# Descent in the injective model structure and descent for simplicial presheaves

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?

• There are some grammatical issues with how you phrased the DHI result. It would be clearer if you edit. Sep 6, 2019 at 14:31
• @DavidWhite Yes I've edited Sep 6, 2019 at 16:17

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.