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[parent] arrow category (Example)

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

  1. an object of $ \mathcal{C}^\textbf{2}$ is an arrow (morphism) of $ \mathcal{C}$
  2. given two objects of $ \mathcal{C}^\textbf{2}$, say $ A \stackrel{f}{\longrightarrow} B$ and $ A' \stackrel{g}{\longrightarrow} B'$, a morphism (of $ \mathcal{C}^\textbf{2}$) from $ f$ to $ g$ consists of an ordered pair $ (h,k)$, where $ A \stackrel{h}{\longrightarrow} A'$ and $ B \stackrel{k}{\longrightarrow} B'$, such that the following diagram
    $\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ A \ar[r]^h \ar[d]_{f} & A' \ar[d]^g \ B \ar[r]_k & B'} } \end{xy}$
    is a commutative diagram.



"arrow category" is owned by CWoo. [ full author list (2) | owner history (1) ]
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Other names:  morphism category

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Cross-references: commutative diagram, ordered pair, morphism, arrow, object, ordinal category, functor category, category
There are 4 references to this entry.

This is version 11 of arrow category, born on 2002-06-01, modified 2007-02-24.
Object id is 2983, canonical name is ExampleOfCategory.
Accessed 4512 times total.

Classification:
AMS MSC: 18A05 (Category theory; homological algebra :: General theory of categories and functors :: Definitions, generalizations)
Pending Errata and Addenda
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