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Homediagonal functor

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# diagonal functor

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

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

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.

## Mathematics Subject Classification

18A05*no label found*18-00

*no label found*

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