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I am interested in the following property, be it on an abstract or concrete category:

$A$ is a substructure of $B$ such that every automorphism of $A$ extends uniquely to an automorphism of $B$. Or we can speak of an embedding $\iota:A\rightarrow B$ such that for every automorphism $\alpha$ of $A$ there exists a unique automorphism $\beta$ of $B$ such that $\iota\alpha=\beta\iota$.

This condition is not too difficult to show up in many combinatorial, algebraic and geometrical settings, at least. So, I wonder if it has been named already.

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  • $\begingroup$ @NoahSchweber I think the OP meant not only that the extension is unique when it exists (which is equivalent to triviality of the group of automorphisms of $B$ fixing $\iota[A]$ pointwise) but also that the extension should always exist. $\endgroup$
    – Andreas Blass
    Commented Nov 24, 2017 at 19:04
  • $\begingroup$ @AndreasBlass Good point, deleted. $\endgroup$
    – Noah Schweber
    Commented Nov 24, 2017 at 19:41
  • $\begingroup$ @AndreasBlass You got it correctly. Also, just to clarify, the small object does not have to be invariant under all automorphisms of the large one. For instance, take $B$ to be a prism over a regular polygon, $A$ to be a base, and morphisms to be rigid motions; in this case, a flip on $A$ cannot extend to $B$, so the embedding is not what I want. On the other hand, if one looks at the 1-skeleton of $B$, and morphisms are graph isomorphisms, then the embedding is like I want. In this case, there are automorphisms of $B$ that don't fix $A$. $\endgroup$
    – Arnaldo Mandel
    Commented Nov 24, 2017 at 20:01

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