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In classical category theory we have a following criterion. If $\mathcal{C}$ and $\mathcal{D}$ are finitely complete categories and $F : \mathcal{C} \to \mathcal{D}$ is a functor which preserves finite limits and reflects isomorphisms, then $F$ is faithful. It follows easily from an observation that an equalizer of two parallel morphisms is an isomorphism if and only if those morphisms are equal.

My question is

Given two finitely complete $(\infty, 1)$-categories $\mathcal{C}$ and $\mathcal{D}$ and a functor $F : \mathcal{C} \to \mathcal{D}$ which preserves finite limits and reflects equivalences, does it follow that $\mathrm{Ho} F : \mathrm{Ho} \mathcal{C} \to \mathrm{Ho} \mathcal{D}$ is faithful?

The observation I mentioned previously doesn't work in higher category theory since for example an equalizer of the identity morphism of an object $X$ with itself is a free loop object on $X$, but I am unable to decide whether this faithfulness criterion is valid.

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up vote 6 down vote accepted

Here's a counterexample. Let $\mathcal{C}=\mathcal{D}$ be the stable $(\infty,1)$-category of perfect complexes over $\mathbb{Z}_{(p)}$, and let $F(X)=X\otimes \mathbb{Z}/p$ (the derived tensor product). Then $F$ is exact, and does not send any nonzero objects to zero and hence reflects equivalences. But $F$ is not faithful, since it kills the multiplication by $p$ map on any object.

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