As I was reading Grothendieck's Tohoku paper(translated by M.L.Barr and M.Barr), I found that the definition of a generator in the category differs from that defined in wikipedia.  

Let $\mathbf{C}$ be a category(It may be necessary that $\mathbf{C}$ is a locally small category), a family of generators {$U\_i$}$\_{i\in I}$ with $I$ being an index set, according to Tohoku paper, are a collection of objects such that for any object $A$ and any subobject $B \neq A$, there is $i\in I$ and a morphism $u\colon U\_i \rightarrow A$ which does not come from $U\_i \rightarrow B$. While in wikipedia, it is defined in a way such that for any $f,g\colon A\rightarrow B$ with $f\neq g$, there is an $i\in I$ and $u\colon U\_i\rightarrow A$, such that $f\circ u \neq g\circ u$.  

What I would like to know is that are these 2 definitions equivalent, or is the definition in Wikipedia stronger than that in Tohoku paper?



Are generators defined in Tohoku paper equivalent to that defined in Wikipedia (Which I believe is a more widely used definition)
In this answer, let me use the terms **generator** and **extremal generator** for the Wikipedia and Tohoku definitions respectively.


In general, these two definitions are not equivalent, and neither implies the other.


**Example 1.** For an example of an extremal generator which is not a generator, consider a non-trivial group seen as a category with one object. The empty family is an extremal generator in this category (because there are no proper subobjects) but not a generator.


**Example 2.** An example of a generator which is not an extremal generator is the singleton family consisting of the terminal object (i.e. the singleton poset) in the category of posets. To see that it is not an extremal generator, observe that the discrete poset $\{0,1\}$ is a proper subobject of the poset $\{0 < 1\}$.


Nevertheless, there are many categories in which these implications do hold.


**Proposition 1.** In any category with equalisers, every extremal generator is a generator.


**Proposition 2.** In any balanced category (i.e. a category in which "monomorphism" + "epimorphism" => "isomorphism"), every generator is an extremal generator.


Hence in any balanced category with equalisers, the two notions of generator and extremal generator coincide. This includes all abelian categories (as in Grothendieck's paper) and all (pre)toposes.


