Definition:Slice Functor
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Definition
Let $\mathbf C$ be a metacategory.
Let $\mathbf{Cat}$ be the category of categories.
The slice functor is the functor $\mathbf C / \cdot: \mathbf C \to \mathbf{Cat}$ defined by:
Object functor: | \(\ds \mathbf C / C := \mathbf C / C \) | ||||||||
Morphism functor: | \(\ds \mathbf C / f := f_* \) |
where $\mathbf C / C$ is a slice category, and $f_*$ is the composition functor defined by $f$.
The effect of $\mathbf C / \cdot$ is captured in the following diagram:
- $\begin{xy} <0em,0em>*+{A} = "a", <4em,0em>*+{B} = "b", <4em,-4em>*+{C}= "c", "a";"b" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f}, "b";"c" **@{-} ?>*@{>} ?<>(.5)*!/_.6em/{g}, "a";"c" **@{-} ?>*@{>} ?<>(.4)*!/^1em/{g \circ f}, "b"+/r4em/+/_3em/;"b"+/r8em/+/_3em/ **@{~} ?>*@2{>} ?*!/_1em/{\mathbf C / \cdot}, "a"+/r13em/*+{\mathbf C / A}="Fa", "b"+/r14em/*+{\mathbf C / B}="Fb", "c"+/r14em/+/_1em/*+{\mathbf C / C}="Fc", "Fa";"Fb" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{f_*}, "Fb";"Fc" **@{-} ?>*@{>} ?<>(.5)*!/_1em/{g_*}, "Fa";"Fc" **@{-} ?>*@{>} ?<>(.7)*!/r3em/{\left({g \circ f}\right)_* \\ = g_* f_*}, \end{xy}$
where $g_* f_*$ denotes a composite functor.
Also see
- Slice Functor is Functor, where it is shown that it is a functor
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.6.4$