Mixing
Diagrams in category theory: formalizing a concept in diagram-chasing
$begingroup$
Lemma 1.6.11. Suppose $f_1,...,f_n$ is a composable sequence - a "path" - of morphisms in a category. If the composite $f_kf_{k-1}...f_{i+1}f_i$ equals $g_m...g_1$ for another composable sequence of morphsism $g_1,...,g_m$, then $f_n...f_1 = f_n...f_{k+1}g_m...g_1f_{i-1}...f_1$
... In such cases, Lemma 1.6.11 and transitivity of equality implies that commutativity of the entire diagram may be checked by establishing commutativity of each minimal subdiagram in the directed graph. Here, a minimal subdiagram corresponds to a composition relation $h_n...h_1 = k_m...k_1$ that cannot be factored into a relation between shorter paths of composable morphisms. The graph corresponding to a minimal relation is a "directed polygon" with a commutative triangle being the simplest case.
This is from a book "Category Theory in Context" by Emily Riehl. As an example, the author gives the case of a commutative cube (a cube of objects and morphisms in a category) such as this:
The formal definition of a diagram in a category $mathcal{C}$ is that it is a functor $Fcolonmathcal{I}tomathcal{C}$ for some category $mathcal{I}$, which is called the shape of the diagram. It is defined the same way in the aforementioned book.
What I wish to know if whether we can formalize the aforementioned concept of a "minimal subdiagram" as it is called by Riehl with respect to an aribtrary functor $Dcolonmathcal{I}tomathcal{C}$ considered as a diagram of shape $mathcal{I}$ in a category $mathcal{C}$.
category-theory diagram-chasing
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
$endgroup$
add a comment |
$begingroup$
Lemma 1.6.11. Suppose $f_1,...,f_n$ is a composable sequence - a "path" - of morphisms in a category. If the composite $f_kf_{k-1}...f_{i+1}f_i$ equals $g_m...g_1$ for another composable sequence of morphsism $g_1,...,g_m$, then $f_n...f_1 = f_n...f_{k+1}g_m...g_1f_{i-1}...f_1$
... In such cases, Lemma 1.6.11 and transitivity of equality implies that commutativity of the entire diagram may be checked by establishing commutativity of each minimal subdiagram in the directed graph. Here, a minimal subdiagram corresponds to a composition relation $h_n...h_1 = k_m...k_1$ that cannot be factored into a relation between shorter paths of composable morphisms. The graph corresponding to a minimal relation is a "directed polygon" with a commutative triangle being the simplest case.
This is from a book "Category Theory in Context" by Emily Riehl. As an example, the author gives the case of a commutative cube (a cube of objects and morphisms in a category) such as this:
The formal definition of a diagram in a category $mathcal{C}$ is that it is a functor $Fcolonmathcal{I}tomathcal{C}$ for some category $mathcal{I}$, which is called the shape of the diagram. It is defined the same way in the aforementioned book.
What I wish to know if whether we can formalize the aforementioned concept of a "minimal subdiagram" as it is called by Riehl with respect to an aribtrary functor $Dcolonmathcal{I}tomathcal{C}$ considered as a diagram of shape $mathcal{I}$ in a category $mathcal{C}$.
category-theory diagram-chasing
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
$endgroup$
add a comment |
$begingroup$
Lemma 1.6.11. Suppose $f_1,...,f_n$ is a composable sequence - a "path" - of morphisms in a category. If the composite $f_kf_{k-1}...f_{i+1}f_i$ equals $g_m...g_1$ for another composable sequence of morphsism $g_1,...,g_m$, then $f_n...f_1 = f_n...f_{k+1}g_m...g_1f_{i-1}...f_1$
... In such cases, Lemma 1.6.11 and transitivity of equality implies that commutativity of the entire diagram may be checked by establishing commutativity of each minimal subdiagram in the directed graph. Here, a minimal subdiagram corresponds to a composition relation $h_n...h_1 = k_m...k_1$ that cannot be factored into a relation between shorter paths of composable morphisms. The graph corresponding to a minimal relation is a "directed polygon" with a commutative triangle being the simplest case.
This is from a book "Category Theory in Context" by Emily Riehl. As an example, the author gives the case of a commutative cube (a cube of objects and morphisms in a category) such as this:
The formal definition of a diagram in a category $mathcal{C}$ is that it is a functor $Fcolonmathcal{I}tomathcal{C}$ for some category $mathcal{I}$, which is called the shape of the diagram. It is defined the same way in the aforementioned book.
What I wish to know if whether we can formalize the aforementioned concept of a "minimal subdiagram" as it is called by Riehl with respect to an aribtrary functor $Dcolonmathcal{I}tomathcal{C}$ considered as a diagram of shape $mathcal{I}$ in a category $mathcal{C}$.
category-theory diagram-chasing
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
$endgroup$
Lemma 1.6.11. Suppose $f_1,...,f_n$ is a composable sequence - a "path" - of morphisms in a category. If the composite $f_kf_{k-1}...f_{i+1}f_i$ equals $g_m...g_1$ for another composable sequence of morphsism $g_1,...,g_m$, then $f_n...f_1 = f_n...f_{k+1}g_m...g_1f_{i-1}...f_1$
... In such cases, Lemma 1.6.11 and transitivity of equality implies that commutativity of the entire diagram may be checked by establishing commutativity of each minimal subdiagram in the directed graph. Here, a minimal subdiagram corresponds to a composition relation $h_n...h_1 = k_m...k_1$ that cannot be factored into a relation between shorter paths of composable morphisms. The graph corresponding to a minimal relation is a "directed polygon" with a commutative triangle being the simplest case.
This is from a book "Category Theory in Context" by Emily Riehl. As an example, the author gives the case of a commutative cube (a cube of objects and morphisms in a category) such as this:
The formal definition of a diagram in a category $mathcal{C}$ is that it is a functor $Fcolonmathcal{I}tomathcal{C}$ for some category $mathcal{I}$, which is called the shape of the diagram. It is defined the same way in the aforementioned book.
What I wish to know if whether we can formalize the aforementioned concept of a "minimal subdiagram" as it is called by Riehl with respect to an aribtrary functor $Dcolonmathcal{I}tomathcal{C}$ considered as a diagram of shape $mathcal{I}$ in a category $mathcal{C}$.
category-theory diagram-chasing
category-theory diagram-chasing
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
asked Oct 16 '18 at 10:08
Jxt921Jxt921
952618
952618
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
I don't understand why the index notation is used. Why not say:
If $f$ is a morphism and $f = gcirc h$, then $a circ f circ b = a circ g circ h circ b$, wherever $a circ f circ b$ is defined.
Since we're in a category, this is equivalent to the lemma given. To reach any path expandable-to from $f$ you just recursively apply this lemma in-place.
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
$endgroup$
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: "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%2f2957702%2fdiagrams-in-category-theory-formalizing-a-concept-in-diagram-chasing%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
$begingroup$
I don't understand why the index notation is used. Why not say:
If $f$ is a morphism and $f = gcirc h$, then $a circ f circ b = a circ g circ h circ b$, wherever $a circ f circ b$ is defined.
Since we're in a category, this is equivalent to the lemma given. To reach any path expandable-to from $f$ you just recursively apply this lemma in-place.
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
$endgroup$
add a comment |
$begingroup$
I don't understand why the index notation is used. Why not say:
If $f$ is a morphism and $f = gcirc h$, then $a circ f circ b = a circ g circ h circ b$, wherever $a circ f circ b$ is defined.
Since we're in a category, this is equivalent to the lemma given. To reach any path expandable-to from $f$ you just recursively apply this lemma in-place.
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
$endgroup$
add a comment |
$begingroup$
I don't understand why the index notation is used. Why not say:
If $f$ is a morphism and $f = gcirc h$, then $a circ f circ b = a circ g circ h circ b$, wherever $a circ f circ b$ is defined.
Since we're in a category, this is equivalent to the lemma given. To reach any path expandable-to from $f$ you just recursively apply this lemma in-place.
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
$endgroup$
I don't understand why the index notation is used. Why not say:
If $f$ is a morphism and $f = gcirc h$, then $a circ f circ b = a circ g circ h circ b$, wherever $a circ f circ b$ is defined.
Since we're in a category, this is equivalent to the lemma given. To reach any path expandable-to from $f$ you just recursively apply this lemma in-place.
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
answered Jan 1 at 22:57
Roll up and smoke AdjointRoll up and smoke Adjoint
9,09552458
9,09552458
add a comment |
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%2f2957702%2fdiagrams-in-category-theory-formalizing-a-concept-in-diagram-chasing%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
