title: An alternative to hypercovers
authors: Andrew W. Macpherson
text: 
  Fingerprint Feature Extraction by Combining Texture, Minutiae, and Frequency Spectrum Using Multi-Task CNN
      Ai Takahashi^1, Yoshinori Koda^1,2, Koichi Ito^1, and Takafumi Aoki^1
  
  1 Graduate School of Information Sciences, Tohoku University, Japan.
  
  2 Biometrics Research Laboratories, NEC Corporation, Japan.
  
  ai@aoki.ecei.tohoku.ac.jp
  
      December 30, 2023
  ========================================================================================================================================================================================================================================
  
  
  
    I introduce a class of diagrams in a Grothendieck site called atlases which can be used to study hyperdescent, and show that hypersheaves take atlases to limits using an indexed `nerve' construction that produces hypercovers from atlases. Atlases have the flexibility to be at the same time more explicit and more universal than hypercovers.
  
  
  
  
  
  
  
  § INTRODUCTION
  
  
  A presheaf F on a topological space X is said to satisfy descent along an open cover {U_i⊆ V}_i:I of an open set V⊆ X if its sections over V can be computed by taking the limit over the Čech nerve
  
  
      ⋯[r][r, shift left=1ex][r, shift right=1ex]    ( ∐_i:IU_i ) ×_X ( ∐_i:IU_i ) 
              [l, shift left=.5ex] [l, shift right=.5ex]    
              [r, shift left = .5ex] [r, shift right = .5ex]    ∐_i:IU_i  [l]
        
        ( ⟶ X)
  
  
  (for sheaves of sets, the first two terms, which appear here explicitly, are enough). It is called a sheaf if it satisfies descent along all open covers. This perspective, popularised in <cit.>, is the starting point for topos theory and the modern approach to numerous flavours of cohomology theory.
  
  
  For computing global sections  — i.e. cohomology — it is useful to be able to 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 ^o(X) of open sets diffeomorphic to ^n. Since these are exactly the contractible open sets, the classifying space of this poset has the same homotopy type as X. Hence, this site is useful for computing homotopy invariants of X, such as the theory of local systems on X.
  
  
  Because contractible subsets of X are not closed under intersection, the Čech nerve construction is not available for every cover in the site ^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 in which a binary intersection of open subsets can be resolved with its own open cover, and again for the intersections of  those open sets, and so on ad infinitum. This is the intuition that the hypercovers of <cit.> attempt to capture.
  
  Today, the theory of hypercovers is woven into the fabric of nearly all flavours of homotopical sheaf theory (with the bulk of <cit.> being a notable exception). They admit an elegant characterisation in terms of the model structure on simplicial presheaves <cit.>. 
  
  In practice, we must often construct hypercovers by conversion from more basic `naturally occurring' data. This process destroys finiteness and, perhaps, intuitiveness. Why not try instead to capture these `naturally occuring' data directly?
  
  D. Quillen is supposed to have said that only when he “freed [himself] from the shackles of the simplicial way of thinking” was he able to discover his Q-construction in algebraic K-theory <cit.>.
  The notion of atlas I now define arose in the search for a similar liberty in the context of descent theory:
  
  
  
  
    Let T be an ∞-category equipped with a Grothendieck topology in the sense of <cit.>, 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 and monotone map α:K→ I, the set
    
      {U_α̃→lim_k:KU_α(k)}_α̃:I_/α
   
    is a covering in (T), where I_/α is the overcategory of <cit.> (also known as the category of left cones over K→ I), and U_α̃ stands for the value of U∘α̃:K^◃→ T_/X on the cone point.
  
  
  
  
  
  
    An ∞-presheaf F:T^→ is said 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:IF(U_i).
  
    (Here, of course, the limit in the ∞-category  of spaces must be understood in the sense of ∞-category theory <cit.>.)
    
  
  
  The relevance of this definition is captured by the following statement:
  
  
  
  
    The following conditions on an ∞-presheaf F:T^→ are equivalent:
    
    
    
    
    * F satisfies descent along atlases.
    
    
    * F satisfies hyperdescent.
    
    
    
    In particular, the full subcategory of _∞(X) spanned by the functors that satisfy descent along atlases is a hypercomplete topos.
  
  
  
  The main application I have for this theory of atlases is a formulation of the universal property of (derived) geometry, which appears in a separate work <cit.>.
  
  In the present paper, I prove a truncated version of Conjecture <ref> that applies to topological spaces — more precisely, to locales — and is sufficient for the application of op. cit..
  
  
  
  
    Let X be a locale with lattice of open sets (X), and let F:(X)^→ be a hypercomplete ∞-sheaf.[More accurately, F is a sheaf on the nerve of the poset (X).] Then F satisfies descent along atlases indexed by posets.
    
  
  
  
  
    It is well-known that hypercompleteness means that F takes hypercovers to limits. See <cit.> (below Definition 3.4.8) for a proof in the context of simplicially enriched categories; to translate into the language of quasi-categories, use  <cit.> and <cit.>.
  
    The data of a hypercover of V⊆ X can be equivalently formulated as a certain kind of diagram JΔ^→(V) indexed by the total space J of a left fibration over Δ^, called its index diagram (<ref>), which has a certain local lifting property (Prop. <ref>).
    
    The result now follows from a nerve construction which associates to each diagram U:I→(V) of open sets of V, indexed by a poset I, a diagram 
    
      (I) [r, "ϵ"] [d, "" description]     I 
      Δ^
  
    such that U∘ϵ:(I)Δ^→(V) has the local lifting properties of a hypercover if and only if U is an atlas of V (Theorem <ref>).
    
    Moreover, ϵ is ∞-cofinal in the sense of <cit.> (Proposition <ref>), whence a colimit over I can be computed on its restriction to (I) <cit.>.
    
    Thus if F is hypercomplete and U is an atlas, then F takes ((I), U∘ϵ) — and therefore also (I,U) — to a limit.
    
  
  
  
  
    
  Acknowledgement Thanks to Adeel Khan whose question exposed an error in an earlier draft of Definition <ref>.
  
  
  
  
  
  § PRELIMINARIES
  
  
  This section contains some background information about posets, 1-categories, and ∞-categories, and discussion of coinitiality and initial objects in these contexts.
  
  [∞-categories and n-categories]
  
  
    Partially ordered sets and classical (-enriched) categories are realised, via the nerve construction, as 0- and 1-categories inside the category of quasi-categories <cit.>.
    
    We do not make any linguistic distinction between a poset (resp. classical category) and its nerve. (This only affects discussions where 0- or 1-categories interact with ∞-categories, i.e. the proof of Proposition <ref>.)
    
    The ∞-category of n-categories, for n:⊔{∞}, is denoted _n.
  
  
  
  [0-Truncation]
  
  
    The inclusion _0→_∞ has a right adjoint h_0 (see <cit.>, although this reference does not provide details).
    
    A realisation of the right adjoint is as follows: given a quasicategory C, h_0C is the 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 that the formation of the left cone (as defined in <cit.>) commutes with h_0.
  
  
  
  
  
  
  
    A subset K_0⊆ K of a poset K is 0-coinitial if every element of K is bounded below by an element of K_0. 
    
    More generally, a functor u:K_0→ K between two categories is called 0-coinitial if the image of K_0 in h_0(K) is 0-coinitial. Equivalently, for every object k:K there exists a k_0:K_0 and a morphism uk_0→ k.
  
  
  
  
  
  
    Let K be a poset, K_0⊆ K a 0-coinitial subset. Then
    
      K_0 [r] [d]     K [d] 
      
            K_0^◃[r]     K^◃
  
    is a pushout in the category of posets.
    
  
  
  
    
    The statement for underlying sets is obvious, so we are just checking that all the order relations on K^◃ = K ⊔{e} factorise as strings of relations that lift to  K_0^◃ 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_0 is 0-coinitial there is some i'≤ j with i'∈ K_0, and the relation factorises as i= e ≤ i' ≤ j with e≤ i' a relation in K_0^◃.
    
  
  
  
  
  
    A 1-category K is said to be left 0-finite if any functor 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; in particular, when K admits an initial object.
    
  
  
  [Finite intersections]
  
    
    Let U:I→ J be a functor into a finitely complete poset J. 
    If K is a left 0-finite category and α:K→ I is a functor, write 
    
      U_αlim_k:K U_α k  ∈  (X)
  
    for the limit of U in (X) over the diagram α.
    
    Since (X) is a poset, this limit can be computed as an intersection
    
      U_α = ⋂_k:K_0U_α k
  
    where K_0⊆ K is any finite 0-coinitial subset. 
    If in particular K admits an initial object e, then of course U_α=U_α (e).
    
  
  
  
  [Simplex boundary]
  
  
    The category of simplices of the simplex boundary _(+)∂Δ_(+)^n (see <ref> for notation) has a 0-coinitial subset 
    
      {Δ^n-1σ_j↪Δ^n}_j=0^n
  
    comprising the facets of Δ^n.
    In particular, ∫_(+)∂Δ_(+)^n is left 0-finite.
    
    For any functor U:I→ J and map τ:_(+)∂Δ^n_(+)→ I, we calculate
    
      U_τ   =   ⋂_j=0^n U_τσ_j.
  
    
  
  
  
  Moving on to the case of cones on a category with initial object:
  
  [Evaluation at initial object is left adjoint to diagonal]
  
  
    Let K be a category with initial object e. Evaluation on e is a left adjoint to the constant functor 
    
      :I →(K,I),    i↦i
  
    with counit induced by applying (-,I) to 
    
      [e→𝕀_K]:Δ^1× K→ K
  
    resulting in a map (K,I)→(Δ^1× K,I)=(K,I)^Δ^1 that transforms ∘_e into the identity.
    
    In particular, from the mapping space formula of the adjunction:
    
      (K,I)(i, ϕ) ≅ I(i, ϕ(e))
  
    for any ϕ:K→ I. (It is natural to identify the left-hand side with the category of cones over ϕ with vertex i <cit.>.)
    
  
  
  
  
  
  
  
    Let K:_1 have an initial object e. The square
    
      (K,i↓ I) [r] [d, "(e)"']    (K,I) [d, "(e)"] 
      
            i↓ I [r]     I
  
    is a pullback in _1.
    
  
  
  
  
    The top-right element can be identified with the category of functors ψ:Δ^1× K → I with a trivialisation ψ|_0≅i, as follows:
    
      (K,i↓ I)    ≅(K,{i}×_II^Δ^1) 
         ≅{i}×_(K,I)(K,I^Δ^1) 
         ≅{i}×_(K,I)(Δ^1× K,I) 
         ≅{i}×_(K,I)(K,I)^Δ^1.
  
    With this identification, the horizontal arrow is the target projection and the vertical is evaluation on e.
    
    Now we observe that both horizontal arrows are left fibrations, so it suffices to check that for each ϕ:K→ I evaluation at Δ^1×{e} induces a bijection
    
      {[      ψ:Δ^1× K→ I; ψ|_0 ≅i, ψ|_1 ≅ϕ ]}≅ I(i,ϕ(e)).
  
    This follows from the adjunction (e)⊣ (<ref>).
    
  
  
  
  
  
  
  
  § ATLASES FOR LOCALES
  
  
  
  If we restrict attention from arbitrary ∞-categories to posets, unsurprisingly we find some substantial simplifications to the general theory. 
  
  [Diagram of opens sets]
  
  
    Let X be a locale with frame of open sets (X) <cit.>.[If the reader prefers, he may instead let X be a topological space without affecting the arguments. Descent theory is in any case mediated through the associated locale.] A diagram of open subsets of X is a monotone map of posets U:I→(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.
  
  
  
  
  
    
    Let U:I→(X) be a diagram of open subsets of X indexed by a poset I. The following conditions are equivalent:
    
      
      
    * X=⋃_i:IU_i, and for any i,j:I, U_i∩ U_j=⋃_k≤ i,jU_k.
      
      
    * For any finite J⊆ I,
      
      ∐_0pt2i:Ii≤ j ∀ j:JU_i 
              ↠⋂_j:JU_j
  
      is a covering.
      
      
    * (The nerve of) U is an atlas.
      
    
    
  
  
  
  
    1⇔ 2 by induction on the cardinality of J.
  
    The criterion 2 is a special case of the definition of atlas; hence 3⇒2. Conversely, suppose that U:I→(X) satisfies 2, and let K→ I be a finite diagram with K some ∞-category. Then
    
      (X)_/α = (X)_/h_0α
  
    because (X) is 0-truncated and the formation K↦ K^◃ of the left cone commutes with truncation (<ref>).
    
    So, replacing K with its truncation h_0K, we may assume that it is a finite poset.
    But then also
    
      (X)_/α = (X)_/α|_K_0
  
    where K_0 is K regarded as a poset with the trivial ordering. 
    
    The atlas condition is now handled by 2.
    
  
  
  The language of atlases is more flexible than that of hypercovers: many hypercovers of interest are obtained by conversion from a naturally arising diagram which may itself already be an atlas.
  
  
  
  
    Let X=U∪ V be a topological space expressed as a union of two open sets.
    The diagram
    
      U∩ V [r] [d]     V 
       U
  
    in (X) is an atlas for X.
    
    Suppose now we have a further decomposition U∩ V= A∪ B with A∩ B=∅. Then the diagram
    
      A [dr, bend left] [dd] 
      
            B [rr, crossing over] [dr, bend right]        U 
          V
  
    is an atlas for X. Notice that the index poset for this atlas has the weak homotopy type of S^1. The reader can no doubt imagine a way to realise this diagram as an atlas of contractible open sets in the case X=S^1.
  
  
  
  [Basis]
    
    A basis I⊆(X) for the topology of X is an atlas for X: condition 2 of Proposition <ref> follows easily from the definition of a basis.
  
  
  
  [Schemes]
  
    A locally ringed space X is a scheme if and only if the inclusion ^aff(X)→(X) of the poset of open immersions from affine schemes is an atlas for X.
    
    Note that this poset is closed under finite intersections (resp. binary intersections) if and only if X is an affine scheme (resp. has affine diagonal). Thus, the Cech nerve construction is not always available to us within this site.
    
  
  
  [Manifolds]
  
    A paracompact Hausdorff space X is a topological manifold if and only if the preorder ^o(X) of open immersions ^n↪ X is an atlas for X.
    
    This preorder is never closed under binary intersections (unless X is a point). This example, or its C^∞ analogue, was what first motivated me to formulate the notion of atlas.
    
  
  
  Atlases give us an easy way to describe certain counterexamples to hypercompleteness:
  
  [Hilbert cube]
  
    Consider the Hilbert cube Q=[0,1]^ as in <cit.>. 
    
    The set of open subsets homeomorphic to Q×[0,1) forms a base for the topology; in particular, it is an atlas. 
    
    Borel-Moore homology defines a sheaf (by excision) whose restriction to this atlas is zero, but whose global sections are nonzero; hence, by Theorem <ref> it cannot be hypercomplete.
  
  
  
  
  
  
  
  § SIMPLICES
  
  
  
  Begin with some preliminary remarks on simplicial sets.
  
  
  [Simplex categories]
  
    As usual, Δ is the geometric simplex category of inhabited totally ordered sets, while Δ_+ is the subcategory consisting of injective maps.
    
    By <cit.>, the inclusion Δ_+→Δ is ∞-coinitial, i.e. for each n, Δ_+↓_ΔΔ^n is weakly contractible.
    
  
  
  [Simplices of a simplicial set]
  
  
    A simplicial set F can be realised as a left fibration in sets by integrating over Δ^.
    We denote this fibration by 
    
      F →Δ^
  
    or, more briefly, FΔ^.
    
    It is called the category of simplices of F.
    
    Similarly, a semisimplicial set G can be realised as a left fibration ∫_+G →Δ_+^. This is called the category of nondegenerate simplices of G.  
    
    Let G be the simplicial envelope of G — that is, simplicial set obtained from G by left Kan extension along Δ_+⊂Δ. As an extension it comes equipped with a canonical functor ∫_+G→∫ G.
    
  
  
  
  [Simplex]
  
  
    We write Δ^n_+ for the semi-simplicial set represented by an ordered set with n+1 elements.
    The category ∫_+Δ_+^n of nondegenerate simplices is the opposite of the poset of inhabited subsets of [n+1] — in particular, it is finite. 
    
    The simplicial envelope of Δ_+^n — the simplicial set represented by the same ordered set considered as an object of Δ — is written Δ^n.
    
    Its category of simplices ∫Δ^n is infinite, but taking the image of a map Δ^k→Δ^n defines a coreflector onto the finite subset ∫_+Δ_+^n of nondegenerate simplices. In particular, it is left finite in that limits over ∫Δ^n are equivalent to limits over a finite category — cf. (<ref>).
    
  
  
  
  [Simplex boundary]
  
  
    Write ∂Δ^n_+ for the semi-simplicial set representing the boundary of the n-simplex, and ∂Δ^n for the associated simplicial set. 
    The category 
    
      _+∂Δ_+^n    =   _+Δ^n ∖{Δ^nid→Δ^n}
  
    of nondegenerate simplices of ∂Δ^n is the poset of inhabited proper subsets of [k+1].
    In particular, 
    
      _+Δ_+^n = (_+∂Δ_+^n)^◃
  
    is a categorical left cone over the category of nondegenerate simplices of ∂Δ_+^n.
    
    Similarly, ∂Δ^n=Δ^n×__+Δ^n_+_+∂Δ^n_+ is the category of simplices equipped with a non-surjective map to Δ^n. 
    
    It is a sieve in Δ^n.
    
    It contains _+∂Δ_+^n as a coreflective subcategory and is therefore left finite.   
  
  
  
  
  [Mixed category of simplices of a simplex]
  
  
    The square
    
      ∫_+∂Δ_+^n [r] [d]    ∫_+Δ_+^n [d] 
      
            ∫∂Δ^n [r]    ∫Δ^n
  
    of fully faithful functors isn't quite a pushout in : rather, the pushout is the full subcategory ∫∂Δ^n∪∫_+Δ_+^n of ∫Δ^n spanned by simplices which are either nondegenerate or factor through the boundary.
    
    This subcategory is coreflective with coreflector fixing the boundary.
    
    Precisely, the value of the coreflector on σ:Δ^k→Δ^n is either σ itself, if the image of σ is contained in the boundary, or Im(σ) otherwise.
    
  
  
  
  
  
  
  
  § INDEX DIAGRAMS
  
  
  
  Hypercoverings of X are defined in <cit.>, <cit.>, <cit.> as certain simplicial objects in the category of presheaves on (X). 
  
  Some manipulation is required to convert these into diagrams in (X) itself.
  
  In this section, we discuss the exchange of a semi-representable presheaf with a mapping indexing its connected components.
  
  
  
  
  
  
    The notion of hypercovering appearing in <cit.> is a bit different, as it takes values in a topos without making reference to any explicit site. 
    A presheaf is a hypercomplete sheaf if and only if it satisfies hyperdescent with respect to all hypercovers in the sheaf topos <cit.>. 
    For the present purposes, it is essential to be able to work with simplicial semi-representable presheaves over 𝒰(X).
    
  
  
  
  
  
  
  
  
    Let C be a poset. Recall from <cit.> 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 _1(C) of presheaves of sets on C spanned by the semi-representable objects is denoted (C).
  
  
  
  [Set indexed objects]
  
  
    An S-indexed element of C, where S:, is a map S→ C. 
    We write 
    
      C^S = (S,C)
  
    for the poset of S-indexed elements of C.
    
    Applying a contravariant Grothendieck construction, the 1-category of all set-indexed elements of C is defined 
    
      ∫_SC^S=∫_S:C^S   _C→
  
    as Cartesian fibration over .
    
    Integrating the projection  (which takes an indexed object to its indexing set) yields a left (co-Cartesian) fibration 
    
      _C ∫_∫_SC^S_C→∫_SC^S,
  
    the universal index set of set-indexed objects of C.
    
  
  
  
  
  
    The functor _-/:∫_S:(S,C) →(C) is constructed as follows:
    
      
    * An object X:S→ C goes to the coproduct presheaf ∐_s:SX_s:_1(C);
      
      
    * A morphism
          
      [column sep = small]
                  S_0 [rr, "f"] [dr, "X_0"']    [d, phantom, "⇒"]     S_1 [dl, "X_1"] 
          C
  
          in ∫_S:C^S gets sent to the map
          
      ∐_s:S_0X_0,s   ≅→∐_s:S_1(_fX_0)_s 
         ≅→∐_s:S_1∐_t:f^-1s X_0,t
         →∐_s:S_1X_1,s
  
          obtained from the universal property of the left Kan extension _f(X_0) of X_0 along f.
    
    
    
    Compatibility with composition follows from uniqueness of the map from the left Kan extension.
    
  
  
  
  
  
  
    The groupoid of representations of a semi-representable presheaf as a coproduct of representables is contractible.
    
    The natural functor 
    
      _-/ : ∫_S:C^S  →̃ (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.
  
  
  
  
  
  
    The functor _-/ is essentially surjective by the definition of (C). Hence, we just have to show that it is fully faithful. If (I,X) and (J,Y) are two set-indexed objects, then maps from X to Y are defined as 
    
      (∫_S:C^S)((I,X),(J,Y)) ≅∐_f:J^I∏_i:I C(X_i, Y_f(i)).
  
    Now, the #I projections
    
      ∐_f:J^I∏_i:IC(X_i, Y_f(i)) [r] [d, "ev_i"] 
              J^I [d, "ev_i"] 
      ∐_j:J C(X_i, Y_j) [r]     J
  
    identify this further with ∏_i:I∐_j:JC(X_i, Y_j).
    
    (This is an application of the Grothendieck construction's commutation with external products; since the objects here are sets, it can be seen by direct computation.)
    
    That  is fully faithful is now visible from the identifications
    
      ∏_i∐_jC(X_i, Y_j)
               ≅∏_i (C) (X_i,∐_jY_j) 
         ≅(C) (∐_i:IX_i, ∐_j:JY_j)
  
    with the first equality following because (C)(X,-) commutes with coproducts for representable X.
    
  
  
  [Tensoring over ]
  
  
    It follows from Proposition <ref> that the coproduct over any set (I_j,X_j)_j:J of objects of ∫_S:C^S is exhibited by the obvious morphisms
    
      (I_j, X_j) →(∐_j:JI_j, ∐_j:JX_j).
  
    In particular, this defines a formula for a (coproduct-preserving) tensoring of ∫_S:C^S over .
    
    If C has a maximal element, as is the case for the example of interest C=(X), then we can alternatively realise this tensoring as Cartesian product inside (C) by embedding  as the full subcategory spanned by formal coproducts of this maximal element.
    
    This perspective will take priority when considering the parametrised version below (<ref>).
    
  
  
  
  
  [Local isomorphisms]
  
  
    Via totalization _-/, a commuting triangle
    
      [column sep = tiny]
            S_0 [rr, "f"] [dr, "X_0"']        S_1 [dl, "X_1"] 
          C
  
    induces a morphism of semi-representable presheaves. A morphism in (C) induced in this way is called a local isomorphism. In other words, local isomorphisms are the morphisms which are Cartesian for the forgetful functor :(C)→.
    
    The Cartesian property means that for any morphism F_0→ F_1 in (C) factorises uniquely into a composite of a map with fixed index — induced by a morphism (i.e. inequality) in ((F_0), C) — followed by a local isomorphism over (F_0)→(F_1).
    
  
  
  
    
    
    The intuition behind this terminology is as follows: a morphism ϕ:K→ L in (C) is called a local isomorphism if for each connected component K'⊆ K, the restriction of ϕ to K' exhibits it as a connected component of L. This concept extends term-wise to s(C).
    
    Beware that this notion of local isomorphism has nothing to do with any auxiliary site structure that may exist on C (such as we have in the case C=(X)): it refers to locality only in the sense of formal coproducts in (C).
    
  
  
  
  
  
  
  
  § INDEX DIAGRAMS IN FAMILIES
  
  
  In this section we study simplicial objects in (C) and, building on <ref> exchange them with a mappings from a left fibration over Δ^.
  
  
  
    
    The 1-category of set-indexed objects of C classifies diagrams
    
      I [r] [d, "" description]     C 
       K
  
    where I→ K is a left fibration in sets. That is, for any 1-category K there is an equivalence of categories
    
      _-/K:
          {·[r] [d, "" description]     C  
      
              K
            }⟶̃(K,∫_SC^S).
  
  
  
  
    Beginning from the alias 
    
      ∫_SC^S = ↓__0{C}
  
    we find
    
      [                                                                                                 (K,∫_SC^S)                                                                                                          ≅                    {[dr, phantom, "⇓" very near end]    [d, hook] 
      
              K [ur] [r, "C"']    _0
            }                                                                                                           ;                                                                                                                                                                                                                     ≅ {[column sep = small]
              ·[rr] [dr, "" description]        
                K × C [dl, "_K"] 
          K
            }                                                                                                    via ∫_K;                                                                                                                                                                                                                     ≅                                                       {·[r] [d, "" description]     C  
      
              K
            } ]
  
    where the expressions in braces stand for categories of diagrams of the specified shape with restrictions indicated by the labels, that is:
     
      
    * named objects (K, C,…) and arrows (C,_K) are equipped with identifications with their eponym;
      
    * arrows marked  are constrained to the category of left fibrations in sets.
      
    
  
  
  [Tensoring over (K,)]
  
  
    Let K:_1. Taking the tensoring of (C) over  (<ref>) pointwise, (K, (C)) is tensored over (K,).
    
    Passing through the equivalence of Proposition <ref>, we find the formula
    
      F ⊗(
          
            E [r] [d]     C  
      
            K
          )
            
          =
            (
          ∫_K F×_K E [r] [d]     C  
      
            K
          )
  
    for tensoring over F:(K,). This is easiest to see by returning to our assumption (<ref>) that C has a greatest element so that the tensoring can be calculated as a Cartesian product in the diagram category.
  
  
  
  [Simplicial semi-representable presheaves]
  
  
    Combining Propositions <ref> and <ref>, a simplicial object U_∙ of (C) yields a simplicial object of ∫_SC^S and hence a diagram
    
      _U [d] [r] [rr, bend left]
              [dr, phantom, "⌟", very near start]    _C [d] [r]     C 
      Δ^[r, "U"]    _S C^S
  
    where _U→Δ^ is a left fibration.
    
    The category _U=∫_Δ^(∘ U) is called the index category or category of indices of the object U_∙, and the data _U:_U→ C is called the index diagram of U_∙.
    
  
  
  
  [Lifting property for simplicial objects]
  
  
    Let K_∙⊆ L_∙ be an inclusion of simplicial sets. A lifting for the diagram 
    
      K [r, "σ"] [d]     J [d] [r, "U"]     C 
       L [r] [ur, dotted]    Δ^
  
    is, removing the integrals throughout, equivalent to the data of an extension
    
      _ KΔ^(U∘σ) [r, "σ"] 
              [d]    _ JΔ^U 
      ·[ur, "" description, dotted]
  
    where the blank is something with index category Δ^nΔ^, the vertical arrow covers the inclusion ∂Δ^n⊂Δ^n, and, as indicated, the extension is required to be a local isomorphism in the sense of <ref>. (Note that this implies that the vertical arrow is also a local isomorphism.)
    
    Hence, remaining in the category of left fibrations handles the commutativity of the lower triangle in lifts of diagrams like (<ref>).
  
  
  
  [Representable objects]
  
    Let V:C. The index diagram of the (simplicially constant) simplicial presheaf on C represented by V is
    
      Δ^[r, "V"] [d, equals]     C 
      Δ^
  
    
  
  
  
  [Tensor-representable objects]
  
  
    Let K_∙ be a simplicial set and V:C. The index diagram of V× K_∙, where this is computed with respect to the tensoring of (C) over s, has the form
    
      K [r, "V"] [d]     C 
      Δ^
  
    If U: K→ C is another diagram, then morphisms K_∙⊗ V→ U are the same as cones over U with vertex V. In particular, if U admits a limit in C, then it admits a final object in the category of maps from tensor-representable objects.
    
  
  
  
  
  
  
  
  
  
  
  
  § LOCAL LIFTING CONDITIONS
  
  
  
  Throughout this section and for the rest of the paper, X will denote a locale with lattice of open subsets (X).
  
  
  
  
  
    A local lifting problem on X is a diagram in _1 of the form
    
      K [r, "σ"] [d]     I [d] [r, "U"]    (X) 
      
            L [r]     J
  
    where, in the examples of interest:
    
      
      
    * K→ L is a fully faithful functor; especially, but not always, equivalent to K⊂ K^◃.
      
      
    * Either J=Δ^ or Δ_+^, in which case the maps from K, L, and I are required to be left fibrations, or J=.
    
    
    
    We say that this problem has a solution, or that the square has the local lifting property (local l.p.), if U_σ is covered by sets of the form U_τ, where τ ranges over completionss
    
      K [r, "σ"] [d]     I [d] 
      
            L [r] [ur, dotted, "τ"]     J
  
    of the square.
    
    If K⊆ L is a functor of 1-categories and U:I J→(X) is a diagram, and moreover any local lifting problem (<ref>) 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.
    
  
    
  
  
  
    Let U:I J→(X) be a diagram of open sets. The set of morphisms K L having the local l.l.p. against (I,U) has the following closure properties:
  
    
    
      
    * It is stable under composition, pushout, and retract. 
      
      
    * Suppose that J=, 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).
      
    
  
  
  
  
  
    The argument for 1 is the same as for any class of arrows defined by a right lifting property. For 2, any lifting problem (<ref>) can be factorised
    
      K [r, equals] [d]     K [r] [d]     I 
      
            L [r]     L'
  
    and so a lift for K L' yields a lift for K L.
    
  
  
  
  
  
  
  
    Let K be a finite poset, K_0⊆ K a 0-coinitial subset. 
    
    Let U:I→(X) be a poset-indexed diagram. 
    
    If K_0⊂ K_0^◃ has the local l.l.p. against U, then so does K⊂ K^◃.
    
  
  
  
  
    By Lemma <ref>, K K^◃ is a pushout of K_0 K_0^◃, and hence inherits the local l.l.p. by Proposition <ref>-1. 
  
  
  
  
  
  
  
    The local r.l.p. for ∫_+∂Δ_+^n∫_+Δ_+^n and for ∫∂Δ^n∫Δ are equivalent.
    
  
  
  
  
    Suppose U has the local r.l.p. against ∫_+∂Δ_+^n∫_+Δ_+^n. By Proposition <ref>-1, it has the local r.l.p. against ∫∂Δ^n∫_+Δ^n_+ ∪∫∂Δ^n. By Example <ref>, this is a retract of ∫∂Δ^n∫Δ fixing the boundary, and so by Proposition <ref>-2 we are done.
  
    Conversely, ∂Δ_+^nΔ_+^n is a retract of ∂Δ^nΔ^n.
    
  
  
  
  
  
  
  
  
  
    The condition <ref>-2 for a diagram (I,U) to be an atlas is the local r.l.p. for inclusions K  K^◃ where K is a finite set.
    
    That is, these morphisms have the local l.l.p. with respect to any atlas (considered over J=).
    
    In particular, the covering condition is the local r.l.p. against ∫_+∂Δ^0 ∫_+Δ^0, and condition 1 is the local r.l.p. for ∫_+∂Δ^1 ∫_+Δ^1.
    
  
  
  
  
  
  
    The condition for a diagram U:I→(X) to be an atlas is equivalent to any of the following local right lifting properties:
    
    
      
    * K K^◃ for K=∅ and K={0,1}.
      
      
    * K K^◃ for all finite sets K.
      
      
    * ∫_+∂Δ_+^n∫_+Δ_+^n for n:{0,1}.
      
      
    * ∫_+∂Δ_+^n∫_+Δ_+^n for all n:.
      
      
    * ∫∂Δ^n∫Δ^n for n:{0,1}.
      
      
    * ∫∂Δ^n∫Δ^n for all n:.
      
    
    
  
  
  
  
    As already observed (cf. Ex. <ref>), 1 and 2 correspond exactly to conditions 1 and 2 of <ref>, respectively.
    
    The implication 2 ⇒ 4 is an application of Corollary <ref> and Example <ref>. Meanwhile 4 ⇒ 3 because 3 is a specialisation of 4, and 1 ⇔ 3 because these are equivalent categories.
  
    The equivalences 3 ⇔ 5 and 4 ⇔ 6 follow from Corollary <ref>.
    
  
  
  
  
  
  
  § HYPERCOVERS VIA LIFTING CONDITIONS
  
  
  
  Hypercovers are defined in terms of local lifting conditions in the category of simplicial presheaves <cit.>. Via the construction of <ref>, these translate almost directly into local lifting conditions for diagrams, i.e. in the sense of Definition <ref>.
  
  The key difference is that the lifting conditions of op. cit. restrict attention to tensor-representable objects (<ref>), while our filling conditions are restricted to local isomorphisms (<ref>). In this section, we define and compare these variants.
  
  Denote by (X) the category of semi-representable presheaves on (X), and by s(X) its category of simplicial objects. 
  
   
  
  
  
    The local lifting property for a diagram
    
      ∫∂Δ^n  [d] [r, "σ"]     K [d, "χ"] [r, "U"]    (X) 
      ∫Δ^n [r]    Δ^
    
    is equivalent to the square
    
      ∂Δ^n ⊗ U_σ[d] [r, "σ"]    _ KΔ^ U [d] 
      Δ^n ⊗ U_σ[r]
  
    admitting local liftings in the sense of <cit.>, where as above U_σ denotes the limit of U∘σ.
    
    In particular, an object U_∙:(X) is a hypercover if and only if its index diagram has the local r.l.p. against ∂Δ^nΔ^n Δ^ for all n:.
    
  
  
  
      
    We will use the same strategy for representing categories of diagrams as in the proof of Proposition <ref>. 
    
    Moreover, we consider (<ref>) in the category of fibrations over Δ^ as in <ref>; hence we may disregard the lower triangle when constructing fillers for (<ref>), (<ref>).
    
    By Lemma <ref>, the fibre of
    
      {∂Δ^n ⊗ V  [d] [r]    _ KΔ^ U [d, equals]  
      Δ^n ⊗ V [r]    _ KΔ^ U
            }  →  {∂Δ^n   [d] [r, "∂τ"]     K [d, equals]  
      Δ^n  [r, "τ"]     K
            },
      
    where τ is a variable, is a truth value equal to 
    
      {[              V              ≤           U∘∂τ in (∂Δ^n, (X));              V              ≤            U∘τ   in (Δ^n,(X)) ]}.
   
    If true, the inequalities factorise the above as: 
    
      [column sep = large]
            ∂Δ^n ⊗ V  [d] [r, ""]    _∂Δ^nΔ^
                (U∘∂τ) [r, ""] [d]    _ KΔ^ U [d, equals] 
      Δ^n ⊗ V [r, ""]    _Δ^nΔ^(U∘τ) 
                [r, ""]     _ KΔ^ U.
  
    Now, specialising to the case ∂τ = σ, we obtain an identification
    
    
      {∂Δ^n ⊗ V  [d] [r, "σ"]    _ KΔ^ U  
      Δ^n ⊗ V [ur]
            }
  
    
    
      ≅  {_Δ^nΔ^(U∘σ) 
                [r] [d]    _ KΔ^ U 
      _Δ^nΔ^ (U∘τ) 
                [ur, "" description]
            |  
            V⊆ U_τ}
  
    
    
    
    The local lifting property for (<ref>) states that the set of V for which a diagram of the left-hand form exists covers U_σ; therefore this set indexes a set of diagrams of the right-hand form whose U_τ cover U_σ, which is the local l.p. for (<ref>).
    
    Conversely, the local l.p. for (<ref>) implies liftings for (<ref>) for V=U_τ covering U_σ.
    
  
  
    
  
  
  
  
    Let f:C→ D be a Cartesian fibration in _1. Then the exponential f∘-:C^Δ^1→ D^Δ^1 is a Cartesian fibration, with Cartesian morphisms u:A→ B those of the form
    
      A_0 [r, "u_0"] [d]     B_0 [d] 
      
            A_1 [r, "u_1"]     B_1
  
    with u_0, u_1 both f-Cartesian.
    
  
  
  
  
    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 Cartesian property for u_0 and u_1 gives us a unique factorisation
    
      A_0' [r] [d]     A_0 [r, "u_0"] [d] [dr, phantom, "↻" description]     B_0 [d] 
      
            A_1' [r] [ur, phantom, "?" description]     A_1 [r, "u_1"]     B_1
      
    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 factorisation of A'_0→ B_1 by u_1 lying over f(A'_0) → f(A_1)→ f(B_1), and hence unique as such by the Cartesian property of u_1.
    
  
  
  
  
  
  
  
  
  
  
  
  
  § NERVES OF ATLASES
  
  
  
  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' construction that takes in an arbitrary diagram of open sets and outputs a diagram indexed by (the category of simplices of) a simplicial set, and which transforms the atlas condition into the hypercover condition. 
  
  
  
    It is easy enough to see that the standard nerve construction <cit.> won't work. Let I be a category, and let f:{0,1}→ I be a functor. This functor admits an extension to [0→ 1] if and only if f(0)→ f(1). The most basic examples of atlases violate this (e.g. <ref>).
  
  
  
  We need to associate a more flexible family of categories to the standard simplices Δ^k. As we will see, the tautological test functor  provided by Grothendieck's theory of test categories <cit.> is good enough.
  
  [Tautological nerve]
  
  
    If regarded as a functor from s into the full subcategory _1×_ of _1 spanned by the categories whose objects have no non-identity automorphisms, Grothendieck integration admits a right adjoint 
    
      : _1×_→ s,   
            K ↦((-),K),
  
    which is (opposite to) the tautological test functor attached to the Grothendieck test category Δ <cit.> and <cit.>. 
    
  
  
  [Tautological refinement]
  
    
    Applying the nerve and then integrating again we obtain a tautological refinement:
    
      ϵ_J:  (J) → J
  
    where the map ϵ_J evaluates a simplex Δ^n→ J at its initial object.
    
    The composite functor ∘ is a right adjoint to the forgetful functor from the category of left fibrations over Δ^
    
      : _1×_ ⇄ LFib(Δ^)  :
  
  
  
  
  
  
  
    
    The square
    
      ∫(i↓ I) [r]  [d, "ϵ"]    ∫(I) [d, "ϵ"] 
      
            i↓ I  [r]       I
  
    is a pullback in _1. Hence, the slice category i↓_I∫(I) of the refinement may be identified with the refinement ∫(i↓ I) of the slice.
  
  
  
  
  
    We may check this is a pullback by factorising
    
      ∫(i↓ I) [r]  [d]    ∫(I) [d] 
      
            i↓ I×Δ^[r] [d] 
              [dr, phantom, "⌟" very near start]      
              I ×Δ^[d] 
      
            i↓ I  [r]       I
  
    and observing that the upper square is a commuting square of co-Cartesian fibrations over Δ^, whence the pullback property can be checked fibre-wise. Hence, restricting to Δ^n:Δ^, we get a square
    
      (Δ^n, i↓ I) [r] [d]    (Δ^n, I) [d] 
      
            i↓ I [r]    
              I
  
    to which we apply Lemma <ref>.
    
  
  
  
  
  
    The tautological refinement ϵ_Δ is ∞-cofinal.
  
  
  
  
  
    Let i:I. We must show that i↓_I(I) is weakly contractible <cit.>.
    
    Using Lemma <ref> we identify this with (i↓ I).
    
    The latter is weakly contractible because i↓ I is weakly contractible and ⊣ induce inverse equivalences of homotopy categories (as Δ is a test category <cit.>).
    
  
  
  
  [Refinement of a diagram]
  
    
    Let U:I→(X) be a diagram. Precomposing U with the counit ϵ, we obtain an indexed diagram
    
      ∫(I) [r, "U∘ϵ"] [d]    (X) 
      Δ^
  
    Denote the associated simplicial semi-representable presheaf by 
    
      ( U)_∙_(I)Δ^(Uϵ)
  
  
  
  
  
  
  
    Let K→ L be a morphism in s.
    
    A diagram U:I→(X) of open sets has the local r.l.p. against K LΔ^ if and only if ((I), Uϵ) has the local r.l.p. against K L.
    
  
  
  
    The adjunction (<ref>) applied to the arrow category (cf. Lemma <ref>) identifies the sets
    
      { K [r] [d]    (I) [d] 
       L [r] [ur, dashed]    Δ^}  ≅  { K [r] [d]     I [d] 
       L [r] [ur, dashed]    }
  
    as well as the vertex 
    
      (U∘ϵ)_τ = lim(U∘ϵ∘τ) = 
            U_ϵ∘τ
  
    for any lift τ of either side. 
    
  
  
  
  
  
  
  
    Let U:I→(X) be a diagram. The following are equivalent: 
      
      
    * U is an atlas.
      
      
    * U has the local r.l.p. for ∫∂Δ^n∫Δ^n, n:.
      
      
    * (U)_∙ is a hypercover.
      
    
    In particular, every atlas restricts along an ∞-cofinal functor to the index diagram of some hypercover.
  
  
  
  
  
    The equivalence 1 ⇔ 2 is Proposition <ref>. Now, 2 ⇔ 3 by the chain of logical equivalences:
    
      
     U satisfies the local filling conditions for ∫∂Δ^n∫Δ^n.
      
      
    ⇔ (U) satisfies the local filling conditions for ∂Δ^nΔ^nΔ^ (Lemma <ref>).
      
      
    ⇔ (U) is a hypercover (Proposition <ref>). 
      
    
  
  
  