In SAG def 1.2.1.4, Lurie defines local rings in topoi. The definition includes that for the inclusion $e:R^\times\hookrightarrow R$, the map $e\amalg (1-e): R^\times \amalg R^\times\to R$ is an effective epimorphism. I'm confused why it says an 'effective' epimorphism since in any topoi it is equivalent to being an epimorphism. Is there a reason for choosing this among many other candidates when defining local rings in more general categories?

@NarukiMasuda In $\infty$-topoi, "effective epimorphism" is not the same as "epimorphism", so perhaps Jacob's choice of terminology is a carry-over from that. There is a generalization of this statement with topoi replaced by $\infty$-topoi and commutative rings replaced by $\mathbb{E}_\infty$-rings, and you definitely need to say "effective" in that case.

@CharlesRezk Thank you. So it is necessary to make the analogous definition of local $\mathbb{E}_\infty$ rings in $\infty$-topoi equivalent to seemingly more naive definition (1.4.2.1: an $\mathbb{E}_\infty$ ring is local if $\pi_0$ is local), is this correct?

yeah

It's not necessary, just makes the analogy clearer.

@CharlesRezk Thanks

As an example the epimorphisms in spaces are the acyclic maps

@SaalHardali Do you have a reference for that?

@DenisNardin arxiv.org/pdf/1711.08898.pdf