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'constant functor'
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| Title of object: |
constant functor |
| Canonical Name: |
ConstantFunctor |
| Type: |
Definition |
| Created on: |
2007-10-24 00:36:33 |
| Modified on: |
2007-10-24 00:36:33 |
| Classification: |
msc:18-00 |
Preamble:
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Content:
Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A \emph{constant functor} from $\mathcal{C}$ to $\mathcal{D}$ is a functor $k:\mathcal{C}\to \mathcal{D}$ such that there is an object $A\in \mathcal{D}$ such that
\begin{itemize}
\item for all objects $X$ in $\mathcal{C}$, $k(X)=A$, and
\item for all morphisms $X\to Y$ in $\mathcal{C}$, $k(X\to Y)=1_A$, the identity morphism of $A$.
\end{itemize}
To see that this is indeed a functor, we merely need to verify that $$k\big((X\to Y)\circ (Y\to Z)\big)=k(X\to Y)\circ k(Y\to Z).$$ But this is obvious, as the left hand side is $k(X\to Z)=1_A$, while the right hand side is $1_A\circ 1_A=1_A$. |
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