PlanetMath: diagonal functor

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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 given by

$\displaystyle \delta(A)=(A)_{i\in I}$ and $\displaystyle \quad \delta(\alpha)=(\alpha)_{i\in I}.$

Here, $\mathcal{C}^I$ denotes the

-fold direct product of the category $\mathcal{C}$ . For any given

, $\delta$ 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 -dimensional “cube”.

More generally, when 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 , 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.

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

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