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?
1 Answer
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.
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$\begingroup$ Thank you for your answer, I would like to know that is there any further reference and proof for the statement? $\endgroup$– jy zOct 26, 2020 at 16:57
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2$\begingroup$ I don’t know a reference for these particular statements, but you might find them or something similar in Chapter 4 of Borceux’s Handbook of Categorical Algebra Vol. 1. The proofs of the two propositions I gave are very simple, and I would encourage you to try to prove them for yourself. $\endgroup$ Oct 26, 2020 at 19:55
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1$\begingroup$ Thank you for your reference, for proposition 1, I proved it successfully, but I was now stuck at proposition 2, I nearly have no idea on how to relate the definition of extremal generators with generators via the definition of balanced category $\endgroup$– jy zOct 26, 2020 at 22:30
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$\begingroup$ The idea is that in a balanced category, if a subobject inclusion $B \to A$ is an epimorphism, then that subobject is not proper. $\endgroup$ Oct 26, 2020 at 22:34
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1$\begingroup$ @jyz: To show the “balanced” implication, I recommend rephrasing the “extremal generator” condition to the contrapositive of Grothendick’s version: show it as “given a subobject $B ≤ A$, if all maps from objects $U_i$ into $A$ factor through $B$, then $B = A$”. This now looks similar to the definition of balancedness: “given a subobject $B≤A$, if its map is epi, then $B = A$”. $\endgroup$ Oct 27, 2020 at 9:02