Descent in the injective model structure and descent for simplicial presheaves

Question

In Jardine's book Local Homotopy Theory P.102, a simplicial presheaf $X$ on a site $C$ is said to satisfy descent if some local injective fibrant replacement $ j : X → Z $ is a sectionwise weak equivalence.

The definition in Dugger Hollander Isaksen defines a presheaf $F$ to satisfy descent if $F(X)$ is weak equivalent to the homotopy limit of $F(U_\bullet)$ for any hypercover $U_\bullet\to X$, that is $F(X)\simeq holim_{\Delta^{op}}F(U_n)$.

Are these two definitions equivalent?

Answer 1

According to DHI's official definition (Def 4.3), a sectionwise fibrant simplicial presheaf $F$ satisfies descent if $F(X) \simeq \text{holim}\, F(U_{\bullet})$ for all hypercovers. If $F$ is not sectionwise fibrant then it satisfies descent if there's a sectionwise weak equivalence $F \to G$ such that $G$ is sectionwise fibrant and satisfies descent.

This definition coincides with Jardine's. Suppose that $F$ satisfies descent in Jardine's sense, so that we have a sectionwise equivalence $F \to G$ with $G$ local injective fibrant. Since $G$ is local injective fibrant, DHI's Theorem 6.2 implies that it's sectionwise fibrant and $G(X) \simeq \text{holim}\, G(U_{\bullet})$ for all hypercovers. So $F$ satisfies descent in DHI's sense.

Conversely, suppose that $F$ satisfies descent in DHI's sense. Taking a fibrant replacement for $F$ in the global injective model structure, let $F \to G$ be a sectionwise weak equivalence with $G$ global injective fibrant. Since $F$ satisfies descent so does $G$, so DHI's Theorem 6.2 implies that $G$ is local injective fibrant. Then $F \to G$ is a local injective fibrant replacement that's a sectionwise weak equivalence, so $F$ satisfies descent in Jardine's sense.