Lemma 21.6.12. The geometric realization functor \(\abs {-}\colon \s \An \to \An \) preserves finite products.
Proof. By definition, this functor is \(\colim _{[n] \in \simp \catop }\colon \Fun (\simp \catop ,\An ) \to \An \), and the claim is equivalent to the statement that the product functor \(- \times -\colon \An \times \An \to \An \) preserves \(\simp \catop \)-indexed colimits. The \(\infty \)-category \(\simp \catop \) is sifted by Proposition 21.6.9. Furthermore, for an anima \(X\) the functor \(X \times - \colon \An \to \An \) admits a right adjoint \(\Hom _{\An }(X,-)\colon \An \to \An \), hence preserves all colimits. This verifies the assumptions of Lemma 21.6.11, proving the claim. □
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