Schreiber relative cohomology


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

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

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

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

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

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

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


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

Let I +={01}I_+ = \{0 \to 1\} and I ={10}I_- = \{1 \to 0\} be the interval category regarded as a Reedy category in two different ways, with degrees as indicated. Regard I +I_+ and I I_- as a SSet-enriched Reedy category in the canonical way. We write just II in cases that apply to either Reedy category structure.

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

Created on August 21, 2009 at 18:22:05. See the history of this page for a list of all contributions to it.