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I was just wondering whether there is already a name for an object of a category as described below.

Recall that an initial object in a category $\mathcal C$ is an object $I\in\mathcal C$ such that for all objects $C\in \mathcal C$ there exists exactly one morphism $I\to C$.

Now if I can express the category as the disjoint union of full subcategories $\mathcal C=\coprod_i \mathcal C_i$ and if $I_i$ is an initial object of $\mathcal C_i$ for some $i$, how would I call $I_i$?

In my situation it is equivalent to say that $I_i$ is an object of $\mathcal C$ such that no (non-identity) morphism has $I_i$ as a target. So I could call it a source, or a pure source (actually this more or less avoids the question rather than to solve it). Another option would be to say that $I_i$ is the initial object of a connected component of $\mathcal C$ motivated by the understanding of the graph or the classifying space of $\mathcal C$.

But I assume somebody must have considered something like that before.

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What's wrong with saying that $I_i$ is the initial object of a connected component of $\mathcal{C}$? There seem to be quite a few instances of this in the literature [at least, as far as I can tell from a quick Google search]. Then again, everything in category theory has a snazzy name! –  Clive Newstead Nov 27 '12 at 16:02
    
@CliveNewstead, this is exactly what I was thinking! If there is a snazzy name I might as well use it. For the time being I am relatively happy with the former though. –  Simon Markett Nov 27 '12 at 16:09
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