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 $C$
Multi tool use
Can anything be seen from the center of the Boötes void? How dark would it be?
What was the first language to use conditional keywords?
Is grep documentation about ignoring case wrong, since it doesn't ignore case in filenames?
How to write the following sign?
Multiple OR (||) Conditions in If Statement
Are all finite dimensional hilbert spaces isomorphic to spaces with Euclidean norms?
Disembodied hand growing fangs
Maximum summed subsequences with non-adjacent items
What is the font for "b" letter?
Localisation of Category
Does the Weapon Master feat grant you a fighting style?
Did Krishna say in Bhagavad Gita "I am in every living being"
Can an alien society believe that their star system is the universe?
The code below, is it ill-formed NDR or is it well formed?
Amount of permutations on an NxNxN Rubik's Cube
Is there hard evidence that the grant peer review system performs significantly better than random?
Generate an RGB colour grid
Why is it faster to reheat something than it is to cook it?
What's the meaning of "fortified infraction restraint"?
When a candle burns, why does the top of wick glow if bottom of flame is hottest?
Why is my ESD wriststrap failing with nitrile gloves on?
Why aren't air breathing engines used as small first stages?
SF book about people trapped in a series of worlds they imagine
How to install press fit bottom bracket into new frame
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
asked 3 hours ago
KarlPeterKarlPeter
7411416
$endgroup$
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
asked 3 hours ago
KarlPeterKarlPeter
7411416
$endgroup$
$begingroup$
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
asked 3 hours ago
KarlPeterKarlPeter
7411416
$endgroup$
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
asked 3 hours ago
KarlPeterKarlPeter
7411416
asked 3 hours ago
KarlPeterKarlPeter
7411416
asked 3 hours ago
KarlPeterKarlPeter
7411416
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.
answered 2 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
$begingroup$
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.
answered 2 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
$begingroup$
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.
answered 2 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
$begingroup$
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.
answered 2 hours ago
Kevin CarlsonKevin Carlson
34k23473
$endgroup$
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 2 hours ago
Kevin CarlsonKevin Carlson
34k23473
answered 2 hours ago
Kevin CarlsonKevin Carlson
34k23473
answered 2 hours ago
Kevin CarlsonKevin Carlson
34k23473
answered 2 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
721kdMElKCWl,ts1wXr NwI 6RR0 0 7,75rikRy ePAQ8tkitYzjsFsJKf,a,1mN20FHfdsMzohg npvh,mruFbz5eYy2jopc5LLm
$begingroup$
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