Can composition of morphisms in a category be carried out on any subgraph of a commutative diagram, in-place?
up vote
0
down vote
favorite
Here is what the rule looks like to us and how we specify it to the app I'm writing.
I was wondering can you take any commutative diagram $J$ and apply this rule to a subgraph matching $A xrightarrow{a} B xrightarrow{b} C$ in place, all the while leaving another commutative diagram. Or does adding in the hypotenuse sometimes break commutativity of a larger diagram?
If it preserves commutativity, what is a simple proof?
category-theory math-software function-and-relation-composition natural-deduction diagram-chasing
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
add a comment |
up vote
0
down vote
favorite
Here is what the rule looks like to us and how we specify it to the app I'm writing.
I was wondering can you take any commutative diagram $J$ and apply this rule to a subgraph matching $A xrightarrow{a} B xrightarrow{b} C$ in place, all the while leaving another commutative diagram. Or does adding in the hypotenuse sometimes break commutativity of a larger diagram?
If it preserves commutativity, what is a simple proof?
category-theory math-software function-and-relation-composition natural-deduction diagram-chasing
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Here is what the rule looks like to us and how we specify it to the app I'm writing.
I was wondering can you take any commutative diagram $J$ and apply this rule to a subgraph matching $A xrightarrow{a} B xrightarrow{b} C$ in place, all the while leaving another commutative diagram. Or does adding in the hypotenuse sometimes break commutativity of a larger diagram?
If it preserves commutativity, what is a simple proof?
category-theory math-software function-and-relation-composition natural-deduction diagram-chasing
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
Here is what the rule looks like to us and how we specify it to the app I'm writing.
I was wondering can you take any commutative diagram $J$ and apply this rule to a subgraph matching $A xrightarrow{a} B xrightarrow{b} C$ in place, all the while leaving another commutative diagram. Or does adding in the hypotenuse sometimes break commutativity of a larger diagram?
If it preserves commutativity, what is a simple proof?
category-theory math-software function-and-relation-composition natural-deduction diagram-chasing
category-theory math-software function-and-relation-composition natural-deduction diagram-chasing
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
asked Dec 1 at 11:33
Roll up and smoke Adjoint
8,96252357
8,96252357
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
Consider any two parallel paths.
Change every occurance of the new arrow $bcirc a$ to the subpath $a, b$.
Since the original diagram commutes, the composition of the two changed paths coincide.
Consequently, the new diagram is commutative, as well.
answered Dec 1 at 14:16
Berci
59.2k23671
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Consider any two parallel paths.
Change every occurance of the new arrow $bcirc a$ to the subpath $a, b$.
Since the original diagram commutes, the composition of the two changed paths coincide.
Consequently, the new diagram is commutative, as well.
answered Dec 1 at 14:16
Berci
59.2k23671
add a comment |
up vote
2
down vote
accepted
Consider any two parallel paths.
Change every occurance of the new arrow $bcirc a$ to the subpath $a, b$.
Since the original diagram commutes, the composition of the two changed paths coincide.
Consequently, the new diagram is commutative, as well.
answered Dec 1 at 14:16
Berci
59.2k23671
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Consider any two parallel paths.
Change every occurance of the new arrow $bcirc a$ to the subpath $a, b$.
Since the original diagram commutes, the composition of the two changed paths coincide.
Consequently, the new diagram is commutative, as well.
answered Dec 1 at 14:16
Berci
59.2k23671
Consider any two parallel paths.
Change every occurance of the new arrow $bcirc a$ to the subpath $a, b$.
Since the original diagram commutes, the composition of the two changed paths coincide.
Consequently, the new diagram is commutative, as well.
answered Dec 1 at 14:16
Berci
59.2k23671
answered Dec 1 at 14:16
Berci
59.2k23671
answered Dec 1 at 14:16
Berci
59.2k23671
answered Dec 1 at 14:16
Berci
59.2k23671
59.2k23671
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3021242%2fcan-composition-of-morphisms-in-a-category-be-carried-out-on-any-subgraph-of-a-c%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
