Schreiber relative cohomology

Idea

Given an (∞,1)-topos $\mathbf{H}$ we get an (∞,1)-category $\mathbf{H}_{rel}$ the arrow (∞,1)-category of $\mathbf{H}$:

• objects are morphisms $\array{A \\ \downarrow \\ A'}$ in $\mathbf{H}$;

• morphisms from $\array{X \\ \downarrow \\ X'}$ to $\array{A \\ \downarrow \\ A'}$ are (homotopy-)commutative
  diagrams

  $$\array{ X &\to& A \\ \downarrow && \downarrow \\ X' &\to& A' } \,.$$

The (∞,1)-category $\mathbf{H}_{rel}$ should be the one presented by the Reedy model structure on

$$Func(I, \mathbf{H}) \,,$$

where $I$ is the interval category $I = \{a \to b\}$.

Definition

Let $SPSh(C)^{loc}$ be a model structure on simplicial presheaves that presents $\mathbf{H}$.

Let $I_+ = \{0 \to 1\}$ and $I_- = \{1 \to 0\}$ be the interval category regarded as a Reedy category in two different ways, with degrees as indicated. Regard $I_+$ and $I_-$ as a SSet-enriched Reedy category in the canonical way. We write just $I$ in cases that apply to either Reedy category structure.

The enriched functor category $[I,SPSh(C)^{loc}]$ – the arrow category of $SPSh(C)$ – carries the Reedy model structure $[I,SPSh(C)^{loc}]_{Reedy}$.

It follows from Angetveit’s theorem that with this model structure and the canonical SSet-enrichment the category $[I,SPSh(C)^{loc}]$ is a simplicial model category.