Pushout of unital non commutative algebras
up vote
1
down vote
favorite
I like to know if there is a pushout in the category of non commutative alegbras with unit and if the answer is "yes", who is it?
category-theory noncommutative-algebra
asked Nov 17 at 20:55
GaSa
536
add a comment |
up vote
1
down vote
favorite
I like to know if there is a pushout in the category of non commutative alegbras with unit and if the answer is "yes", who is it?
category-theory noncommutative-algebra
asked Nov 17 at 20:55
GaSa
536
1
Yes. It's a variant of the amalgamated free product, adapted to algebras.
– Qiaochu Yuan
Nov 17 at 23:49
@QiaochuYuan, do you have some reference? I would like to check the construction carefully and if your reference has it with $ast$-algebras, it would be better. I'd really appreciate that
– GaSa
Nov 18 at 1:35
1
You might not find a reference that's amenable to careful checking. Careful proofs of a theorem like this are more likely to be done in much more generality-for arbitrary "categories of algebras," for instance.
– Kevin Carlson
Nov 18 at 1:39
ok @KevinCarlson. Well, it would be ok if I just see who is this algebra and how I can define the morphisms involved in the diagrams.
– GaSa
Nov 18 at 1:54
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I like to know if there is a pushout in the category of non commutative alegbras with unit and if the answer is "yes", who is it?
category-theory noncommutative-algebra
asked Nov 17 at 20:55
GaSa
536
I like to know if there is a pushout in the category of non commutative alegbras with unit and if the answer is "yes", who is it?
category-theory noncommutative-algebra
category-theory noncommutative-algebra
asked Nov 17 at 20:55
GaSa
536
asked Nov 17 at 20:55
GaSa
536
asked Nov 17 at 20:55
GaSa
536
asked Nov 17 at 20:55
GaSa
536
asked Nov 17 at 20:55
GaSa
536
536
1
Yes. It's a variant of the amalgamated free product, adapted to algebras.
– Qiaochu Yuan
Nov 17 at 23:49
@QiaochuYuan, do you have some reference? I would like to check the construction carefully and if your reference has it with $ast$-algebras, it would be better. I'd really appreciate that
– GaSa
Nov 18 at 1:35
1
You might not find a reference that's amenable to careful checking. Careful proofs of a theorem like this are more likely to be done in much more generality-for arbitrary "categories of algebras," for instance.
– Kevin Carlson
Nov 18 at 1:39
ok @KevinCarlson. Well, it would be ok if I just see who is this algebra and how I can define the morphisms involved in the diagrams.
– GaSa
Nov 18 at 1:54
add a comment |
1
Yes. It's a variant of the amalgamated free product, adapted to algebras.
– Qiaochu Yuan
Nov 17 at 23:49
@QiaochuYuan, do you have some reference? I would like to check the construction carefully and if your reference has it with $ast$-algebras, it would be better. I'd really appreciate that
– GaSa
Nov 18 at 1:35
1
You might not find a reference that's amenable to careful checking. Careful proofs of a theorem like this are more likely to be done in much more generality-for arbitrary "categories of algebras," for instance.
– Kevin Carlson
Nov 18 at 1:39
ok @KevinCarlson. Well, it would be ok if I just see who is this algebra and how I can define the morphisms involved in the diagrams.
– GaSa
Nov 18 at 1:54
1
1
Yes. It's a variant of the amalgamated free product, adapted to algebras.
– Qiaochu Yuan
Nov 17 at 23:49
Yes. It's a variant of the amalgamated free product, adapted to algebras.
– Qiaochu Yuan
Nov 17 at 23:49
@QiaochuYuan, do you have some reference? I would like to check the construction carefully and if your reference has it with $ast$-algebras, it would be better. I'd really appreciate that
– GaSa
Nov 18 at 1:35
@QiaochuYuan, do you have some reference? I would like to check the construction carefully and if your reference has it with $ast$-algebras, it would be better. I'd really appreciate that
– GaSa
Nov 18 at 1:35
1
1
You might not find a reference that's amenable to careful checking. Careful proofs of a theorem like this are more likely to be done in much more generality-for arbitrary "categories of algebras," for instance.
– Kevin Carlson
Nov 18 at 1:39
You might not find a reference that's amenable to careful checking. Careful proofs of a theorem like this are more likely to be done in much more generality-for arbitrary "categories of algebras," for instance.
– Kevin Carlson
Nov 18 at 1:39
ok @KevinCarlson. Well, it would be ok if I just see who is this algebra and how I can define the morphisms involved in the diagrams.
– GaSa
Nov 18 at 1:54
ok @KevinCarlson. Well, it would be ok if I just see who is this algebra and how I can define the morphisms involved in the diagrams.
– GaSa
Nov 18 at 1:54
add a comment |
1 Answer
1
active
oldest
votes
up vote
3
down vote
Given unital $R$-algebras $Aleftarrow Bto C$, the pushout $A star_B C$ is generated as an $R$-algebra by generators of $A$ and of $C$, modulo the union of the relations in $A$ and in $C$, as well as further relations identifying the two resulting images of each element of $B$. This immediately gives the canonical maps from $A$ and $C$. You can describe this construction as a quotient of the free $R$-module on words with letters from $A$ and $C$, if you like.
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
3
down vote
Given unital $R$-algebras $Aleftarrow Bto C$, the pushout $A star_B C$ is generated as an $R$-algebra by generators of $A$ and of $C$, modulo the union of the relations in $A$ and in $C$, as well as further relations identifying the two resulting images of each element of $B$. This immediately gives the canonical maps from $A$ and $C$. You can describe this construction as a quotient of the free $R$-module on words with letters from $A$ and $C$, if you like.
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
add a comment |
up vote
3
down vote
Given unital $R$-algebras $Aleftarrow Bto C$, the pushout $A star_B C$ is generated as an $R$-algebra by generators of $A$ and of $C$, modulo the union of the relations in $A$ and in $C$, as well as further relations identifying the two resulting images of each element of $B$. This immediately gives the canonical maps from $A$ and $C$. You can describe this construction as a quotient of the free $R$-module on words with letters from $A$ and $C$, if you like.
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
add a comment |
up vote
3
down vote
up vote
3
down vote
Given unital $R$-algebras $Aleftarrow Bto C$, the pushout $A star_B C$ is generated as an $R$-algebra by generators of $A$ and of $C$, modulo the union of the relations in $A$ and in $C$, as well as further relations identifying the two resulting images of each element of $B$. This immediately gives the canonical maps from $A$ and $C$. You can describe this construction as a quotient of the free $R$-module on words with letters from $A$ and $C$, if you like.
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
Given unital $R$-algebras $Aleftarrow Bto C$, the pushout $A star_B C$ is generated as an $R$-algebra by generators of $A$ and of $C$, modulo the union of the relations in $A$ and in $C$, as well as further relations identifying the two resulting images of each element of $B$. This immediately gives the canonical maps from $A$ and $C$. You can describe this construction as a quotient of the free $R$-module on words with letters from $A$ and $C$, if you like.
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
answered Nov 18 at 8:18
Kevin Carlson
32.2k23270
32.2k23270
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%2f3002797%2fpushout-of-unital-non-commutative-algebras%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
1
Yes. It's a variant of the amalgamated free product, adapted to algebras.
– Qiaochu Yuan
Nov 17 at 23:49
@QiaochuYuan, do you have some reference? I would like to check the construction carefully and if your reference has it with $ast$-algebras, it would be better. I'd really appreciate that
– GaSa
Nov 18 at 1:35
1
You might not find a reference that's amenable to careful checking. Careful proofs of a theorem like this are more likely to be done in much more generality-for arbitrary "categories of algebras," for instance.
– Kevin Carlson
Nov 18 at 1:39
ok @KevinCarlson. Well, it would be ok if I just see who is this algebra and how I can define the morphisms involved in the diagrams.
– GaSa
Nov 18 at 1:54