Definition 0. Let $(\mathbf{C},U)$ denote a concrete category. Let "splits" be a desirable property that pointed $\mathbf{C}$-objects may or may not satisfy. Let $(X,x)$ be a pointed $\mathbf{C}$-object. Then a splitting object of $(X,x)$ is a Galois-initial object in the category of coslices of $(X,x)$ whose underlying pointed $\mathbf{C}$-object splits.
By a "Galois-initial object," I mean what here I refer to as a nearly-initial object; Garner and Hirschowitz refer to this simply as a "Galois object." But I've settled on using the word "Galois" to mean "weak, insofar as its unique up to isomorphism, but not necessarily up to unique isomorphism."
For example, splitting fields are the case where $\mathbf{C}$ is the category of fields, $U$ is defined by asserting that, for each field $k$, $U(k)$ is the set of non-constant univariate polynomials in $k$. Ergo a "pointed field," in this case, is a field equipped with a non-constant univariate polynomial. And a pointed field $(k,P)$ in this sense is said to "split" iff $P$ can be written as a finite product of degree $1$ polynomials. For example, a splitting field of $(\mathbb{R},x^2+1)$ is $(\mathbb{C},(x-i)(x+i)).$
Definition 1. Let $(\mathbf{C},U)$ denote a concrete category. Let "splits" be a desirable property that pointed $\mathbf{C}$-objects may or may not satisfy. Let $X$ be a $\mathbf{C}$-object. Then an algebraic closure of $X$ is a Galois-initial object in the category of coslices of $X$ such that every way of making the relevant object into a pointed object splits.
For example, the algebraic closure applied to fields (as above) is just the usual algebraic closure.
Questions.
Q0. Are there general conditions under which splitting objects exist, generalizing the fact that splitting fields exist?
(It doesn't have to be super-general... I just want the theorem to include a few other instructive examples from algebra, so I can feel like splitting fields are less mysterious, because they form part of a coherent, larger system of ideas.)
Q1. Are there general conditions under which we can infer, from the existence of splitting objects, the existence of algebraic closures?
