Is there a categorical way (in terms of diagrams, limits, lifting properties etc) to formulate the requirement that for every pair of monos $f,g:C \to D$ there should be an endomorphism $h:D \to D$, s.t. $h\circ f = g$? I found a couple of specific monos (regular, strict,strong,orthogonal..) but nothing seems to fit for my problem..
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2$\begingroup$ Well.. Isn't what you wrote already in terms of category theory? Sets and Vector spaces are examples. Do you have in mind any other examples? $\endgroup$– BerciOct 7, 2015 at 18:19
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$\begingroup$ Technically yes, but I thought that there is maybe a more general concept "explaining" this typical property... $\endgroup$– Bipolar MindsOct 7, 2015 at 18:39
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$\begingroup$ Also, "commute" is not the term I'd use for this either. $\endgroup$– Qiaochu YuanOct 8, 2015 at 20:29
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1 Answer
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This is not a condition on monomorphisms at all; as formulated it's a condition on $C$ and $D$. Here is the most closely related thing I know.
A topological space $X$ is sometimes said to be homogeneous if for every $x, y \in X$ there is a homeomorphism $f : X \to X$ such that $f(x) = y$. This is the special case of your situation in $\text{Top}$ where $C$ is a point and $h$ is required to be an isomorphism. Most spaces are not homogeneous but, for example, all connected manifolds without boundary are.
In general this condition seems hard to satisfy. I don't expect there's anything particularly nice to say about it.
answered Oct 7, 2015 at 23:31
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$\begingroup$ Okay thx! At least now I have a name for such categories ;) $\endgroup$ Oct 8, 2015 at 7:41
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$\begingroup$ @Bipolar: I think that's a bad term to use for the category. I'm explicitly using it for $D$, with a fixed choice of $C$, and with a restriction on $h$. $\endgroup$ Oct 8, 2015 at 16:51