Enlargement of (fibered) derivators

Abstract

We show that the theory of derivators (or, more generally, of fibered multiderivators) on all small categories is equivalent to this theory on partially ordered sets, in the following sense: Every derivator (more generally, every fibered multiderivator) defined on partially ordered sets has an enlargement to all small categories that is unique up to equivalence of derivators. Furthermore, extending a theorem of Cisinski, we show that every bifibration of multi-model categories (basically a collection of model categories, and Quillen adjunctions in several variables between them) gives rise to a left and right fibered multiderivator on all small categories.1 Small categories Along the lines we also prove that a left (or right) fibered multiderivator defined on a smaller diagram category can be intrinsically enlarged to a larger diagram category, whenever some nervelike construction is available, relating the two diagram categories. This is more of theoretical interest because most derivators occurring in nature come from model categories. One concrete result of the construction is the following: Corollary 1.2. Let D → S be a left (resp. right) fibered multiderivator with domain Invpos 2 (resp. Dirpos 3 ) such that S is defined on Cat and such that also (FDer0 right) (resp. (FDer0 left)) holds. Then there exists an enlargement of D to a left (resp. right) fibered multiderivator E → S with domain Cat, such that its restriction to Invpos (resp. Dirpos) is equivalent to D. Any other such enlargement is equivalent to E.Note that this holds, in particular, for usual derivators (take for S the final pre-derivator) and closed monoidal derivators (take for S the final pre-multiderivator).Proof. In the left case, apply the machine of Theorem 4.1 twice using the functors N constructed in Proposition 2.5, firstly for the pair (Invpos ⊂ Cat ○ ), and secondly for the pair (Inv ⊂ Cat). Similarly for the right case.If we start with a fibered multiderivator on all of Pos, however, we show that the two extensions to Cat agree. Therefore we arrive at the following Corollary 1.3. Let D → S be a left and right fibered multiderivator with domain Pos such that S is defined on Cat. Then there exists a canonical enlargement of D to a left and right fibered multiderivator E → S with domain Cat, such that its restriction to Pos is equivalent to D. The enlargement is unique up to equivalence of fibered multiderivators over S.Actually, here Pos can be even replaced by the smallest diagram category containing both Invpos and Dirpos.Proof. We consider this time the pairs (Pos ⊂ Cat ○ ) and (Cat ○ ⊂ Cat). For each of these pairs we dispose of functors N as in 2.3 for the left and the right case simultaneously by Proposition 2.5 (by enlarging Dia ′ we only weaken the axioms). Hence we may conclude by applying Proposition 4.10 twice.For the reader mainly interested in plain (left and right) derivators, we state explicitly: Corollary 1.4. Let D be a derivator with domain Pos. Then there exists a canonical enlargement of D to a derivator E with doma...

Six-Functor-Formalisms on Higher Stacks

“…Proof. This is an easy consequence of the fact [25] that the theory of left (resp. right) fibered derivators with domain Cat is in a certain sense equivalent to its restriction to Dirlf (resp.…”

“…6. Extend D (k) to Cat using the theory of [25] and establish that D (k) is also a right fibered (multi)derivator using Brown representability.…”

Higher stacks as diagrams

“…Let S be a small idempotent complete 13 category with Grothendieck pretopology and consider the left Bousfield localization SET S op ×∆ op loc . Its cofibrant objects are the same as the cofibrant objects in SET S op ×∆ op and thus these are in the essential image of S ∐,∆ op because S is idempotent complete.…”

“…Hence, if N (α) is a Čech weak equivalence, α must be in the localizer. 13 For example S is idempotent complete if it has finite limits or finite colimits.…”

“…Corollary 5. 13. Let M be one of the model categories in 5.1 and let S ∈ S. Then a morphism α ∶ D 1 → D 2 in Cat(S) is a D M -equivalence over S if and only if N (α) is a weak equivalence in M. In particular, the classes of morphisms α such that N (α) is a weak equivalence form a localizer in each of the cases w.r.t.…”