CatDat

Implication Details

Claim: If a category is core-connected, then it has an extremal generator.

Proof: Let $G$ be any object of a core-connected category. Then certainly $G$ is a generator: Let $f, g : X \rightrightarrows Y$ be morphisms such that $f \circ h = g \circ h$ for all $h : G \to X$. Choose an isomorphism $h : G \to X$. Then $h$ is an epimorphism, so that $f = g$.
To see $G$ is in fact an extremal generator: Let $f : X \to Y$ be a morphism such that $$f \circ {-} : \Hom(G, X) \to \Hom(G, Y)$$ is a bijection. Then for any object $T$, $$f \circ {-} : \Hom(T, X) \to \Hom(T, Y)$$ is a bijection since $T \cong G$, and the Yoneda embedding of $f$ is a natural transformation $\Hom({-}, X) \to \Hom({-}, Y)$. By the Yoneda Lemma, $f$ is therefore an isomorphism.