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.

  • $\begingroup$ 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
  • 1
    $\begingroup$ I would suggest an atom. $\endgroup$ – J.-E. Pin May 18 '16 at 10:43
  • 2
    $\begingroup$ 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
  • 3
    $\begingroup$ 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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.