Definition:Slice Functor

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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