Category theory & geometric measure theory?
$begingroup$
My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research interests, too.
Q. Is there any link between these areas (GMT, PDEs) and category theory? Could categories be useful to study, e.g. fine properties of BV functions? Or to understand the concept of entropy solution to a non-linear conservation law?
I have looked for similar questions, but I have not found anything as "explicit" as I want. I am not interested into possible definitions of category theory, nor I am looking for some apologies of this area or of that area (everything is math and deserved to be studied). What I would like to know is if it is possible to frame some "fine" definitions/theorems of the areas I am working in by means of the language of CT.
ct.category-theory soft-question measure-theory geometric-measure-theory
asked 6 hours ago
RomeoRomeo
13912
$endgroup$
add a comment |
$begingroup$
My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research interests, too.
Q. Is there any link between these areas (GMT, PDEs) and category theory? Could categories be useful to study, e.g. fine properties of BV functions? Or to understand the concept of entropy solution to a non-linear conservation law?
I have looked for similar questions, but I have not found anything as "explicit" as I want. I am not interested into possible definitions of category theory, nor I am looking for some apologies of this area or of that area (everything is math and deserved to be studied). What I would like to know is if it is possible to frame some "fine" definitions/theorems of the areas I am working in by means of the language of CT.
ct.category-theory soft-question measure-theory geometric-measure-theory
asked 6 hours ago
RomeoRomeo
13912
$endgroup$
2
$begingroup$
Category theory hasn't really penetrated analysis, so I doubt it.
$endgroup$
– Harry Gindi
5 hours ago
6
$begingroup$
@HarryGindi, I'd disagree with a blanket claim about the irrelevance of category theory to analysis. I include this in my graduate real analysis course the point that the "correct" topology on spaces of smooth functions is demonstrably not a matter of whim, since it must be a (projective) limit of $C^k$ functions. Even more primitively, the "coarseness" of the product topology is explained by its unequivocal categorical definition. The topology on test functions must be the (strict) colimit topology. So I think the viewpoint, if not big theorems, of category theory is very relevant.
$endgroup$
– paul garrett
4 hours ago
2
$begingroup$
see eg Michal Marvan A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno ncatlab.org/nlab/files/MarvanJetComonad.pdf for something on the PDE side, it might be interesting to push this in the direction of geometric measure theory.
$endgroup$
– David Roberts
3 hours ago
$begingroup$
@HarryGindi may not have penetrated to the extent it has algebraic geometry, but that is not to say that it hasn't got some underappreciated connections.
$endgroup$
– David Roberts
3 hours ago
add a comment |
$begingroup$
My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research interests, too.
Q. Is there any link between these areas (GMT, PDEs) and category theory? Could categories be useful to study, e.g. fine properties of BV functions? Or to understand the concept of entropy solution to a non-linear conservation law?
I have looked for similar questions, but I have not found anything as "explicit" as I want. I am not interested into possible definitions of category theory, nor I am looking for some apologies of this area or of that area (everything is math and deserved to be studied). What I would like to know is if it is possible to frame some "fine" definitions/theorems of the areas I am working in by means of the language of CT.
ct.category-theory soft-question measure-theory geometric-measure-theory
asked 6 hours ago
RomeoRomeo
13912
$endgroup$
My background is essentially Geometric Measure Theory and its application to partial differential equations (e.g. linear and non-linear hyperbolic conservation laws). These are currently my research interests, too.
Q. Is there any link between these areas (GMT, PDEs) and category theory? Could categories be useful to study, e.g. fine properties of BV functions? Or to understand the concept of entropy solution to a non-linear conservation law?
I have looked for similar questions, but I have not found anything as "explicit" as I want. I am not interested into possible definitions of category theory, nor I am looking for some apologies of this area or of that area (everything is math and deserved to be studied). What I would like to know is if it is possible to frame some "fine" definitions/theorems of the areas I am working in by means of the language of CT.
ct.category-theory soft-question measure-theory geometric-measure-theory
ct.category-theory soft-question measure-theory geometric-measure-theory
asked 6 hours ago
RomeoRomeo
13912
asked 6 hours ago
RomeoRomeo
13912
asked 6 hours ago
RomeoRomeo
13912
asked 6 hours ago
RomeoRomeo
13912
asked 6 hours ago
RomeoRomeo
13912
13912
2
$begingroup$
Category theory hasn't really penetrated analysis, so I doubt it.
$endgroup$
– Harry Gindi
5 hours ago
6
$begingroup$
@HarryGindi, I'd disagree with a blanket claim about the irrelevance of category theory to analysis. I include this in my graduate real analysis course the point that the "correct" topology on spaces of smooth functions is demonstrably not a matter of whim, since it must be a (projective) limit of $C^k$ functions. Even more primitively, the "coarseness" of the product topology is explained by its unequivocal categorical definition. The topology on test functions must be the (strict) colimit topology. So I think the viewpoint, if not big theorems, of category theory is very relevant.
$endgroup$
– paul garrett
4 hours ago
2
$begingroup$
see eg Michal Marvan A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno ncatlab.org/nlab/files/MarvanJetComonad.pdf for something on the PDE side, it might be interesting to push this in the direction of geometric measure theory.
$endgroup$
– David Roberts
3 hours ago
$begingroup$
@HarryGindi may not have penetrated to the extent it has algebraic geometry, but that is not to say that it hasn't got some underappreciated connections.
$endgroup$
– David Roberts
3 hours ago
add a comment |
2
$begingroup$
Category theory hasn't really penetrated analysis, so I doubt it.
$endgroup$
– Harry Gindi
5 hours ago
6
$begingroup$
@HarryGindi, I'd disagree with a blanket claim about the irrelevance of category theory to analysis. I include this in my graduate real analysis course the point that the "correct" topology on spaces of smooth functions is demonstrably not a matter of whim, since it must be a (projective) limit of $C^k$ functions. Even more primitively, the "coarseness" of the product topology is explained by its unequivocal categorical definition. The topology on test functions must be the (strict) colimit topology. So I think the viewpoint, if not big theorems, of category theory is very relevant.
$endgroup$
– paul garrett
4 hours ago
2
$begingroup$
see eg Michal Marvan A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno ncatlab.org/nlab/files/MarvanJetComonad.pdf for something on the PDE side, it might be interesting to push this in the direction of geometric measure theory.
$endgroup$
– David Roberts
3 hours ago
$begingroup$
@HarryGindi may not have penetrated to the extent it has algebraic geometry, but that is not to say that it hasn't got some underappreciated connections.
$endgroup$
– David Roberts
3 hours ago
2
2
$begingroup$
Category theory hasn't really penetrated analysis, so I doubt it.
$endgroup$
– Harry Gindi
5 hours ago
$begingroup$
Category theory hasn't really penetrated analysis, so I doubt it.
$endgroup$
– Harry Gindi
5 hours ago
6
6
$begingroup$
@HarryGindi, I'd disagree with a blanket claim about the irrelevance of category theory to analysis. I include this in my graduate real analysis course the point that the "correct" topology on spaces of smooth functions is demonstrably not a matter of whim, since it must be a (projective) limit of $C^k$ functions. Even more primitively, the "coarseness" of the product topology is explained by its unequivocal categorical definition. The topology on test functions must be the (strict) colimit topology. So I think the viewpoint, if not big theorems, of category theory is very relevant.
$endgroup$
– paul garrett
4 hours ago
$begingroup$
@HarryGindi, I'd disagree with a blanket claim about the irrelevance of category theory to analysis. I include this in my graduate real analysis course the point that the "correct" topology on spaces of smooth functions is demonstrably not a matter of whim, since it must be a (projective) limit of $C^k$ functions. Even more primitively, the "coarseness" of the product topology is explained by its unequivocal categorical definition. The topology on test functions must be the (strict) colimit topology. So I think the viewpoint, if not big theorems, of category theory is very relevant.
$endgroup$
– paul garrett
4 hours ago
2
2
$begingroup$
see eg Michal Marvan A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno ncatlab.org/nlab/files/MarvanJetComonad.pdf for something on the PDE side, it might be interesting to push this in the direction of geometric measure theory.
$endgroup$
– David Roberts
3 hours ago
$begingroup$
see eg Michal Marvan A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno ncatlab.org/nlab/files/MarvanJetComonad.pdf for something on the PDE side, it might be interesting to push this in the direction of geometric measure theory.
$endgroup$
– David Roberts
3 hours ago
$begingroup$
@HarryGindi may not have penetrated to the extent it has algebraic geometry, but that is not to say that it hasn't got some underappreciated connections.
$endgroup$
– David Roberts
3 hours ago
$begingroup$
@HarryGindi may not have penetrated to the extent it has algebraic geometry, but that is not to say that it hasn't got some underappreciated connections.
$endgroup$
– David Roberts
3 hours ago
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You might want to look at the notion of magnitude:
The magnitude of a metric space: from category theory to geometric measure theory by Tom Leinster and Mark W. Meckes
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
$endgroup$
add a comment |
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: "504"
};
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%2fmathoverflow.net%2fquestions%2f322001%2fcategory-theory-geometric-measure-theory%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
$begingroup$
You might want to look at the notion of magnitude:
The magnitude of a metric space: from category theory to geometric measure theory by Tom Leinster and Mark W. Meckes
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
$endgroup$
add a comment |
$begingroup$
You might want to look at the notion of magnitude:
The magnitude of a metric space: from category theory to geometric measure theory by Tom Leinster and Mark W. Meckes
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
$endgroup$
add a comment |
$begingroup$
You might want to look at the notion of magnitude:
The magnitude of a metric space: from category theory to geometric measure theory by Tom Leinster and Mark W. Meckes
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
$endgroup$
You might want to look at the notion of magnitude:
The magnitude of a metric space: from category theory to geometric measure theory by Tom Leinster and Mark W. Meckes
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
answered 5 hours ago
Thomas KalinowskiThomas Kalinowski
2,47911118
2,47911118
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f322001%2fcategory-theory-geometric-measure-theory%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
2
$begingroup$
Category theory hasn't really penetrated analysis, so I doubt it.
$endgroup$
– Harry Gindi
5 hours ago
6
$begingroup$
@HarryGindi, I'd disagree with a blanket claim about the irrelevance of category theory to analysis. I include this in my graduate real analysis course the point that the "correct" topology on spaces of smooth functions is demonstrably not a matter of whim, since it must be a (projective) limit of $C^k$ functions. Even more primitively, the "coarseness" of the product topology is explained by its unequivocal categorical definition. The topology on test functions must be the (strict) colimit topology. So I think the viewpoint, if not big theorems, of category theory is very relevant.
$endgroup$
– paul garrett
4 hours ago
2
$begingroup$
see eg Michal Marvan A note on the category of partial differential equations, in Differential geometry and its applications, Proceedings of the Conference August 24-30, 1986, Brno ncatlab.org/nlab/files/MarvanJetComonad.pdf for something on the PDE side, it might be interesting to push this in the direction of geometric measure theory.
$endgroup$
– David Roberts
3 hours ago
$begingroup$
@HarryGindi may not have penetrated to the extent it has algebraic geometry, but that is not to say that it hasn't got some underappreciated connections.
$endgroup$
– David Roberts
3 hours ago