# 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.

• Surely for vector spaces, but also in general: what a bout "a spanning subset" or, as you suggested implicitly, "a generating set"? May 15, 2021 at 11:09
• Doesn't 'generating set' fit your requirements? I see no reason to invent anything fancier. May 15, 2021 at 11:41
• You don't need to put the identity in as a generator for unital algebras. It's part of the signature. May 15, 2021 at 12:34
• Another example why generating set does not fit: In the category of topological space with continuous functions, a "base" would be a dense subset. I don't think one would usually say that the dense subset "generates" the space. May 15, 2021 at 12:34
• If you allow nonunital homomorphisms then 1 is not part of the generating set because you are just working in the category of not necessarily unital rings and so you would have to include it as a generator. But then I'd don't think 0 would be a base because you can map $\mathbb C$ to itself by the zero map and the identity and these both agree on zero and are X- algebra homomorphisms in the nonunital sense in that f(cx)=cf(x) for all complex numbers c. But you have to define your morphisms clearly May 15, 2021 at 13:52

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.

• +1, but “jointly epimorphic family” is a much more common term for this than “epi-sink”, at least in my experience. May 15, 2021 at 22:01
• This is true (I have added it), perhaps also because "jointly monomorphic family" is much more common than "mono-source" (as defined in Joy of Cats), since "source" already has a different meaning in category theory. May 15, 2021 at 23:12
• There is no Definition 10.26 in the linked Joy of Cats. Maybe it's a typo in the number? May 16, 2021 at 7:52
• Yes I corrected the typo. May 16, 2021 at 9:21

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.