Localisation of Category Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Category Theory with and without ObjectsWhat is a “foo” in category theory?Question on category theoryN-Tuples or N-functions in category theoryCan the choice of definition of morphisms for a slice category be justified categorically?Morphisms as zigzags, composition as concatenationSome doubts in Category TheoryCoslice category in 2-categoriesStatus of pairs/tuples in category theoryA new category $C^*$ from a given category $C$
Project Euler #1 in C++
What is the topology associated with the algebras for the ultrafilter monad?
Why aren't air breathing engines used as small first stages?
Selecting user stories during sprint planning
A term for a woman complaining about things/begging in a cute/childish way
What are the out-of-universe reasons for the references to Toby Maguire-era Spider-Man in Into the Spider-Verse?
Does the Weapon Master feat grant you a fighting style?
Drawing without replacement: why the order of draw is irrelevant?
How do I use the new nonlinear finite element in Mathematica 12 for this equation?
Should I follow up with an employee I believe overracted to a mistake I made?
Most bit efficient text communication method?
Source for Esri sample data from 911 Hot Spot Analysis
Using audio cues to encourage good posture
Export Xpubkey from Bitcoin Core
Sum letters are not two different
How to install press fit bottom bracket into new frame
Maximum summed subsequences with non-adjacent items
How to compare two different files line by line in unix?
How fail-safe is nr as stop bytes?
Multiple OR (||) Conditions in If Statement
Do I really need to have a message in a novel to appeal to readers?
An adverb for when you're not exaggerating
What was the first language to use conditional keywords?
Why wasn't DOSKEY integrated with COMMAND.COM?
Localisation of Category
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Category Theory with and without ObjectsWhat is a “foo” in category theory?Question on category theoryN-Tuples or N-functions in category theoryCan the choice of definition of morphisms for a slice category be justified categorically?Morphisms as zigzags, composition as concatenationSome doubts in Category TheoryCoslice category in 2-categoriesStatus of pairs/tuples in category theoryA new category $C^*$ from a given category $C$
$begingroup$
I have a quite broad question about localisations of categories:
Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?
category-theory
$endgroup$
add a comment |
$begingroup$
I have a quite broad question about localisations of categories:
Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?
category-theory
$endgroup$
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago
add a comment |
$begingroup$
I have a quite broad question about localisations of categories:
Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?
category-theory
$endgroup$
I have a quite broad question about localisations of categories:
Often I encountered that the construction of such indeced category is motivated by considering zigzag morphisms. Could anybody explain the connection/ the essence behind this motivation?
category-theory
category-theory
asked 3 hours ago
KarlPeterKarlPeter
7411416
7411416
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago
add a comment |
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.
$endgroup$
$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago
add a comment |
Your Answer
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%2f3192674%2flocalisation-of-category%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.
$endgroup$
$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago
add a comment |
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.
$endgroup$
$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago
add a comment |
$begingroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.
$endgroup$
If you wanted to localize a noncommutative ring, you'd need fractions like $ab^-1cd^-1...$ There's no way to simplify this into a traditional fraction-it need not be equal to $fracacb^-1d^-1$, for instance, because of noncommutativity. The same thing happens in localizing a category. You want to add inverses to things that don't have inverses freely, so what you get is words in the things you originally had together with certain formal inverses, subject to certain relations. This is exactly a zigzag, just viewed diagrammatically rather than syntactically.
answered 3 hours ago
Kevin CarlsonKevin Carlson
34k23473
34k23473
$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago
add a comment |
$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago
$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
For categories, you have not just noncommutativity but compositions which only make sense in one order, since the morphisms have different domains and codomains.
$endgroup$
– Eric Wofsey
2 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago
$begingroup$
@EricWofsey Yes, of course. I wasn't sure whether it aided or occluded the intuition to add that.
$endgroup$
– Kevin Carlson
2 hours ago
add a comment |
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%2f3192674%2flocalisation-of-category%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
$begingroup$
Not sure what you mean by "motivated by considering zigzag morphisms". Zigzag morphisms are the construction, not motivation for it.
$endgroup$
– Eric Wofsey
2 hours ago