class, one learns about covering maps $p : E to B$. A natural first question to ask is if the composition of two covering maps is also a covering map, and the answer is no. The classical counterexample is the covering map involving a "countable series of hawaiian earrings" (see here), and in fact the only counterexample I've ever heard of.
However, to me it seems that the fact that the covering maps don't form a category should be a more substantial fact. I feel like there should be a better / more high-level explanation than "look at this crazy space!" So I suppose my questions are:
Is there some property of the counter-example in question that is key in the failing of the composability, and can this point us in the direction of other counter-examples?
Along the lines of the above, is there some kind of restriction we can make on the property of the spaces (or the maps) which does indeed make covering spaces into a category?
general-topology soft-question covering-spaces
general-topology soft-question covering-spaces
asked Jan 21 at 17:33
MCTMCT
14.5k42668
asked Jan 21 at 17:33
MCTMCT
14.5k42668
asked Jan 21 at 17:33
MCTMCT
14.5k42668
asked Jan 21 at 17:33
MCTMCT
14.5k42668
asked Jan 21 at 17:33
MCTMCT
14.5k42668
14.5k42668
$begingroup$
See Rob Arthan's anwer to math.stackexchange.com/q/146976.
$endgroup$
– Paul Frost
Jan 22 at 16:11
add a comment |
$begingroup$
See Rob Arthan's anwer to math.stackexchange.com/q/146976.
$endgroup$
– Paul Frost
Jan 22 at 16:11
$begingroup$
See Rob Arthan's anwer to math.stackexchange.com/q/146976.
$endgroup$
– Paul Frost
Jan 22 at 16:11
$begingroup$
See Rob Arthan's anwer to math.stackexchange.com/q/146976.
$endgroup$
– Paul Frost
Jan 22 at 16:11
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3082148%2fare-there-more-intuitive-and-or-illustrative-reason-for-why-covering-maps-do-not%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3082148%2fare-there-more-intuitive-and-or-illustrative-reason-for-why-covering-maps-do-not%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
See Rob Arthan's anwer to math.stackexchange.com/q/146976.
$endgroup$
– Paul Frost
Jan 22 at 16:11