In Proposition 7.2.1.14 of *Higher Topos Theory*, Lurie asserts the following:



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> Let $\mathcal{X}$ be an $\infty$-topos and let $\tau\_{\leq0}:\mathcal{X}\to\tau\_{\leq0}\mathcal{X}$ denote a left adjoint to the inclusion. A morphism $\phi:U\to X$ is an effective epimorphism if and only if $\tau\_{\leq 0}$ is an effective epimorphism in the ordinary topos $\operatorname{h}(\tau\_{\leq0}\mathcal{X}$).
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His proof relies on Lemma 7.2.1.13, which in turn relies on Proposition 6.5.1.20. His proof of Proposition 6.5.1.20, in turn, relies on Proposition 7.2.1.14. **This is circular.**


Hopefully, the proof of Proposition 6.5.1.20 only uses the following weaker version of Lemma 7.2.1.13:



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> ($\ast$) Let $\mathcal{X}$ be an $\infty$-topos and let $f:X\to Y$ be a morphism of $\mathcal{X}$. Suppose that $X$ and $Y$ are $1$-connective. Then $f$ is an effective epimorphism.
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So my question is: **Does anyone know how to prove ($\ast$) without using Proposition 7.2.1.14?** Thanks in advance.



Effective epimorphisms and 0-truncations (HTT, 7.2.1.14)
Here's an easy way to resolve the circularity. Proposition 7.2.1.13 is only used in the proof of 7.2.1.14 to establish the following statement:


(1) If $f\colon V\to X$ is a monomorphism and is surjective on $\tau\_{\leq 0}$, then it is an isomorphism.


This in turn follows from:


(2) If $f\colon V\to X$ is a monomorphism, then it is pulled back from $\tau\_{\leq 0}f\colon \tau\_{\leq 0}V\to\tau\_{\leq 0}X$.


Now (2) is obviously preserved by left exact localizations, and it holds in $\infty$-categories of presheaves since it holds in $\mathcal S$, so it holds in any $\infty$-topos.


This gives a direct proof of 7.2.1.14 and in particular of (\*).


