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Definition. Call an object $X$ of a category $\mathbf{C}$ nearly initial iff firstly, it is weakly initial, and secondly, for all objects $Y$ and all morphisms $f,g : X \rightarrow Y$, there exists an automorphism $\alpha$ of $X$ such that $g = f \circ \alpha$.

It's straightforward to show that any two nearly initial objects of a category $\mathbf{C}$ are isomorphic (though not necessarily up to unique isomorphism.) For instance, if I'm not mistaken, the algebraic closure of a field is, by definition, the unique (up to isomorphism) nearly-initial algebraically-closed extension of that field.

Question. What are "nearly initial" objects really called?

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    $\begingroup$ I've not seen this notion myself, but it reminds me of the notion of injective hull: ncatlab.org/nlab/show/injective+hull Algebraic closures are an example. $\endgroup$ – Todd Trimble Dec 4 '16 at 23:11
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    $\begingroup$ I don't have a better name, but I do have a fun fact to share (a straightforward application of Quillen's Theorem A): If $\mathbf{C}$ has a nearly initial object $X$ and all morphisms out of $X$ are monomorphisms, then the classifying space of $\mathbf{C}$ is a $K(G,1)$ where $G = \mathbf{Aut}(X)$. $\endgroup$ – Tim Campion Dec 5 '16 at 1:22
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    $\begingroup$ Also, the dual concept is also important. For example, saturated models and Fraisse limits have this "nearly terminal" property, at least with respect to other objects which are "small". To tie into Todd's remark, such objects can be constructed as injective hulls / Fraisse-type constructions, using back-and-forth arguments. I wrote a note about this here once. $\endgroup$ – Tim Campion Dec 5 '16 at 1:32
  • $\begingroup$ Another example is the universal covering of a space, in the category of coverings. Accordingly, $G$ in the category of $G$-sets is such. $\endgroup$ – მამუკა ჯიბლაძე Dec 5 '16 at 5:30
  • $\begingroup$ @მამუკაჯიბლაძე, I think you mean $G$ in the category of inhabited $G$-sets? And, is that really true? $\endgroup$ – goblin Dec 5 '16 at 5:53
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In Definition 4.1 of their paper Shapely monads and analytic functors, Richard Garner and Tom Hirschowitz call such an object a "Galois object".

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    $\begingroup$ What is the precise relation to Galois extensions of fields? $\endgroup$ – HeinrichD Dec 5 '16 at 9:05
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On reflection, maybe I do have some terminological suggestions, though I'm not sure how much I like them.

  • Model theorists would call a weakly terminal object "universal", so you might call a weakly initial object "co-universal". Bleh.

  • Model theorists would call an object with the dual of your automorphism property "homogeneous", so you might use the term "co-homogeneous". Maybe a little less bleh.

  • The term "saturated" for model theorists is roughly equivalent to universal + homogeneous, i.e. to "nearly terminal". So you might use the term "co-saturated". This strikes me as a little dangerous because the correspondence in the dual case is not exact.

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