Name for a set of elements that fully determine a morphism In a concrete category (i.e., where the morphisms are functions between sets), I define a base of an object $A$ to be a set of elements $M$ of $A$ such that for any morphisms $F,G:A\to B$ that coincide on $M$, we have $F=G$.
Question: Is there an established name for a base in that sense?
Examples: In the category of vectors spaces, generating sets are bases. In the category of sets, $A$ is the only base of $A$.
Note: The above definition does not really need a concrete category (an initial object is enough), but I decided to formulate it in a concrete category for simplicity.
 A: The term "base" should not be used, since, as you say, you are actually generalizing the notion of a generating set.
It is an epi-sink, also known as jointly epimorphic family. See Joy of Cats, Definition 10.62 and (dual of) Definition 10.5. A family of morphisms $(f_i : A_i \to A)$ is called an epi-sink when for $u,v : A \to B$ we have $\forall i (u \circ f_i = v \circ f_i) \implies u=v$. When the coproduct $\coprod_i A_i$ exists, this means that we have an epimorphism $\coprod_{i \in I} A_i \to A$.
If you have a terminal object $1$, morphisms $1 \to A$ are called global elements, and we can look at epi-sinks consisting of global elements of $A$.
For many categories, though, global elements are not enough. When we have a forgetful functor $U$ to $\mathbf{Set}$ with a left adjoint $F$, we have $U(-) \cong \hom(F(1),-)$, so that elements of the underlying set can be seen as morphisms on $F(1)$, and we can talk about epi-sinks on $F(1)$.
But the most general form does not put any restrictrions on the domains at all.
A: At least in the context of von Neumann algebras, separating is used for this concept. Confer [Takesaki], Definition II.3.16 (slightly reformulated):
Definition. Let $\mathcal M$ be a von Neumann algebra on $\mathfrak H$. A subset $\mathfrak U$ of $\mathfrak H$ is called separating for $\mathcal M$ iff for all $a\in\mathcal M$, $a\xi=0$ for all $\xi\in\mathfrak U$ implies $a=0$.
(But note also the definition of a separating set in nLab which is related by a different concept.)
[Takesaki] Takesaki, Masamichi, Theory of operator algebras I, New York, Heidelberg, Berlin: Springer-Verlag. VII, 415 p. DM 79.00; $ 44.30 (1979). ZBL0436.46043.
