What is the fundamental category?
$begingroup$
Given a category $mathcal{C}$, we have a nerve functor
$$mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$$
that assigns to $mathcal{C}$ its nerve $mathrm{N}(mathcal{C})$. This functor seems to have a left adjoint
$$tau_1 colon mathbf{Set}_{Delta} to mathbf{Cat}$$
that assigns to a simplicial set $X$ its fundamental category, as in Joyal's Notes on Quasi-Categories.
There it also states that the fundamental grouped $pi_1 X$ is obtained by inverting the arrows of $tau_1 X$, but there is no construction of $tau_1 X$.
What is the construction/definition of the fundamental category of a simplicial set $tau_1 X$? What are its objects and morphisms?
reference-request algebraic-topology category-theory simplicial-stuff higher-category-theory
asked Dec 28 '18 at 16:55
user313212user313212
363520
$endgroup$
add a comment |
$begingroup$
Given a category $mathcal{C}$, we have a nerve functor
$$mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$$
that assigns to $mathcal{C}$ its nerve $mathrm{N}(mathcal{C})$. This functor seems to have a left adjoint
$$tau_1 colon mathbf{Set}_{Delta} to mathbf{Cat}$$
that assigns to a simplicial set $X$ its fundamental category, as in Joyal's Notes on Quasi-Categories.
There it also states that the fundamental grouped $pi_1 X$ is obtained by inverting the arrows of $tau_1 X$, but there is no construction of $tau_1 X$.
What is the construction/definition of the fundamental category of a simplicial set $tau_1 X$? What are its objects and morphisms?
reference-request algebraic-topology category-theory simplicial-stuff higher-category-theory
asked Dec 28 '18 at 16:55
user313212user313212
363520
$endgroup$
add a comment |
$begingroup$
Given a category $mathcal{C}$, we have a nerve functor
$$mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$$
that assigns to $mathcal{C}$ its nerve $mathrm{N}(mathcal{C})$. This functor seems to have a left adjoint
$$tau_1 colon mathbf{Set}_{Delta} to mathbf{Cat}$$
that assigns to a simplicial set $X$ its fundamental category, as in Joyal's Notes on Quasi-Categories.
There it also states that the fundamental grouped $pi_1 X$ is obtained by inverting the arrows of $tau_1 X$, but there is no construction of $tau_1 X$.
What is the construction/definition of the fundamental category of a simplicial set $tau_1 X$? What are its objects and morphisms?
reference-request algebraic-topology category-theory simplicial-stuff higher-category-theory
asked Dec 28 '18 at 16:55
user313212user313212
363520
$endgroup$
Given a category $mathcal{C}$, we have a nerve functor
$$mathrm{N} colon mathbf{Cat} to mathbf{Set}_{Delta}$$
that assigns to $mathcal{C}$ its nerve $mathrm{N}(mathcal{C})$. This functor seems to have a left adjoint
$$tau_1 colon mathbf{Set}_{Delta} to mathbf{Cat}$$
that assigns to a simplicial set $X$ its fundamental category, as in Joyal's Notes on Quasi-Categories.
There it also states that the fundamental grouped $pi_1 X$ is obtained by inverting the arrows of $tau_1 X$, but there is no construction of $tau_1 X$.
What is the construction/definition of the fundamental category of a simplicial set $tau_1 X$? What are its objects and morphisms?
reference-request algebraic-topology category-theory simplicial-stuff higher-category-theory
reference-request algebraic-topology category-theory simplicial-stuff higher-category-theory
asked Dec 28 '18 at 16:55
user313212user313212
363520
asked Dec 28 '18 at 16:55
user313212user313212
363520
asked Dec 28 '18 at 16:55
user313212user313212
363520
asked Dec 28 '18 at 16:55
user313212user313212
363520
asked Dec 28 '18 at 16:55
user313212user313212
363520
363520
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The best presentation that I know of is in Riehl and Verity: 1.1.10&11
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
$endgroup$
$begingroup$
Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently
$endgroup$
– user313212
Dec 28 '18 at 17:38
$begingroup$
Yes, as proved in 1.1.11.
$endgroup$
– Ivan Di Liberti
Dec 28 '18 at 17:40
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: "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%2f3055065%2fwhat-is-the-fundamental-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
$begingroup$
The best presentation that I know of is in Riehl and Verity: 1.1.10&11
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
$endgroup$
$begingroup$
Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently
$endgroup$
– user313212
Dec 28 '18 at 17:38
$begingroup$
Yes, as proved in 1.1.11.
$endgroup$
– Ivan Di Liberti
Dec 28 '18 at 17:40
add a comment |
$begingroup$
The best presentation that I know of is in Riehl and Verity: 1.1.10&11
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
$endgroup$
$begingroup$
Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently
$endgroup$
– user313212
Dec 28 '18 at 17:38
$begingroup$
Yes, as proved in 1.1.11.
$endgroup$
– Ivan Di Liberti
Dec 28 '18 at 17:40
add a comment |
$begingroup$
The best presentation that I know of is in Riehl and Verity: 1.1.10&11
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
$endgroup$
The best presentation that I know of is in Riehl and Verity: 1.1.10&11
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
answered Dec 28 '18 at 17:05
Ivan Di LibertiIvan Di Liberti
2,60311123
2,60311123
$begingroup$
Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently
$endgroup$
– user313212
Dec 28 '18 at 17:38
$begingroup$
Yes, as proved in 1.1.11.
$endgroup$
– Ivan Di Liberti
Dec 28 '18 at 17:40
add a comment |
$begingroup$
Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently
$endgroup$
– user313212
Dec 28 '18 at 17:38
$begingroup$
Yes, as proved in 1.1.11.
$endgroup$
– Ivan Di Liberti
Dec 28 '18 at 17:40
$begingroup$
Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently
$endgroup$
– user313212
Dec 28 '18 at 17:38
$begingroup$
Thank you for the reference! Just to clarify, then homotopy category and fundamental category refer to the same thing, right? It seems that there are so many different notations and terminology that depending on where you look everything is named differently
$endgroup$
– user313212
Dec 28 '18 at 17:38
$begingroup$
Yes, as proved in 1.1.11.
$endgroup$
– Ivan Di Liberti
Dec 28 '18 at 17:40
$begingroup$
Yes, as proved in 1.1.11.
$endgroup$
– Ivan Di Liberti
Dec 28 '18 at 17:40
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%2f3055065%2fwhat-is-the-fundamental-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
