Mixing
Is there a category whose morphisms between A and B are diagrams of type “A -> B -> A”?
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I'm reading about spans in Category theory, which denote categories with finite colimits whose morphisms are equivalence classes of diagrams "A <- X -> B".
There's also their categorical dual, cospans, with the arrows reversed.
Is there a construction where arrows would form a sequence A -> B -> A?
Composition of Hom(A, B) (elements are A -> B -> A) and Hom(B, C) (elements are B -> C -> B) would then yield a an element of Hom(A, C) which is a diagram A -> B -> C -> B -> A.
It seems this is a category (associativity and identity hold), but I'm not sure has it been discussed in the literature somewhere.
I'm not even sure how to search for something like this.
category-theory
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
$endgroup$
|
show 2 more comments
$begingroup$
I'm reading about spans in Category theory, which denote categories with finite colimits whose morphisms are equivalence classes of diagrams "A <- X -> B".
There's also their categorical dual, cospans, with the arrows reversed.
Is there a construction where arrows would form a sequence A -> B -> A?
Composition of Hom(A, B) (elements are A -> B -> A) and Hom(B, C) (elements are B -> C -> B) would then yield a an element of Hom(A, C) which is a diagram A -> B -> C -> B -> A.
It seems this is a category (associativity and identity hold), but I'm not sure has it been discussed in the literature somewhere.
I'm not even sure how to search for something like this.
category-theory
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
$endgroup$
$begingroup$
Equivalent how? By coherent objectwise isos?
$endgroup$
– Randall
Jan 15 at 18:51
$begingroup$
Actually, I think the "equivalence classes" isn't needed in the title. I thought the composition in this category was associative only up to isomorphism at one point, but it seems it's strictly associative and unital. Does this answer your question?
$endgroup$
– Bruno Gavranovic
Jan 15 at 19:08
1
$begingroup$
It seems to me that your construction is isomorphic to the full subcategory of the product $Ctimes C^{op}$ defined by objects with the two components equal.
$endgroup$
– Arnaud D.
Jan 15 at 19:13
1
$begingroup$
I think it's also the functor category $mathcal{C}^mathcal{I}$ where $mathcal{I}$ is the little category $0 stackrel{alpha}{to} 1 stackrel{beta}{to} 0$.
$endgroup$
– Randall
Jan 15 at 19:21
$begingroup$
@Randall The objects of your $C^{mathcal I}$ would be the morphisms of the category in the question.
$endgroup$
– Andreas Blass
Jan 15 at 20:08
|
show 2 more comments
$begingroup$
I'm reading about spans in Category theory, which denote categories with finite colimits whose morphisms are equivalence classes of diagrams "A <- X -> B".
There's also their categorical dual, cospans, with the arrows reversed.
Is there a construction where arrows would form a sequence A -> B -> A?
Composition of Hom(A, B) (elements are A -> B -> A) and Hom(B, C) (elements are B -> C -> B) would then yield a an element of Hom(A, C) which is a diagram A -> B -> C -> B -> A.
It seems this is a category (associativity and identity hold), but I'm not sure has it been discussed in the literature somewhere.
I'm not even sure how to search for something like this.
category-theory
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
$endgroup$
I'm reading about spans in Category theory, which denote categories with finite colimits whose morphisms are equivalence classes of diagrams "A <- X -> B".
There's also their categorical dual, cospans, with the arrows reversed.
Is there a construction where arrows would form a sequence A -> B -> A?
Composition of Hom(A, B) (elements are A -> B -> A) and Hom(B, C) (elements are B -> C -> B) would then yield a an element of Hom(A, C) which is a diagram A -> B -> C -> B -> A.
It seems this is a category (associativity and identity hold), but I'm not sure has it been discussed in the literature somewhere.
I'm not even sure how to search for something like this.
category-theory
category-theory
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
edited Jan 15 at 19:05
Bruno Gavranovic
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
asked Jan 15 at 18:49
Bruno GavranovicBruno Gavranovic
434
434
$begingroup$
Equivalent how? By coherent objectwise isos?
$endgroup$
– Randall
Jan 15 at 18:51
$begingroup$
Actually, I think the "equivalence classes" isn't needed in the title. I thought the composition in this category was associative only up to isomorphism at one point, but it seems it's strictly associative and unital. Does this answer your question?
$endgroup$
– Bruno Gavranovic
Jan 15 at 19:08
1
$begingroup$
It seems to me that your construction is isomorphic to the full subcategory of the product $Ctimes C^{op}$ defined by objects with the two components equal.
$endgroup$
– Arnaud D.
Jan 15 at 19:13
1
$begingroup$
I think it's also the functor category $mathcal{C}^mathcal{I}$ where $mathcal{I}$ is the little category $0 stackrel{alpha}{to} 1 stackrel{beta}{to} 0$.
$endgroup$
– Randall
Jan 15 at 19:21
$begingroup$
@Randall The objects of your $C^{mathcal I}$ would be the morphisms of the category in the question.
$endgroup$
– Andreas Blass
Jan 15 at 20:08
|
show 2 more comments
$begingroup$
Equivalent how? By coherent objectwise isos?
$endgroup$
– Randall
Jan 15 at 18:51
$begingroup$
Actually, I think the "equivalence classes" isn't needed in the title. I thought the composition in this category was associative only up to isomorphism at one point, but it seems it's strictly associative and unital. Does this answer your question?
$endgroup$
– Bruno Gavranovic
Jan 15 at 19:08
1
$begingroup$
It seems to me that your construction is isomorphic to the full subcategory of the product $Ctimes C^{op}$ defined by objects with the two components equal.
$endgroup$
– Arnaud D.
Jan 15 at 19:13
1
$begingroup$
I think it's also the functor category $mathcal{C}^mathcal{I}$ where $mathcal{I}$ is the little category $0 stackrel{alpha}{to} 1 stackrel{beta}{to} 0$.
$endgroup$
– Randall
Jan 15 at 19:21
$begingroup$
@Randall The objects of your $C^{mathcal I}$ would be the morphisms of the category in the question.
$endgroup$
– Andreas Blass
Jan 15 at 20:08
$begingroup$
Equivalent how? By coherent objectwise isos?
$endgroup$
– Randall
Jan 15 at 18:51
$begingroup$
Equivalent how? By coherent objectwise isos?
$endgroup$
– Randall
Jan 15 at 18:51
$begingroup$
Actually, I think the "equivalence classes" isn't needed in the title. I thought the composition in this category was associative only up to isomorphism at one point, but it seems it's strictly associative and unital. Does this answer your question?
$endgroup$
– Bruno Gavranovic
Jan 15 at 19:08
$begingroup$
Actually, I think the "equivalence classes" isn't needed in the title. I thought the composition in this category was associative only up to isomorphism at one point, but it seems it's strictly associative and unital. Does this answer your question?
$endgroup$
– Bruno Gavranovic
Jan 15 at 19:08
1
1
$begingroup$
It seems to me that your construction is isomorphic to the full subcategory of the product $Ctimes C^{op}$ defined by objects with the two components equal.
$endgroup$
– Arnaud D.
Jan 15 at 19:13
$begingroup$
It seems to me that your construction is isomorphic to the full subcategory of the product $Ctimes C^{op}$ defined by objects with the two components equal.
$endgroup$
– Arnaud D.
Jan 15 at 19:13
1
1
$begingroup$
I think it's also the functor category $mathcal{C}^mathcal{I}$ where $mathcal{I}$ is the little category $0 stackrel{alpha}{to} 1 stackrel{beta}{to} 0$.
$endgroup$
– Randall
Jan 15 at 19:21
$begingroup$
I think it's also the functor category $mathcal{C}^mathcal{I}$ where $mathcal{I}$ is the little category $0 stackrel{alpha}{to} 1 stackrel{beta}{to} 0$.
$endgroup$
– Randall
Jan 15 at 19:21
$begingroup$
@Randall The objects of your $C^{mathcal I}$ would be the morphisms of the category in the question.
$endgroup$
– Andreas Blass
Jan 15 at 20:08
$begingroup$
@Randall The objects of your $C^{mathcal I}$ would be the morphisms of the category in the question.
$endgroup$
– Andreas Blass
Jan 15 at 20:08
|
show 2 more comments
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$begingroup$
Equivalent how? By coherent objectwise isos?
$endgroup$
– Randall
Jan 15 at 18:51
$begingroup$
Actually, I think the "equivalence classes" isn't needed in the title. I thought the composition in this category was associative only up to isomorphism at one point, but it seems it's strictly associative and unital. Does this answer your question?
$endgroup$
– Bruno Gavranovic
Jan 15 at 19:08
1
$begingroup$
It seems to me that your construction is isomorphic to the full subcategory of the product $Ctimes C^{op}$ defined by objects with the two components equal.
$endgroup$
– Arnaud D.
Jan 15 at 19:13
1
$begingroup$
I think it's also the functor category $mathcal{C}^mathcal{I}$ where $mathcal{I}$ is the little category $0 stackrel{alpha}{to} 1 stackrel{beta}{to} 0$.
$endgroup$
– Randall
Jan 15 at 19:21
$begingroup$
@Randall The objects of your $C^{mathcal I}$ would be the morphisms of the category in the question.
$endgroup$
– Andreas Blass
Jan 15 at 20:08