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This has always been a strange thing to beginning category theorists (and maybe still for the rest of us). The usual example seems to be the algebraic closure of a field (although that's an injective hull only in a slightly unobvious category, now noted on our page). Any two algebraic closures are isomorphic, but not canonically so, and this applies to injective hulls in general.
It should tell us something, but I don't know what.
Added to injective hull a general definition in terms of a class of morphisms rather than a class of objects, taken from the paper Essential weak factorization systems by Tholen (which Patrick Schultz just told me about).
I am somewhat confused by the current status of this page. It has an “Idea” section which puts itself in the context of a general concrete category, but then refers to the page essential embedding whose definition is stated only for modules. There is no “Definition” section and it then plunges into Examples, with a Generalization section afterwards. Can we put one or more general definitions at the top and then explain exactly how the examples are examples of this definition?
Also, I wonder how are all the definitions related to each other? I think that an $\mathcal{H}$-injective hull in the sense of the second definition should also be a $\mathcal{C}$-hull in the sense of the first definition where $\mathcal{C}$ is the class of $\mathcal{H}$-injectives, but I don’t immediately see any way to go the other direction. Which of these definitions is more directly a generalization of the more concrete one?
I wrote the original version of essential embedding. I am not sure what is the intrinsic characterization of the categories where this makes sense so I wrote in some categories of modules (say, over operator algebras). It seems that the categories with pullbacks and zero object suffice at least for the definition.
Does it agree with the general definition in terms of a class of morphisms $\mathcal{H}$?
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