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I would like to know if there is a name for an object $X$ in a (finitely complete and cocomplete) category $\mathcal{C}$ which has the following property:

$X$ is non-empty and for every sub-object $Y$ of $X$ (by which I mean $Y \cong Y\times_ X Y$), either $Y$ is empty, or $Y=X$.

By non-empty I mean not initial.

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  • $\begingroup$ What does "empty" mean ? $\endgroup$ – Captain Lama May 18 '16 at 9:48
  • $\begingroup$ ...not initial. I have edited the question. $\endgroup$ – L.K.C. May 18 '16 at 9:49
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    $\begingroup$ I would suggest an atom. $\endgroup$ – J.-E. Pin May 18 '16 at 10:43
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    $\begingroup$ See this: ncatlab.org/nlab/show/simple+object I'm not sure this property has a name, it's equivalent to being simple if your category is abelian. $\endgroup$ – Najib Idrissi May 18 '16 at 13:53
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    $\begingroup$ In most categories this seems like too strong a condition to be interesting. For example, the point is the only topological space satisfying this condition, but of course there are many connected spaces that aren't points. $\endgroup$ – Qiaochu Yuan May 18 '16 at 17:13
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Define a partial order on the objects of $\mathcal{C}$ by declaring that $X \le Y$ if there exists a monomorphism $X \to Y$, i.e. if $X$ can be viewed as a subobject of $Y$. Then the initial object is the least element of $\operatorname{ob}\mathcal{C}$, and an object satisfies your condition exactly when it is an atom for this partial order. So I guess you can call these objects "atoms".

If your category is abelian, then this is equivalent to being a simple object, a better-known property of objects. If the category is not abelian then the conditions are no longer equivalent (the $n$Lab mentions the example of the category of groups: a simple group has no nontrivial quotients, i.e. it has no nontrivial normal subgroups, but it can still have non-normal subgroups).

This has actually little to do with being a connected object: in the category of topological spaces, an object is connected in the categorical sense iff it is connected as a space (fortunately). But only a singleton space has no nontrivial subspace.

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