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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.

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    $\begingroup$ Surely for vector spaces, but also in general: what a bout "a spanning subset" or, as you suggested implicitly, "a generating set"? $\endgroup$
    – Uri Bader
    Commented May 15, 2021 at 11:09
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    $\begingroup$ Doesn't 'generating set' fit your requirements? I see no reason to invent anything fancier. $\endgroup$
    – Denis T
    Commented May 15, 2021 at 11:41
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    $\begingroup$ You don't need to put the identity in as a generator for unital algebras. It's part of the signature. $\endgroup$
    – Benjamin Steinberg
    Commented May 15, 2021 at 12:34
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    $\begingroup$ 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. $\endgroup$
    – Dominique Unruh
    Commented May 15, 2021 at 12:34
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    $\begingroup$ 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 $\endgroup$
    – Benjamin Steinberg
    Commented May 15, 2021 at 13:52

2 Answers 2

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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.

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    $\begingroup$ +1, but “jointly epimorphic family” is a much more common term for this than “epi-sink”, at least in my experience. $\endgroup$
    – Peter LeFanu Lumsdaine
    Commented May 15, 2021 at 22:01
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    $\begingroup$ 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. $\endgroup$
    – Martin Brandenburg
    Commented May 15, 2021 at 23:12
  • $\begingroup$ There is no Definition 10.26 in the linked Joy of Cats. Maybe it's a typo in the number? $\endgroup$
    – Dominique Unruh
    Commented May 16, 2021 at 7:52
  • $\begingroup$ Yes I corrected the typo. $\endgroup$
    – Martin Brandenburg
    Commented May 16, 2021 at 9:21
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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.

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