Mixing
How is this a commutative diagram?
$begingroup$
So, I'm rather new to category theory (well, Abstract Algebra in general as well), so I decided to pick up Paulo Aluffi's "Algebra: Chapter 0". However, there is one example in the introductory chapter to Categories that I don't quite understand. He's attempting to construct a slice category $C_{A,B}$ in the following way:
Let $C$ be a given category, and let $A$ and $B$ be objects of $C$. Then, he constructs a category $C_{A,B}$ by specifying that the objects of $C_{A,B}$ will be diagrams (see picture below), and that the morphisms of $C_{A,B}$ will be commutative diagrams
My question is, based off of the diagrams he has given, how are the morphisms of $C_{A,B}$ commutative diagrams? My understanding was that a commutative diagram had to have all paths essentially going to the same place, but clearly in the the diagram Aluffi has given us, there are paths going to object $A$ and object $B$ of category $C$. Could someone please clarify this?
abstract-algebra category-theory slice-category
asked Jan 2 at 19:18
user516079user516079
280210
$endgroup$
add a comment |
$begingroup$
So, I'm rather new to category theory (well, Abstract Algebra in general as well), so I decided to pick up Paulo Aluffi's "Algebra: Chapter 0". However, there is one example in the introductory chapter to Categories that I don't quite understand. He's attempting to construct a slice category $C_{A,B}$ in the following way:
Let $C$ be a given category, and let $A$ and $B$ be objects of $C$. Then, he constructs a category $C_{A,B}$ by specifying that the objects of $C_{A,B}$ will be diagrams (see picture below), and that the morphisms of $C_{A,B}$ will be commutative diagrams
My question is, based off of the diagrams he has given, how are the morphisms of $C_{A,B}$ commutative diagrams? My understanding was that a commutative diagram had to have all paths essentially going to the same place, but clearly in the the diagram Aluffi has given us, there are paths going to object $A$ and object $B$ of category $C$. Could someone please clarify this?
abstract-algebra category-theory slice-category
asked Jan 2 at 19:18
user516079user516079
280210
$endgroup$
$begingroup$
it just means the top and bottom triangles in the lowest diagram commute.
$endgroup$
– Cheerful Parsnip
Jan 2 at 19:23
2
$begingroup$
One definition of commutative diagram is that the diagram is commutative if the composition of morphisms along any path between two objects is independent of the path chosen between the two objects.
$endgroup$
– jgon
Jan 2 at 19:23
$begingroup$
Oh okay! This makes sense - thank you!
$endgroup$
– user516079
Jan 2 at 19:25
add a comment |
$begingroup$
So, I'm rather new to category theory (well, Abstract Algebra in general as well), so I decided to pick up Paulo Aluffi's "Algebra: Chapter 0". However, there is one example in the introductory chapter to Categories that I don't quite understand. He's attempting to construct a slice category $C_{A,B}$ in the following way:
Let $C$ be a given category, and let $A$ and $B$ be objects of $C$. Then, he constructs a category $C_{A,B}$ by specifying that the objects of $C_{A,B}$ will be diagrams (see picture below), and that the morphisms of $C_{A,B}$ will be commutative diagrams
My question is, based off of the diagrams he has given, how are the morphisms of $C_{A,B}$ commutative diagrams? My understanding was that a commutative diagram had to have all paths essentially going to the same place, but clearly in the the diagram Aluffi has given us, there are paths going to object $A$ and object $B$ of category $C$. Could someone please clarify this?
abstract-algebra category-theory slice-category
asked Jan 2 at 19:18
user516079user516079
280210
$endgroup$
So, I'm rather new to category theory (well, Abstract Algebra in general as well), so I decided to pick up Paulo Aluffi's "Algebra: Chapter 0". However, there is one example in the introductory chapter to Categories that I don't quite understand. He's attempting to construct a slice category $C_{A,B}$ in the following way:
Let $C$ be a given category, and let $A$ and $B$ be objects of $C$. Then, he constructs a category $C_{A,B}$ by specifying that the objects of $C_{A,B}$ will be diagrams (see picture below), and that the morphisms of $C_{A,B}$ will be commutative diagrams
My question is, based off of the diagrams he has given, how are the morphisms of $C_{A,B}$ commutative diagrams? My understanding was that a commutative diagram had to have all paths essentially going to the same place, but clearly in the the diagram Aluffi has given us, there are paths going to object $A$ and object $B$ of category $C$. Could someone please clarify this?
abstract-algebra category-theory slice-category
abstract-algebra category-theory slice-category
asked Jan 2 at 19:18
user516079user516079
280210
asked Jan 2 at 19:18
user516079user516079
280210
asked Jan 2 at 19:18
user516079user516079
280210
asked Jan 2 at 19:18
user516079user516079
280210
asked Jan 2 at 19:18
user516079user516079
280210
280210
$begingroup$
it just means the top and bottom triangles in the lowest diagram commute.
$endgroup$
– Cheerful Parsnip
Jan 2 at 19:23
2
$begingroup$
One definition of commutative diagram is that the diagram is commutative if the composition of morphisms along any path between two objects is independent of the path chosen between the two objects.
$endgroup$
– jgon
Jan 2 at 19:23
$begingroup$
Oh okay! This makes sense - thank you!
$endgroup$
– user516079
Jan 2 at 19:25
add a comment |
$begingroup$
it just means the top and bottom triangles in the lowest diagram commute.
$endgroup$
– Cheerful Parsnip
Jan 2 at 19:23
2
$begingroup$
One definition of commutative diagram is that the diagram is commutative if the composition of morphisms along any path between two objects is independent of the path chosen between the two objects.
$endgroup$
– jgon
Jan 2 at 19:23
$begingroup$
Oh okay! This makes sense - thank you!
$endgroup$
– user516079
Jan 2 at 19:25
$begingroup$
it just means the top and bottom triangles in the lowest diagram commute.
$endgroup$
– Cheerful Parsnip
Jan 2 at 19:23
$begingroup$
it just means the top and bottom triangles in the lowest diagram commute.
$endgroup$
– Cheerful Parsnip
Jan 2 at 19:23
2
2
$begingroup$
One definition of commutative diagram is that the diagram is commutative if the composition of morphisms along any path between two objects is independent of the path chosen between the two objects.
$endgroup$
– jgon
Jan 2 at 19:23
$begingroup$
One definition of commutative diagram is that the diagram is commutative if the composition of morphisms along any path between two objects is independent of the path chosen between the two objects.
$endgroup$
– jgon
Jan 2 at 19:23
$begingroup$
Oh okay! This makes sense - thank you!
$endgroup$
– user516079
Jan 2 at 19:25
$begingroup$
Oh okay! This makes sense - thank you!
$endgroup$
– user516079
Jan 2 at 19:25
add a comment |
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$begingroup$
it just means the top and bottom triangles in the lowest diagram commute.
$endgroup$
– Cheerful Parsnip
Jan 2 at 19:23
2
$begingroup$
One definition of commutative diagram is that the diagram is commutative if the composition of morphisms along any path between two objects is independent of the path chosen between the two objects.
$endgroup$
– jgon
Jan 2 at 19:23
$begingroup$
Oh okay! This makes sense - thank you!
$endgroup$
– user516079
Jan 2 at 19:25