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Suppose in an Abelian category $\mathscr C$, the direct sum of any family of objects exists, then is $\bigoplus_{i\in\varnothing}A_i$ equal to 0 or meaningless?

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An empty direct sum in an abelian category is a zero object.

In general:

An empty product is a terminal object (provided that the category has terminal objects).

An empty coproduct is an initial object (provided that the category has initial objects).

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