Category:Definitions/Slice Categories

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This category contains definitions related to Slice Categories.
Related results can be found in Category:Slice Categories.


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:

$\begin{xy} <-3em,0em>*+{X} = "X", <3em,0em>*+{X'} = "X2", <0em,-4em>*+{C} = "C", "X";"X2" **@{-} ?>*@{>} ?*!/_1em/{a}, "X";"C" **@{-} ?>*@{>} ?<>(.3)*!/^1em/{f}, "X2";"C" **@{-} ?>*@{>} ?<>(.3)*!/_1em/{f'}, \end{xy}$

Pages in category "Definitions/Slice Categories"

The following 2 pages are in this category, out of 2 total.