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categorical direct sum (Definition)

Let $ \{C_i\}_{i \in I}$ be a set of objects in a category $ \mathcal{C}$. A direct sum of the collection $ \{C_i\}_{i \in I}$ is an object $ \coprod_{i \in I} C_i$ of $ \mathcal{C}$, with morphisms $ \iota_i: C_i \longrightarrow \coprod_{j \in I} C_j$ for each $ i \in I$, such that:

For every object $ A$ in $ \mathcal{C}$, and any collection of morphisms $ f_i: C_i \longrightarrow A$ for every $ i \in I$, there exists a unique morphism $ f: \coprod_{i \in I} C_i \longrightarrow A$ making the following diagram commute for all $ i \in I$.

$\displaystyle \begin{xy} *!C\xybox{ \xymatrix{ C_i \ar[dr]_{\iota_i} \ar[rr]^{f_i} & & A \ & \coprod_{j \in I} C_j \ar@{-->}[ur]_{f} } } \end{xy}$



"categorical direct sum" is owned by djao.
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See Also: categorical direct product, direct sum

Other names:  direct sum, coproduct

Attachments:
counterexamples for products and coproduct (Example) by Algeboy
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Cross-references: morphisms, collection, category, objects
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This is version 4 of categorical direct sum, born on 2002-04-21, modified 2002-04-24.
Object id is 2859, canonical name is CategoricalDirectSum.
Accessed 6048 times total.

Classification:
AMS MSC18A30 (Category theory; homological algebra :: General theory of categories and functors :: Limits and colimits )

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