Definition:Slice Category
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Definition
Let $\mathbf C$ be a metacategory.
Let $C \in \mathbf C_0$ be an object of $\mathbf C$.
The slice category of $\mathbf C$ over $C$, denoted $\mathbf C / C$, is defined as follows:
Objects: | $f: X \to C$, i.e. the morphisms of $\mathbf C$ with codomain $C$ | |
Morphisms: | $a: f \to f'$, for all morphisms $a \in \mathbf C_1$ with $f' \circ a = f$ | |
Composition: | $a \circ b$ is defined precisely as in $\mathbf C$ | |
Identity morphisms: | $\operatorname{id}_f := \operatorname{id}_X$, for $f: X \to C$ |
The morphisms can be displayed using a commutative diagram as follows:
- $\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.3)*!/^1em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{f'}, \end{xy}$
Also see
- Slice Category is Category
- Results about slice categories can be found here.
- Coslice Category
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.6.4$