PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
Owner confidence rating: Very high Entry average rating: No information on entry rating
[parent] diagonal functor (Definition)

Let $ \mathcal{C}$ be a category. A diagonal functor on $ \mathcal{C}$ is a functor $ \delta:\mathcal{C}\to \mathcal{C}^I$ for some set $ I$ given by

$\displaystyle \delta(A)=(A)_{i\in I}$    and $\displaystyle \quad \delta(\alpha)=(\alpha)_{i\in I}.$
Here, $ \mathcal{C}^I$ denotes the $ I$-fold direct product of the category $ \mathcal{C}$. For any given $ I$, $ \delta$ is unique.

$ \delta$ is faithful. Its image, $ \delta(\mathcal{C})$, is the subcategory of $ \mathcal{C}^I$ whose objects are $ (A)_{i\in I}$ and morphisms are $ (\alpha)_{i\in I}$. $ \delta(\mathcal{C})$ is isomorphic to $ \mathcal{C}$, and may be pictured as the great diagonal of an $ I$-dimensional “cube”.

More generally, when $ I$ is a category, then the diagonal functor is just a functor $ \delta$ that sends each object $ A\in \mathcal{C}$ to the constant functor $ \delta(A):I\to \mathcal{C}$ with fixed value $ A$, and every morphism $ \alpha:A\to B$ to the natural transformation $ \delta(\alpha):\delta(A)\dot{\to} \delta(B)$, which sends every object $ i\in I$ to $ \alpha$. A routine verification shows that $ \delta(\alpha)$ is indeed a natural transformation.



"diagonal functor" is owned by CWoo.
(view preamble)

View style:


This object's parent.
Log in to rate this entry.
(view current ratings)

Cross-references: natural transformation, fixed, constant functor, diagonal, morphisms, objects, subcategory, image, functor, category
There are 3 references to this entry.

This is version 2 of diagonal functor, born on 2007-01-24, modified 2007-10-24.
Object id is 8818, canonical name is DiagonalFunctor.
Accessed 552 times total.

Classification:
AMS MSC18-00 (Category theory; homological algebra :: General reference works )
 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)

Pending Errata and Addenda
None.
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)