… in Grothendieck abelian categories (like derivators associated to fields, rings, or schemes),
and homotopy … By duality it suffices to show the equivalence of the first four statements. The …

… also algebras over operads, only up to a homotopy equivalence. If we work with a monoidal
… over operads up to a weak equivalence in the categorical homotopy-theoretic sense [14]. …

… ’s homotopy colimit theorem. Some of these results are suggestive of the (∞, 2)categorical
functoriality of the Grothendieck … Of course, by duality the contravariant Grothendieck …

… of Grothendieck duality the functor f! is obtained constructively through local computation.
Following the work of A. Neeman, we generalize Grothendieck duality … We call Z the homotopy …

… (in which the bottom row is any standard triangle, and the homotopy equivalence ϕ becomes
an isomorphism) as soon as one checks that the top row is in fact of the form specified by (…

… (iv) Hq(Hom*(Q*(F) ,Q*(G» is the group of homotopy classes of morphisms which anti-…
dependS only on the homotopy class of u. Therefore, there exist pairings RPT(F) x Hq(Hom*…

… homotopy equivalence for infinity-morphisms, which gives a simple description of the homotopy
… Grothendieck [16] and introduced the notion of model category [27]. A derived functor, …

… whose cohomologies are quasi-coherent and vanish in sufficiently negative degrees and
whose morphisms are homotopy equivalence classes of maps of complexes will be denoted K …

… , the stable homotopy groups of which coincide with the negative Grothendieck-Witt groups.
In … This section introduces the general yoga of duality on the dg category of strictly perfect …

… For instance, we record an ∞-categorical version of Thomason's homotopy colimit theorem.
Some of … Of course, by duality the contravariant Grothendieck construction enjoys analogous …