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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, 2016 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, 2016 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, 2016 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. $\endgroup$ Dec 5, 2016 at 5:30
  • $\begingroup$ @მამუკაჯიბლაძე, I think you mean $G$ in the category of inhabited $G$-sets? And, is that really true? $\endgroup$ Dec 5, 2016 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, 2016 at 9:05
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    $\begingroup$ From the paper: "(Our nomenclature draws on one of the basic examples of a Galois object: if $k \subset K$ is a Galois field extension, and $\mathscr A$ is the category of intermediate field extensions, then $K$ is Galois in $\mathscr A^{\text{op}}$.)" $\endgroup$
    – varkor
    Dec 24, 2020 at 13:06
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Assuming uniqueness of the automorphism, an earlier reference than Garner–Hirschowitz's paper is Tholen's MacNeille completion of concrete categories with local properties (1979), in which these objects are called quasi-initial (Definition 1.1), defined as those objects that are weakly initial and prequasi-initial (which is exactly the automorphism condition you describe).

Another earlier reference is Huq's Semilimits in Categories (1991), where these (or their duals) are called semiterminal objects, again assuming uniqueness of the automorphism.

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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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I think you are interested in the notion of poly-colimits. This was originally introduced by Lamarche as a generalization of the notion of multi-colimits.

  • Michel Hebert, Syntactic characterizations of closure under pullbacks and of locally polypresentable categories.
  • Francois Lamarche, Modelling polymorphism with categories.

More recently, Paul Taylor, and the group of categorical model theory in Brno (Rosicky, Leiberman) have brought back the topic to the general interest. The Brno group applied this technology to Abstract elementary classes with intersection.

  • Lieberman, Rosicky, Vasey, Universal Abstract Elementary classes and Locally multi-presentable categories.

  • Taylor, Locally Finitely Poly-Presentable Categories.

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