aa r X i v : . [ m a t h . C T ] O c t An alternative to hypercovers
Andrew W. MacphersonOctober 13, 2020
Abstract
I introduce a class of diagrams in a Grothendieck site called atlases which can be used to study hyperdescent, and show that hypersheavestake atlases to limits using an indexed ‘nerve’ construction that pro-duces hypercovers from atlases. Atlases have the flexibility to be at thesame time more explicit and more universal than hypercovers.
Contents Introduction
A presheaf F on a topological space X is said to satisfy descent along an opencover { U i ⊆ V } i : I of an open set V ⊆ X if its sections over V can be computedby taking the limit over the ˇCech nerve · · · ( ` i : I U i ) × X ( ` i : I U i ) ` i : I U i ← →← →← →←→ ←→ ← →← →←→ ( −→ X ) (1.0.1)(for sheaves of sets, the first two terms, which appear here explicitly, areenough). It is called a sheaf if it satisfies descent along all open covers. Thisperspective, popularised in [SGA4], is the starting point for topos theory andthe modern approach to numerous flavours of cohomology theory.For computing global sections — i.e. cohomology — it is useful to be ableto restrict attention to covers belonging to a certain site for X . For example,if X is a manifold, then a convenient site is the poset U o ( X ) of open setsdiffeomorphic to R n . Since these are exactly the contractible open sets, theclassifying space of this poset has the same homotopy type as X . Hence, thissite is useful for computing homotopy invariants of X , such as the theory oflocal systems on X .Because contractible subsets of X are not closed under intersection, theˇCech nerve construction is not available for every cover in the site U o ( X ).The preceding definition of descent along open covers does not, therefore,translate easily into the new context. What we need is a generalisation inwhich a binary intersection of open subsets can be resolved with its ownopen cover, and again for the intersections of those open sets, and so on adinfinitum . This is the intuition that the hypercovers of [SGA4, Exp. V, 7.3]attempt to capture.Today, the theory of hypercovers is woven into the fabric of nearly allflavours of homotopical sheaf theory (with the bulk of [HTT, Chap. 6] beinga notable exception). They admit an elegant characterisation in terms of themodel structure on simplicial presheaves [DHI04].In practice, we must often construct hypercovers by conversion frommore basic ‘naturally occurring’ data. This process destroys finiteness and,perhaps, intuitiveness. Why not try instead to capture these ‘naturally oc-curing’ data directly? D. Quillen is supposed to have said that only whenhe “freed [himself] from the shackles of the simplicial way of thinking” washe able to discover his Q-construction in algebraic K -theory [Gra13]. Thenotion of atlas I now define arose in the search for a similar liberty in thecontext of descent theory: 2 tlas/definition (Atlas) . Let T be an ∞ -category equipped with a Grothendiecktopology in the sense of [HTT, Def. 6.2.2.1], X : T an object. An ∞ -functor U : I → T / X is said to be an atlas for X if for any finite direct category K andmonotone map α : K → I , the set { U ˜ α → lim k : K U α ( k ) } ˜ α : I / α is a covering in PSh( T ), where I / α is the overcategory of [HTT, §1.2.9] (alsoknown as the category of left cones over K → I ), and U ˜ α stands for the valueof U ◦ ˜ α : K ⊳ → T / X on the cone point. atlas/descent/definition (Descent along atlases) . An ∞ -presheaf F : T op → Spc issaid to satisfy descent along atlases if each open V ⊆ X and atlas ( I , U ) of V induces an equivalence of spaces F ( V ) ˜ → lim i : I F ( U i ).(Here, of course, the limit in the ∞ -category Spc of spaces must be under-stood in the sense of ∞ -category theory [HTT, Def. 1.2.13.4].)The relevance of this definition is captured by the following statement: main/conjecture The following conditions on an ∞ -presheaf F : T op → Spc are equivalent:1. F satisfies descent along atlases.2. F satisfies hyperdescent.In particular, the full subcategory of
PSh ∞ ( X ) spanned by the functors thatsatisfy descent along atlases is a hypercomplete topos. The main application I have for this theory of atlases is a formulationof the universal property of (derived) geometry, which appears in a separatework [Mac17]. In the present paper, I prove a truncated version of Conjec-ture 1.3 that applies to topological spaces — more precisely, to locales — andis sufficient for the application of op. cit ..3 ain/descent-theorem
Let X be a locale with lattice of open sets U ( X ) , and let F : U ( X ) op → Spc be a hypercomplete ∞ -sheaf. Then F satisfies descent alongatlases indexed by posets.Proof.
It is well-known that hypercompleteness means that F takes hyper-covers to limits. See [HAG-I, §3] (below Definition 3.4.8) for a proof in thecontext of simplicially enriched categories; to translate into the language ofquasi-categories, use [HTT, Prop. 4.2.4.4] and [HTT, Rmk. 6.5.2.15].The data of a hypercover of V ⊆ X can be equivalently formulated as acertain kind of diagram J / ∆ op → U ( V ) indexed by the total space J of a leftfibration over ∆ op , called its index diagram (6.3), which has a certain locallifting property (Prop. 8.1).The result now follows from a nerve construction which associates to eachdiagram U : I → U ( V ) of open sets of V , indexed by a poset I , a diagram ∫ ˜ N ( I ) I ∆ op ← → ǫ ←→ l − fib such that U ◦ ǫ : ∫ ˜ N ( I ) / ∆ op → U ( V ) has the local lifting properties of a hyper-cover if and only if U is an atlas of V (Theorem 9.8). Moreover, ǫ is ∞ -cofinalin the sense of [HTT, §4.1] (Proposition 9.5), whence a colimit over I can becomputed on its restriction to ∫ ˜ N ( I ) [HTT, Prop. 4.1.1.8]. Thus if F is hyper-complete and U is an atlas, then F takes ( ∫ ˜ N ( I ), U ◦ ǫ ) — and therefore also( I , U ) — to a limit. Acknowledgement
Thanks to Adeel Khan whose question exposed an er-ror in an earlier draft of Definition 1.1.
This section contains some background information about posets, 1-categories,and ∞ -categories, and discussion of coinitiality and initial objects in thesecontexts. More accurately, F is a sheaf on the nerve of the poset U ( X ). ategory ( ∞ -categories and n -categories) . Partially ordered sets and classical (
Set -enriched) categories are realised, via the nerve construction, as 0- and 1-categories inside the category of quasi-categories [HTT, Ex. 2.3.4.3, Prop. 2.3.4.5].We do not make any linguistic distinction between a poset (resp. classi-cal category) and its nerve. (This only affects discussions where 0- or 1-categories interact with ∞ -categories, i.e. the proof of Proposition 3.2.) The ∞ -category of n -categories, for n : N ⊔ { ∞ } , is denoted Cat n . category/truncation (0-Truncation) . The inclusion
Cat → Cat ∞ has a right adjoint h (see[HTT, Rmk. 2.3.4.13], although this reference does not provide details). Arealisation of the right adjoint is as follows: given a quasicategory C , h C isthe poset whose elements are the vertices of C and whose relations are x ≤ y if and only if C ( x , y ) is nonempty. From this description it is also clear thatthe formation of the left cone (as defined in [HTT, Not. 1.2.8.4]) commuteswith h . (0-coinitial) . A subset K ⊆ K of a poset K is ifevery element of K is bounded below by an element of K .More generally, a functor u : K → K between two categories is called 0-coinitial if the image of K in h ( K ) is 0-coinitial. Equivalently, for everyobject k : K there exists a k : K and a morphism uk → k . util/cone/reduce-to-coinitial (Pushout of a cone along a coinitial map) . Let K be a poset,K ⊆ K a -coinitial subset. ThenK KK ⊳ K ⊳ ← →←→ ←→← → is a pushout in the category of posets.Proof. The statement for underlying sets is obvious, so we are just checkingthat all the order relations on K ⊳ = K ⊔ { e } factorise as strings of relationsthat lift to K ⊳ and K . The only relations i ≤ j in K ⊳ that do not lift to K or { e } are those for which i = e is the cone point and j ∈ K . In this case, as K is 0-coinitial there is some i ′ ≤ j with i ′ ∈ K , and the relation factorises as i = e ≤ i ′ ≤ j with e ≤ i ′ a relation in K ⊳ .5 eft-0-finite (Left 0-finite) . A 1-category K is said to be left 0-finite if anyfunctor from K into a finitely complete poset admits a limit cone.For example, this is the case for any K admitting a 0-coinitial subset; inparticular, when K admits an initial object. finite-limit (Finite intersections) . Let U : I → J be a functor into a finitely completeposet J . If K is a left 0-finite category and α : K → I is a functor, write U α : = lim k : K U α k ∈ U ( X )for the limit of U in U ( X ) over the diagram α . Since U ( X ) is a poset, thislimit can be computed as an intersection U α = \ k : K U α k where K ⊆ K is any finite 0-coinitial subset. If in particular K admits aninitial object e , then of course U α = U α ( e ) . finite-limit/example/simplex-boundary (Simplex boundary) . The category of simplices of the simplexboundary ∫ ( + ) ∂ ∆ n ( + ) (see §4 for notation) has a 0-coinitial subset { ∆ n − σ j , → ∆ n } nj = comprising the facets of ∆ n . In particular, R ( + ) ∂ ∆ n ( + ) is left 0-finite.For any functor U : I → J and map τ : ∫ ( + ) ∂ ∆ n ( + ) → I , we calculate U τ = n \ j = U τσ j .Moving on to the case of cones on a category with initial object: util/cone/initial-object/adjunction (Evaluation at initial object is left adjoint to diagonal) . Let K be a cat-egory with initial object e . Evaluation on e is a left adjoint to the constantfunctor const : I → Fun( K , I ), i i with counit induced by applying Fun( − , I ) to[ e → id K ] : ∆ × K → K K , I ) → Fun( ∆ × K , I ) = Fun( K , I ) ∆ that transformsconst ◦ ev e into the identity.In particular, from the mapping space formula of the adjunction:Fun( K , I )( i , φ ) ∼= I ( i , φ ( e ))for any φ : K → I . (It is natural to identify the left-hand side with the cate-gory of cones over φ with vertex i [Mac13, p. 67].) util/cone/initial-object/pullback Let K : Cat have an initial object e. The square Fun( K , i ↓ I ) Fun( K , I ) i ↓ I I ← →←→ ev( e ) ←→ ev( e ) ← → is a pullback in Cat .Proof. The top-right element can be identified with the category of functors ψ : ∆ × K → I with a trivialisation ψ | ∼= i , as follows:Fun( K , i ↓ I ) ∼= Fun( K , { i } × I I ∆ ) ∼= { i } × Fun( K , I ) Fun( K , I ∆ ) ∼= { i } × Fun( K , I ) Fun( ∆ × K , I ) ∼= { i } × Fun( K , I ) Fun( K , I ) ∆ .With this identification, the horizontal arrow is the target projection and thevertical is evaluation on e .Now we observe that both horizontal arrows are left fibrations, so it suf-fices to check that for each φ : K → I evaluation at ∆ × { e } induces a bijection ½ ψ : ∆ × K → I ψ | ∼= i , ψ | ∼= φ ¾ ∼= I ( i , φ ( e )).This follows from the adjunction ev( e ) ⊣ const (2.8). atlas If we restrict attention from arbitrary ∞ -categories to posets, unsurprisinglywe find some substantial simplifications to the general theory.7 tlas/of-locale (Diagram of opens sets) . Let X be a locale with frame of open sets U ( X )[Joh82]. A diagram of open subsets of X is a monotone map of posets U : I → U ( X ). We often abbreviate these data as a tuple ( I , U ). We write U i ⊆ X for the open set associated to i : I by U . atlas/of-locale/criterion (Atlas for a locale) . Let U : I → U ( X ) be a diagram of opensubsets of X indexed by a poset I. The following conditions are equivalent:1. X = S i : I U i , and for any i , j : I, U i ∩ U j = S k ≤ i , j U k .2. For any finite J ⊆ I, a i : Ii ≤ j ∀ j : J U i ։ \ j : J U j is a covering.3. (The nerve of) U is an atlas.Proof. ⇔ J .The criterion 2 is a special case of the definition of atlas; hence 3 ⇒ U : I → U ( X ) satisfies 2, and let K → I be a finitediagram with K some ∞ -category. Then U ( X ) / α = U ( X ) / h α because U ( X ) is 0-truncated and the formation K K ⊳ of the left cone com-mutes with truncation (2.2). So, replacing K with its truncation h K , wemay assume that it is a finite poset. But then also U ( X ) / α = U ( X ) / α | K where K is K regarded as a poset with the trivial ordering. The atlas con-dition is now handled by 2.The language of atlases is more flexible than that of hypercovers: manyhypercovers of interest are obtained by conversion from a naturally arisingdiagram which may itself already be an atlas. If the reader prefers, he may instead let X be a topological space without affecting thearguments. Descent theory is in any case mediated through the associated locale. tlas/example/basic Let X = U ∪ V be a topological space expressed as a union oftwo open sets. The diagram U ∩ V VU ← →←→ in U ( X ) is an atlas for X .Suppose now we have a further decomposition U ∩ V = A ∪ B with A ∩ B =; . Then the diagram AB UV ← → ←→← → ← → is an atlas for X . Notice that the index poset for this atlas has the weakhomotopy type of S . The reader can no doubt imagine a way to realise thisdiagram as an atlas of contractible open sets in the case X = S . (Basis) . A basis I ⊆ U ( X ) for the topology of X is an atlas for X : condition 2 of Proposition 3.2 follows easily from the definition of a basis. (Schemes) . A locally ringed space X is a scheme if and only ifthe inclusion U aff ( X ) → U ( X ) of the poset of open immersions from affineschemes is an atlas for X . Note that this poset is closed under finite in-tersections (resp. binary intersections) if and only if X is an affine scheme(resp. has affine diagonal). Thus, the Cech nerve construction is not alwaysavailable to us within this site. (Manifolds) . A paracompact Hausdorff space X is a topologicalmanifold if and only if the preorder U o ( X ) of open immersions R n , → X is anatlas for X . This preorder is never closed under binary intersections (unless X is a point). This example, or its C ∞ analogue, was what first motivatedme to formulate the notion of atlas.Atlases give us an easy way to describe certain counterexamples to hy-percompleteness: 9 .7 Example (Hilbert cube) . Consider the Hilbert cube Q = [0, 1] N as in [HTT,Ex. 6.5.4.8]. The set of open subsets homeomorphic to Q × [0, 1) forms a basefor the topology; in particular, it is an atlas. Borel-Moore homology definesa sheaf (by excision) whose restriction to this atlas is zero, but whose globalsections are nonzero; hence, by Theorem 1.4 it cannot be hypercomplete. sset Begin with some preliminary remarks on simplicial sets. (Simplex categories) . As usual, ∆ is the geometric simplex category ofinhabited totally ordered sets, while ∆ + is the subcategory consisting of in-jective maps. By [HTT, Lemma 6.5.3.7], the inclusion ∆ + → ∆ is ∞ -coinitial,i.e. for each n , ∆ + ↓ ∆ ∆ n is weakly contractible. sset/simplices (Simplices of a simplicial set) . A simplicial set F can be realised as a leftfibration in sets by integrating over ∆ op . We denote this fibration by ∫ F → ∆ op or, more briefly, ∫ F / ∆ op . It is called the category of simplices of F .Similarly, a semisimplicial set G can be realised as a left fibration R + G → ∆ op + . This is called the category of nondegenerate simplices of G .Let lke ∆ + / ∆ G be the simplicial envelope of G — that is, simplicial setobtained from G by left Kan extension along ∆ + ⊂ ∆ . As an extension itcomes equipped with a canonical functor R + G → R lke ∆ + / ∆ G . sset/example/simplex (Simplex) . We write ∆ n + for the semi-simplicial set representedby an ordered set with n + R + ∆ n + of nondegeneratesimplices is the opposite of the poset of inhabited subsets of [ n +
1] — inparticular, it is finite.The simplicial envelope of ∆ n + — the simplicial set represented by thesame ordered set considered as an object of ∆ — is written ∆ n . Its categoryof simplices R ∆ n is infinite, but taking the image of a map ∆ k → ∆ n definesa coreflector onto the finite subset R + ∆ n + of nondegenerate simplices. In par-ticular, it is left finite in that limits over R ∆ n are equivalent to limits over afinite category — cf. (2.6). 10 set/example/simplex-boundary (Simplex boundary) . Write ∂ ∆ n + for the semi-simplicial set rep-resenting the boundary of the n -simplex, and ∂ ∆ n for the associated simpli-cial set. The category ∫ + ∂ ∆ n + = ∫ + ∆ n \ { ∆ n id → ∆ n } of nondegenerate simplices of ∂ ∆ n is the poset of inhabited proper subsets of[ k + ∫ + ∆ n + = ¡ ∫ + ∂ ∆ n + ¢ ⊳ is a categorical left cone over the category of nondegenerate simplices of ∂ ∆ n + .Similarly, ∫ ∂ ∆ n = ∫ ∆ n × ∫ + ∆ n + ∫ + ∂ ∆ n + is the category of simplices equippedwith a non-surjective map to ∆ n . It is a sieve in ∫ ∆ n . It contains ∫ + ∂ ∆ n + as acoreflective subcategory and is therefore left finite. sset/example/mixed (Mixed category of simplices of a simplex) . The square R + ∂ ∆ n + R + ∆ n + R ∂ ∆ n R ∆ n ← →←→ ←→← → of fully faithful functors isn’t quite a pushout in Cat : rather, the pushoutis the full subcategory R ∂ ∆ n ∪ R + ∆ n + of R ∆ n spanned by simplices which areeither nondegenerate or factor through the boundary. This subcategory iscoreflective with coreflector fixing the boundary. Precisely, the value of thecoreflector on σ : ∆ k → ∆ n is either σ itself, if the image of σ is contained inthe boundary, or Im( σ ) otherwise. index-diagram Hypercoverings of X are defined in [SGA4, Exp. V, §7.3], [DHI04, Def. 4.2],[HAG-I, Def. 3.4.8] as certain simplicial objects in the category of presheaveson U ( X ). Some manipulation is required to convert these into diagrams in U ( X ) itself.In this section, we discuss the exchange of a semi-representable presheafwith a mapping indexing its connected components.11 emi-representable/definition (Semi-representable presheaves) . Let C be a poset. Re-call from [SGA4, Exp. V, 7.3] that a presheaf of sets on C is said to be semi-representable if it can be represented as a coproduct of representables.The full subcategory of the 1-category PSh ( C ) of presheaves of sets on C spanned by the semi-representable objects is denoted SR( C ). indexed-object/definition (Set indexed objects) . An S -indexed element of C , where S : Set , is a map S → C . We write C S = Fun( S , C )for the poset of S -indexed elements of C . Applying a contravariant Grothendieckconstruction, the 1-category of all set-indexed elements of C is defined Z S C S = Z S : Set C S idx C → Set as Cartesian fibration over
Set . Integrating the projection idx (which takesan indexed object to its indexing set ) yields a left (co-Cartesian) fibration f idx C : = Z R S C S idx C → Z S C S , (5.2.1) indexed-object/index-set/universalindexed-object/index-set/universal the universal index set of set-indexed objects of C . indexed-object/lke The functor lke − / pt : R S : Set
Fun( S , C ) → SR( C ) is constructed as follows:• An object X : S → C goes to the coproduct presheaf ` s : S X s : PSh ( C );• A morphism S S C ← → f ←→ X ⇒ ←→ X in R S : Set C S gets sent to the map a s : S X s ∼= a s : S (lke f X ) s ∼= a s : S a t : f − s X t → a s : S X s f ( X )of X along f .Compatibility with composition follows from uniqueness of the map from theleft Kan extension. indexed-object/lke/is-equivalence The groupoid of representations of a semi-representablepresheaf as a coproduct of representables is contractible. The natural functor lke − / pt : Z S : Set C S ˜ → SR( C ) from the 1-category of set-indexed objects of C to the 1-category of semi-representable presheaves on C is an equivalence of 1-categories.Proof. The functor lke − / pt is essentially surjective by the definition of SR( C ).Hence, we just have to show that it is fully faithful. If ( I , X ) and ( J , Y ) aretwo set-indexed objects, then maps from X to Y are defined as µZ S : Set C S ¶ (( I , X ), ( J , Y )) ∼= a f : J I Y i : I C ( X i , Y f ( i ) ).Now, the I projections ` f : J I Q i : I C ( X i , Y f ( i ) ) J I ` j : J C ( X i , Y j ) J ← →←→ ev i ←→ ev i ← → identify this further with Q i : I ` j : J C ( X i , Y j ). (This is an application of theGrothendieck construction’s commutation with external products; since theobjects here are sets, it can be seen by direct computation.)That lke is fully faithful is now visible from the identifications Y i a j C ( X i , Y j ) ∼= Y i SR( C ) Ã X i , a j Y j ! ∼= SR( C ) Ãa i : I X i , a j : J Y j ! with the first equality following because SR( C )( X , − ) commutes with coprod-ucts for representable X . 13 ndexed-object/tensoring (Tensoring over Set ) . It follows from Proposition 5.4 that the coprod-uct over any set ( I j , X j ) j : J of objects of R S : Set C S is exhibited by the obviousmorphisms ( I j , X j ) → Ãa j : J I j , a j : J X j ! .In particular, this defines a formula for a (coproduct-preserving) tensoringof R S : Set C S over Set .If C has a maximal element, as is the case for the example of interest C = U ( X ), then we can alternatively realise this tensoring as Cartesian productinside SR( C ) by embedding Set as the full subcategory spanned by formalcoproducts of this maximal element. This perspective will take priority whenconsidering the parametrised version below (6.2). semi-representable/local-isomorphism (Local isomorphisms) . Via totalization lke − / pt , a commuting triangle S S C ← → f ←→ X ←→ X induces a morphism of semi-representable presheaves. A morphism in SR( C )induced in this way is called a local isomorphism . In other words, local iso-morphisms are the morphisms which are Cartesian for the forgetful functoridx : SR( C ) → Set .The Cartesian property means that for any morphism F → F in SR( C )factorises uniquely into a composite of a map with fixed index — induced bya morphism (i.e. inequality) in Fun(idx( F ), C ) — followed by a local isomor-phism over idx( F ) → idx( F ). (Terminology) . The intuition behind this terminology is as fol-lows: a morphism φ : K → L in SR( C ) is called a local isomorphism if foreach connected component K ′ ⊆ K , the restriction of φ to K ′ exhibits it as aconnected component of L . This concept extends term-wise to s SR( C ).Beware that this notion of local isomorphism has nothing to do with anyauxiliary site structure that may exist on C (such as we have in the case C = U ( X )): it refers to locality only in the sense of formal coproducts inSR( C ). 14 Index diagrams in families
In this section we study simplicial objects in SR( C ) and, building on §5 ex-change them with a mappings from a left fibration over ∆ op . indexed-object/family (Families of set indexed objects) . The 1-category of set-indexed objects of C classifies diagramsI CK ← →←→ l − fib where I → K is a left fibration in sets. That is, for any 1-category K there isan equivalence of categories lke − / K : · CK ← →←→ l − fib ˜ −→ Fun µ K , Z S C S ¶ . (6.1.1) Proof.
Beginning from the alias Z S C S = Set ↓ Cat { C } we find Fun ¡ K , R S C S ¢ ∼= Set K Cat ⇓ ← - → ← → ← → C ∼= · K × CK ← → ←→ l − fib ← → pr K via R K ∼= · CK ← →←→ l − fib where the expressions in braces stand for categories of diagrams of the spec-ified shape with restrictions indicated by the labels, that is:15 named objects ( K , C , . . .) and arrows ( C ,pr K ) are equipped with identi-fications with their eponym;• arrows marked l − fib are constrained to the category of left fibrationsin sets. indexed-object/family/tensoring (Tensoring over Fun( K , Set )) . Let K : Cat . Taking the tensoring ofSR( C ) over Set (5.5) pointwise, Fun( K , SR( C )) is tensored over Fun( K , Set ).Passing through the equivalence of Proposition 6.1, we find the formula F ⊗ E CK ← →←→ = R K F × K E CK ← →←→ (6.2.1) indexed-object/family/tensoring/formulaindexed-object/family/tensoring/formula for tensoring over F : Fun( K , Set ). This is easiest to see by returning to ourassumption (5.5) that C has a greatest element so that the tensoring can becalculated as a Cartesian product in the diagram category. semi-representable/index-diagram (Simplicial semi-representable presheaves) . Combining Propositions 5.4and 6.1, a simplicial object U • of SR( C ) yields a simplicial object of R S C S andhence a diagram idx U f idx C C ∆ op ∫ S C S ←→ ← → ← → y ←→ ← →← → U where idx U → ∆ op is a left fibration. The category idx U = R ∆ op (idx ◦ U ) is calledthe index category or category of indices of the object U • , and the data item U :idx U → C is called the index diagram of U • . lifting-property/for-sset (Lifting property for simplicial objects) . Let K • ⊆ L • be an inclusion ofsimplicial sets. A lifting for the diagram ∫ K ∫ J C ∫ L ∆ op ← → σ ←→ ←→ ← → U ← → ← → (6.4.1)16s, removing the integrals throughout, equivalent to the data of an extensionlke ∫ K / ∆ op ( U ◦ σ ) lke ∫ J / ∆ op U · ← → σ ←→ ← → local-iso (6.4.2) lifting-property/dhi/local-isolifting-property/dhi/local-iso where the blank is something with index category ∫ ∆ n / ∆ op , the verticalarrow covers the inclusion ∂ ∆ n ⊂ ∆ n , and, as indicated, the extension is re-quired to be a local isomorphism in the sense of 5.6. (Note that this impliesthat the vertical arrow is also a local isomorphism.) Hence, remaining in thecategory of left fibrations handles the commutativity of the lower triangle inlifts of diagrams like (7.1.1). (Representable objects) . Let V : C . The index diagram of the(simplicially constant) simplicial presheaf on C represented by V is ∆ op C ∆ op ← → V ⇐⇐ semi-representable/tensor-representable (Tensor-representable objects) . Let K • be a simplicial set and V : C . Theindex diagram of V × K • , where this is computed with respect to the tensoringof SR( C ) over s Set , has the form ∫ K C ∆ op ← → V ←→ If U : ∫ K → C is another diagram, then morphisms K • ⊗ V → U are the sameas cones over U with vertex V . In particular, if U admits a limit in C , thenit admits a final object in the category of maps from tensor-representableobjects. lifting-property Throughout this section and for the rest of the paper, X will denote a localewith lattice of open subsets U ( X ). 17 ifting-property/definition (Local lifting conditions) . A local lifting problem on X is adiagram in Cat of the form K I U ( X ) L J ← → σ ←→ ←→ ← → U ← → (7.1.1) lifting-property/diagramlifting-property/diagram where, in the examples of interest:• K → L is a fully faithful functor; especially, but not always, equivalentto K ⊂ K ⊳ .• Either J = ∆ op or ∆ op + , in which case the maps from K , L , and I arerequired to be left fibrations, or J = pt .We say that this problem has a solution , or that the square has the locallifting property (local l.p.), if U σ is covered by sets of the form U τ , where τ ranges over completionss K IL J ← → σ ←→ ←→← → ← → τ of the square.If K ⊆ L is a functor of 1-categories and U : I / J → U ( X ) is a diagram,and moreover any local lifting problem (7.1.1) has a solution, we say that K / L has the local left lifting property or local l.l.p. against ( I / J , U ), or that( I / J , U ) has the local right lifting property or local r.l.p. against K / L . lifting-property/closure-properties Let U : I / J → U ( X ) be a diagram of open sets. The setof morphisms K / L having the local l.l.p. against ( I , U ) has the followingclosure properties:1. It is stable under composition, pushout, and retract.2. Suppose that J = pt , L → L ′ under K , and K / L ′ has the local l.l.p. against ( I , U ) . Then K / L has the local l.l.p. against ( I , U ) . roof. The argument for 1 is the same as for any class of arrows defined bya right lifting property. For 2, any lifting problem (7.1.1) can be factorised
K K IL L ′ ⇐ ⇐←→ ← →←→← → and so a lift for K / L ′ yields a lift for K / L . lifting-property/reduce-to-coinitial (Local lifting property reduces to a coinitial subset) . Let K bea finite poset, K ⊆ K a 0-coinitial subset. Let U : I → U ( X ) be a poset-indexeddiagram. If K ⊂ K ⊳ has the local l.l.p. against U , then so does K ⊂ K ⊳ .Proof. By Lemma 2.4, K / K ⊳ is a pushout of K / K ⊳ , and hence inherits thelocal l.l.p. by Proposition 7.2-1. lifting-property/sset-to-ssset (Local lifting properties for simplices versus semi-simplices) . The local r.l.p. for R + ∂ ∆ n + / R + ∆ n + and for R ∂ ∆ n / R ∆ are equivalent.Proof. Suppose U has the local r.l.p. against R + ∂ ∆ n + / R + ∆ n + . By Proposition7.2-1, it has the local r.l.p. against R ∂ ∆ n / R + ∆ n + ∪ R ∂ ∆ n . By Example 4.5, thisis a retract of R ∂ ∆ n / R ∆ fixing the boundary, and so by Proposition 7.2-2 weare done.Conversely, ∫ ∂ ∆ n + / ∫ ∆ n + is a retract of ∫ ∂ ∆ n / ∫ ∆ n . lifting-property/example/atlas The condition 3.2-2 for a diagram ( I , U ) to be an atlas is thelocal r.l.p. for inclusions K / K ⊳ where K is a finite set. That is, these mor-phisms have the local l.l.p. with respect to any atlas (considered over J = pt ).In particular, the covering condition is the local r.l.p. against R + ∂ ∆ / R + ∆ ,and condition 1 is the local r.l.p. for R + ∂ ∆ / R + ∆ . lifting-property/atlas (Atlases via lifting properties) . The condition for a diagramU : I → U ( X ) to be an atlas is equivalent to any of the following local rightlifting properties:1. K / K ⊳ for K = ; and K = {
0, 1 } .2. K / K ⊳ for all finite sets K .3. R + ∂ ∆ n + / R + ∆ n + for n : {
0, 1 } . . R + ∂ ∆ n + / R + ∆ n + for all n : N .5. R ∂ ∆ n / R ∆ n for n : {
0, 1 } .6. R ∂ ∆ n / R ∆ n for all n : N .Proof. As already observed (cf. Ex. 7.5), 1 and 2 correspond exactly to condi-tions 1 and 2 of 3.2, respectively.The implication 2 ⇒ ⇒ ⇔ ⇔ ⇔ hypercover Hypercovers are defined in terms of local lifting conditions in the category ofsimplicial presheaves [DHI04, §3]. Via the construction of 6.3, these trans-late almost directly into local lifting conditions for diagrams, i.e. in the senseof Definition 7.1. The key difference is that the lifting conditions of op. cit . re-strict attention to tensor-representable objects (6.6), while our filling condi-tions are restricted to local isomorphisms (5.6). In this section, we defineand compare these variants.Denote by SR( X ) the category of semi-representable presheaves on U ( X ),and by s SR( X ) its category of simplicial objects. hypercover/fill-condition The local lifting property for a diagram R ∂ ∆ n ∫ K U ( X ) R ∆ n ∆ op ←→ ← → σ ←→ χ ← → U ← → (8.1.1) lifting-property/simplexlifting-property/simplex is equivalent to the square ∂ ∆ n ⊗ U σ lke ∫ K / ∆ op U ∆ n ⊗ U σ pt ←→ ← → σ ←→← → (8.1.2) lifting-property/dhilifting-property/dhi dmitting local liftings in the sense of [DHI04, §3], where as above U σ denotesthe limit of U ◦ σ .In particular, an object U • : SR( X ) is a hypercover if and only if its indexdiagram has the local r.l.p. against ∫ ∂ ∆ n / ∫ ∆ n / ∆ op for all n : N .Proof. We will use the same strategy for representing categories of diagramsas in the proof of Proposition 6.1. Moreover, we consider (8.1.1) in the cat-egory of fibrations over ∆ op as in 6.4; hence we may disregard the lowertriangle when constructing fillers for (8.1.1), (8.1.2).By Lemma 8.2, the fibre of ∂ ∆ n ⊗ V lke ∫ K / ∆ op U ∆ n ⊗ V lke ∫ K / ∆ op U ←→ ← → ⇐⇐← → idx → ∫ ∂ ∆ n ∫ K ∫ ∆ n ∫ K ←→ ← → ∂τ ⇐⇐← → τ ,where τ is a variable, is a truth value equal to ½ V ≤ U ◦ ∂τ in Fun( ∫ ∂ ∆ n , U ( X )) V ≤ U ◦ τ in Fun( ∫ ∆ n , U ( X )) ¾ .If true, the inequalities factorise the above as: ∂ ∆ n ⊗ V lke ∫ ∂ ∆ n / ∆ op ( U ◦ ∂τ ) lke ∫ K / ∆ op U ∆ n ⊗ V lke ∫ ∆ n / ∆ op ( U ◦ τ ) lke ∫ K / ∆ op U . ←→ ← → fixed-idx ← → local-iso ←→ ⇐⇐← → fixed-idx ← → local-iso Now, specialising to the case ∂τ = σ , we obtain an identification ∂ ∆ n ⊗ V lke ∫ K / ∆ op U ∆ n ⊗ V ←→ ← → σ ← → ∼= lke ∫ ∆ n / ∆ op ( U ◦ σ ) lke ∫ K / ∆ op U lke ∫ ∆ n / ∆ op ( U ◦ τ ) ← →←→ ← → local-iso | V ⊆ U τ V for which a dia-gram of the left-hand form exists covers U σ ; therefore this set indexes a setof diagrams of the right-hand form whose U τ cover U σ , which is the locall.p. for (8.1.1). Conversely, the local l.p. for (8.1.1) implies liftings for (8.1.2)for V = U τ covering U σ . util/fibration/power (Powers of Cartesian fibrations) . Let f : C → D be a Cartesianfibration in
Cat . Then the exponential f ◦ − : C ∆ → D ∆ is a Cartesianfibration, with Cartesian morphisms u : A → B those of the formA B A B ← → u ←→ ←→← → u with u , u both f -Cartesian.Proof. Assume u : A → B is Cartesian on each component. Given a map u ′ : A ′ → B and a factorisation f ( A ′ ) → f ( A ) f ( u ) → f ( B ) of f ( u ′ ), the Cartesianproperty for u and u gives us a unique factorisation A ′ A B A ′ A B ← →←→ ← → u ←→ (cid:8) ←→← → ? ← → u whereupon it will suffice to prove that the left-hand square is commutative.This follows from the fact that either path around the square is a factori-sation of A ′ → B by u lying over f ( A ′ ) → f ( A ) → f ( B ), and hence uniqueas such by the Cartesian property of u . nerve To generate a hypercover — a kind of simplicial diagram — from an atlas —which may be indexed by any category — we need a kind of ‘nerve’ construc-tion that takes in an arbitrary diagram of open sets and outputs a diagramindexed by (the category of simplices of) a simplicial set, and which trans-forms the atlas condition into the hypercover condition.22 .1 Remark (Why not just use the standard nerve?) . It is easy enough tosee that the standard nerve construction [HTT, p. 9] won’t work. Let I be acategory, and let f : {
0, 1 } → I be a functor. This functor admits an extensionto [0 →
1] if and only if f (0) → f (1). The most basic examples of atlasesviolate this (e.g. 3.3).We need to associate a more flexible family of categories to the standardsimplices ∆ k . As we will see, the tautological test functor ˜ N provided byGrothendieck’s theory of test categories [Mal05] is good enough. nerve/of-category/definition (Tautological nerve) . If regarded as a functor from s Set into the full sub-category
Cat × Gpd
Set of Cat spanned by the categories whose objects haveno non-identity automorphisms, Grothendieck integration admits a right ad-joint ˜ N : Cat × Gpd
Set → s Set , K Hom( ∫ ( − ), K ), (9.2.1) nerve/formulanerve/formula which is (opposite to) the tautological test functor attached to the Grothendiecktest category ∆ [Mal05, Def. 1.3.7] and [Mal05, Prop. 1.5.13]. nerve/refinement (Tautological refinement) . Applying the nerve and then integrating againwe obtain a tautological refinement : ǫ J : ∫ ˜ N ( J ) → J (9.3.1) nerve/counitnerve/counit where the map ǫ J evaluates a simplex ∫ ∆ n → J at its initial object. Thecomposite functor ∫ ◦ ˜ N is a right adjoint to the forgetful functor from thecategory of left fibrations over ∆ op ∫ ˜ N : Cat × Gpd
Set ⇄ LFib( ∆ op ) : forget (9.3.2) test/adjunction/varianttest/adjunction/variantnerve/refinement/slice (Slices of the refinement) . The square R ˜ N ( i ↓ I ) R ˜ N ( I ) i ↓ I I ← →←→ ǫ ←→ ǫ ← → is a pullback in Cat . Hence, the slice category i ↓ I R ˜ N ( I ) of the refinementmay be identified with the refinement R ˜ N ( i ↓ I ) of the slice. roof. We may check this is a pullback by factorising R ˜ N ( i ↓ I ) R ˜ N ( I ) i ↓ I × ∆ op I × ∆ op i ↓ I I ← →←→ ←→← →←→ y ←→← → and observing that the upper square is a commuting square of co-Cartesianfibrations over ∆ op , whence the pullback property can be checked fibre-wise.Hence, restricting to ∆ n : ∆ op , we get a squareFun( ∫ ∆ n , i ↓ I ) Fun( ∫ ∆ n , I ) i ↓ I I ← →←→ ←→← → to which we apply Lemma 2.9. nerve/counit/is-cofinal
The tautological refinement ǫ ∆ is ∞ -cofinal.Proof. Let i : I . We must show that i ↓ I ∫ ˜ N ( I ) is weakly contractible [HTT,Thm. 4.1.3.1]. Using Lemma 9.4 we identify this with ∫ ˜ N ( i ↓ I ). The latteris weakly contractible because i ↓ I is weakly contractible and ˜ N ⊣ ∫ induceinverse equivalences of homotopy categories (as ∆ is a test category [Mal05,Prop. 1.6.14]). nerve/of-diagram/definition (Refinement of a diagram) . Let U : I → U ( X ) be a diagram. Precompos-ing U with the counit ǫ , we obtain an indexed diagram R ˜ N ( I ) U ( X ) ∆ op ← → U ◦ ǫ ←→ (9.6.1)Denote the associated simplicial semi-representable presheaf by˜ N ( U ) • : = lke ∫ ˜ N ( I )/ ∆ op ( U ǫ ) (9.6.2)24 erve/fill-condition Let K → L be a morphism in s
Set . A diagram U : I → U ( X ) of open sets has the local r.l.p. against ∫ K / ∫ L / ∆ op if and only if ( ∫ ˜ N ( I ), U ǫ ) has the local r.l.p. against ∫ K / ∫ L.Proof.
The adjunction (9.3.2) applied to the arrow category (cf. Lemma 8.2)identifies the sets ∫ K ∫ ˜ N ( I ) ∫ L ∆ op ← →←→ ←→← → ← → ∼= ∫ K I ∫ L pt ← →←→ ←→← → ← → as well as the vertex ( U ◦ ǫ ) τ = lim( U ◦ ǫ ◦ τ ) = U ǫ ◦ τ for any lift τ of either side. nerve/theorem Let U : I → U ( X ) be a diagram. The following are equivalent:1. U is an atlas.2. U has the local r.l.p. for R ∂ ∆ n / R ∆ n , n : N .3. ˜ N ( U ) • is a hypercover.In particular, every atlas restricts along an ∞ -cofinal functor to the indexdiagram of some hypercover.Proof. The equivalence 1 ⇔ ⇔ U satisfies the local filling conditions for R ∂ ∆ n / R ∆ n . ⇔ ∫ ˜ N ( U ) satisfies the local filling conditions for ∫ ∂ ∆ n / ∫ ∆ n / ∆ op (Lemma9.7). ⇔ ˜ N ( U ) is a hypercover (Proposition 8.1).25 eferences [DHI04] Daniel Dugger, Sharon Hollander, and Daniel C Isaksen. “Hyper-covers and simplicial presheaves”. In: Mathematical Proceedingsof the Cambridge Philosophical Society . Vol. 136. 1. CambridgeUniversity Press. 2004, pp. 9–51. arXiv: math/0205027 [math.AT] .[Gra13] Daniel R Grayson. “Quillen’s work in algebraic K-theory”. In:
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