Definition:Domain Functor
Definition
Let $\mathbf C$ be a metacategory.
Let $\mathbf C^\to$ be its morphism category.
The domain functor is the functor $\operatorname{\mathbf{dom}}: \mathbf C^\to \to \mathbf C$ defined by:
Object functor: | \(\ds \operatorname{\mathbf{dom} } f := \operatorname{dom} f \) | ||||||||
Morphism functor: | \(\ds \operatorname{\mathbf{dom} } \left({g_1, g_2}\right) := g_1 \) |
That it is in fact a functor is shown on Domain Functor is Functor.
The functor $\mathbf{dom}$ can be represented as follows:
- $\begin{xy}
<0em,2em>*+{A} = "A", <0em,-2em>*+{B} = "B", <4em,2em>*+{A'} = "A2", <4em,-2em>*+{B'} = "B2",
"A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2},
<6em,0em>;<10em,0em> **@{~} ?>*@2{>} ?*!/_1em/{\mathbf{dom}},
<12em,2em>*+{A} = "AA", <16em,2em>*+{A'} = "AA2",
"AA";"AA2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, \end{xy}$
It is thus seen to be an example of a forgetful functor.
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.6.3$