Do any authors take the sheaf-theoretic viewpoint on multivalued functions and/or indefinite integrals?












9












$begingroup$


It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.



For example:




  • The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$


  • The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$


  • The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$


  • If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$


  • Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$




Question. Do any published books or articles take this point of view?







calculus real-analysis complex-analysis sheaf-theory multivalued-functions







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
    $endgroup$
    – Cory Griffith
    Jun 29 '18 at 17:49










  • $begingroup$
    @CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
    $endgroup$
    – goblin
    Jul 4 '18 at 15:15


















9












$begingroup$


It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.



For example:




  • The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$


  • The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$


  • The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$


  • If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$


  • Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$




Question. Do any published books or articles take this point of view?







calculus real-analysis complex-analysis sheaf-theory multivalued-functions







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
    $endgroup$
    – Cory Griffith
    Jun 29 '18 at 17:49










  • $begingroup$
    @CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
    $endgroup$
    – goblin
    Jul 4 '18 at 15:15
















9












9








9


0



$begingroup$


It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.



For example:




  • The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$


  • The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$


  • The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$


  • If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$


  • Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$




Question. Do any published books or articles take this point of view?







calculus real-analysis complex-analysis sheaf-theory multivalued-functions







share|cite|improve this question









$endgroup$




It seems to me that multivalued functions and/or indefinite integrals can be thought of as sheaves.



For example:




  • The real square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{R}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{R}$ satisfying $forall x in U(f(x)^2 = x).$


  • The complex square-root function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(f(z)^2 = z).$


  • The complex logarithm function can be viewed as the sheaf $mathcal{F}$ defined on the open sets of $mathbb{C}$ by letting $mathcal{F}(U)$ denote the set of all continuous function $f : U rightarrow mathbb{C}$ satisfying $forall z in U(e^{f(z)} = z).$


  • If $f$ is a continuous real-valued function, we can define $int f(x) dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable functions $F$ on $U$ satifying $F'=f$. More generally, if $mathfrak{f}$ is a sheaf of continuous functions, we can define $int mathfrak{f}(x)dx$ to be the sheaf $mathcal{F}$ such that $mathcal{F}(U)$ is the set of all differentiable function $F : U rightarrow mathbb{R}$ satisfying $F' in mathfrak{f}(U).$


  • Similar statements to the above probably hold for the complex case, allowing us to prove claims like $$log(z) subseteq int frac{1}{z}dz,$$ etc. where $log$ is viewed as the sheaf-theoretic inverse of $z mapsto e^z$ as described above, and the $subseteq$ in this context really means something like: for all open $U subseteq mathbb{C}$, we have $$(z mapsto log(z))(U) subseteq left(z mapsto int frac{1}{z}dzright)(U).$$




Question. Do any published books or articles take this point of view?







calculus real-analysis complex-analysis sheaf-theory multivalued-functions




calculus real-analysis complex-analysis sheaf-theory multivalued-functions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Jun 9 '18 at 3:51









goblingoblin

36.8k1159192




36.8k1159192








  • 1




    $begingroup$
    I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
    $endgroup$
    – Cory Griffith
    Jun 29 '18 at 17:49










  • $begingroup$
    @CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
    $endgroup$
    – goblin
    Jul 4 '18 at 15:15
















  • 1




    $begingroup$
    I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
    $endgroup$
    – Cory Griffith
    Jun 29 '18 at 17:49










  • $begingroup$
    @CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
    $endgroup$
    – goblin
    Jul 4 '18 at 15:15










1




1




$begingroup$
I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49




$begingroup$
I don't have a positive or negative answer, but I would expect any utility from this in the complex case to be subsumed by the theory of branched coverings.
$endgroup$
– Cory Griffith
Jun 29 '18 at 17:49












$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15






$begingroup$
@CoryGriffith, can you elaborate a bit? For example, what is the definition of a multivalued function in the theory of branched coverings? Can multivalued functions be composed? Do they form a category? If $f$ and $g$ are multivalued self-maps of the complex plane, does $f+g$ make sense?
$endgroup$
– goblin
Jul 4 '18 at 15:15












1 Answer
1






active

oldest

votes


















2












$begingroup$

Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks. I'm excited to get it!
    $endgroup$
    – goblin
    Dec 15 '18 at 5:58











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2813129%2fdo-any-authors-take-the-sheaf-theoretic-viewpoint-on-multivalued-functions-and-o%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









2












$begingroup$

Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks. I'm excited to get it!
    $endgroup$
    – goblin
    Dec 15 '18 at 5:58
















2












$begingroup$

Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Thanks. I'm excited to get it!
    $endgroup$
    – goblin
    Dec 15 '18 at 5:58














2












2








2





$begingroup$

Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.






share|cite|improve this answer









$endgroup$



Yes. L. Ahlfors, Complex analysis, McGraw Hill 1979. This is a standard graduate text on complex analysis in the USA.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Dec 15 '18 at 5:26









Alexandre EremenkoAlexandre Eremenko

2,415921




2,415921












  • $begingroup$
    Thanks. I'm excited to get it!
    $endgroup$
    – goblin
    Dec 15 '18 at 5:58


















  • $begingroup$
    Thanks. I'm excited to get it!
    $endgroup$
    – goblin
    Dec 15 '18 at 5:58
















$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58




$begingroup$
Thanks. I'm excited to get it!
$endgroup$
– goblin
Dec 15 '18 at 5:58


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2813129%2fdo-any-authors-take-the-sheaf-theoretic-viewpoint-on-multivalued-functions-and-o%23new-answer', 'question_page');
}
);

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







-

Popular posts from this blog

Bressuire

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more

Cabo Verde

Image
  Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
Read more

Gyllenstierna

Image
.mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
Read more