Definition:Morphism Category

Definition

Let $\mathbf C$ be a metacategory.

Its morphism category, denoted $\mathbf C^\to$, is defined as follows:

 Objects: The morphisms $\mathbf C_1$ of $\mathbf C$ Morphisms: $g: f \to f'$ is a pair $\tuple {g_1, g_2}$ of morphisms of $\mathbf C$ such that $g_2 \circ f = f' \circ g_1$ Composition: $\tuple {h_1, h_2} \circ \tuple {g_1, g_2} := \tuple {h_1 \circ g_1, h_2 \circ g_2}$, whenever this is defined Identity morphisms: $\operatorname{id}_f := \tuple {\operatorname{id}_C, \operatorname{id}_D}$ for $f: C \to D$

The morphisms of $\mathbf C^\to$ can be made more intuitive by the following diagram:

$\begin{xy} <-2em,0em>*+{f} = "f", <2em,0em>*+{f'} = "f2", "f";"f2" **@{-} ?>*@{>} ?*!/_1em/{\scriptstyle \tuple {g_1, g_2} }, <3em,0em>*{:}, <7em,2em>*+{A} = "A", <7em,-2em>*+{B} = "B", <11em,2em>*+{A'} = "A2", <11em,-2em>*+{B'} = "B2", "A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2} \end{xy}$

The composition likewise benefits from a diagrammatic representation:

$\begin{xy} <4em,5em>*{\tuple {h_1, h_2} \circ \tuple {g_1, g_2} }, <0em,2em>*+{A} = "A", <0em,-2em>*+{B} = "B", <4em,2em>*+{A'} = "A2", <4em,-2em>*+{B'} = "B2", "A";"B" **@{-} ?>*@{>} ?*!/^1em/{f}, "A";"A2" **@{-} ?>*@{>} ?*!/_1em/{g_1}, "A2";"B2" **@{-} ?>*@{>} ?*!/_1em/{f'}, "B";"B2" **@{-} ?>*@{>} ?*!/^1em/{g_2}, <8em,2em>*+{A''} = "A3", <8em,-2em>*+{B''} = "B3", "A2";"A3" **@{-} ?>*@{>} ?*!/_1em/{h_1}, "B2";"B3" **@{-} ?>*@{>} ?*!/^1em/{h_2}, "A3";"B3" **@{-} ?>*@{>} ?*!/_1em/{f''}, <12em,5em>*{=}, <10em,0em>;<14em,0em> **@{~} ?>*@2{>}, <20em,5em>*+{\tuple {h_1 \circ g_1, h_2 \circ g_2} }, <16em,2em>*+{A} = "AA", <16em,-2em>*+{B} = "BB", <24em,2em>*+{A''} = "AA3", <24em,-2em>*+{B''} = "BB3", "AA";"BB" **@{-} ?>*@{>} ?*!/^1em/{f}, "AA";"AA3" **@{-} ?>*@{>} ?*!/_1em/{h_1 \circ g_1}, "AA3";"BB3" **@{-} ?>*@{>} ?*!/_1em/{f''}, "BB";"BB3" **@{-} ?>*@{>} ?*!/^1em/{h_2 \circ g_2}, \end{xy}$

Also known as

The morphism category $\mathbf C^\to$ is also called the arrow category.