I am totally green in category theory, so please excuse me it this is easily googlable. Is there a name for a free object in a given concrete category such that for every other member of the category there is an epimorphism from that object? The name universal free object seems to be reserved for something else.
Example. For the category of countable (Abelian) groups take the free (Abelian) group on countably many generators.
This is quite uncommon and I'm not aware of any applications even when they do exist, e.g. for the categories of all groups, all vector spaces, etc. there is no such object because objects have unbounded cardinality.
More common is the concept of a generator; one possible definition (and there are many) is that an object $c$ is a generator if every other object admits an epimorphism from some coproduct of copies of $c$. These are quite common, e.g. the category of algebraic objects of a given type (groups, vector spaces, etc.; formally, the category of models of a Lawvere theory) admits a generator given by the free object on one generator in the usual sense.
Sometimes people talk about an object satisfying some property P which is "universal" in the sense that all other objects satisfying P embed into it, which goes the other way; e.g. every compact metrizable space embeds into the Hilbert cube $[0, 1]^{\mathbb{N}}$. "Universal" is in quotes here because this is not a universal property. I'm not aware of any applications of this sort of thing either but it comes up sometimes at least.
