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# arrow category

Let $\mathcal{C}$ be a category. The *arrow category* of $\mathcal{C}$ is the functor category $\mathcal{C}^{\textbf{2}}$. Here, 2 is the ordinal category consisting of $0,1$. Specifically, the contents of $\mathcal{C}^{\textbf{2}}$ are:

1. 2. given two objects of $\mathcal{C}^{\textbf{2}}$, say $A\stackrel{f}{\longrightarrow}B$ and $A^{{\prime}}\stackrel{g}{\longrightarrow}B^{{\prime}}$, a morphism (of $\mathcal{C}^{\textbf{2}}$) from $f$ to $g$ consists of an ordered pair $(h,k)$, where $A\stackrel{h}{\longrightarrow}A^{{\prime}}$ and $B\stackrel{k}{\longrightarrow}B^{{\prime}}$, such that the following diagram

$\xymatrix{A\ar[r]^{h}\ar[d]_{{f}}&A^{{\prime}}\ar[d]^{g}\\ B\ar[r]_{k}&B^{{\prime}}}$ is a commutative diagram.

Synonym:

morphism category

Major Section:

Reference

Type of Math Object:

Example

Parent:

## Mathematics Subject Classification

18A05*no label found*

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