TOPOLOGICAL HYPERCOVERS AND 𝔸^{1}-REALIZATIONS

DANIEL DUGGER AND DANIEL C. ISAKSEN

Abstract. We show that if U_{∗} is a hypercover of a topological space X then the natural map hocolim U_{∗} → X is a weak equivalence. This fact is used to construct topological realization functors for the 𝔸^{1}-homotopy theory of schemes over real and complex fields. In an appendix, we also prove a theorem about computing homotopy colimits of spaces that are not cofibrant.

1. Introduction

Let X be a topological space, and let 𝒰 = {U_{a}} be an open cover of X. From this data one may build the Čech complex Č(𝒰)_{∗}, which is the simplicial space

\xymatrix{∐ U_{a_{0}} & ∐ U_{a_{0}a_{1}} \ar[l] \ar[l] & ∐ U_{a_{0}a_{1}a_{2}} ⋯ \ar[l] \ar[l] \ar[l]}

Here U_{a_{0}···a_{n}} = U_{a_{0}} ∩⋯∩U_{a_{n}}, and the face maps are obtained by omitting indices— we have chosen not to draw the degeneracies for typographical reasons. Segal [S1] proved that if X has a partition of unity subordinate to 𝒰 then the map |Č(𝒰)_{∗}| → X is a homotopy equivalence, where |−| denotes geometric realization. Our first goal in this paper is to generalize this result to the following theorem. Theorem 1.1. For every open cover 𝒰 of X, the map hocolim Č(𝒰)_{∗} → X is a weak equivalence (where the hocolim is taken over the simplicial indexing category).

There are two steps in the argument. First, we prove that |Č(𝒰)_{∗}| → X is a weak equivalence for arbitrary open covers. It is possible to deduce this from Segal’s result, making use of the fact that weak equivalences are detected by spheres, and spheres always have partitions of unity. But instead of going this route we give a proof that avoids Segal’s theorem completely, and is quite elementary.

The second step is to deal with the difference between |Č(𝒰)_{∗}| and hocolim Č(𝒰)_{∗}. For any simplicial object W_{∗} in a model category, there are general criteria for when its geometric realization agrees with its homotopy colimit (cf. [H, Th. 18.7.4]); unfortunately these criteria apply only when the objects W_{n} are all cofibrant, and we are definitely not assuming that the open sets U_{a} and their intersections are cofibrant. It turns out that this cofibrancy issue never matters, which is somewhat of a surprise. Here is one specific result which we will use later: Theorem 1.2. Let U_{∗} be a simplicial space that has free degeneracies (see Definition A.4), but do not assume that each U_{n} is cofibrant. Then the homotopy colimit of U_{∗} is weakly equivalent to the realization of U_{∗}.

1991 Mathematics Subject Classification. 55U35, 14F20, 14F42.

Key words and phrases. hypercover, homotopy colimit, geometric realization, motivic homotopy theory.

The second author was supported by an NSF Postdoctoral Research Fellowship.

This result is similar to [S2, A.5], but different in that it is about weak equivalences rather than homotopy equivalences. We in fact prove that when taking homotopy colimits for any diagram of topological spaces one doesn’t first have to make all the spaces involved cofibrant; the usual formulas are already homotopyinvariant. This is definitely a non-standard fact, but we’ve banished it to an appendix so it won’t distract the reader from the general theme of the paper. On the other hand it is a useful result, and we’d like to call attention to it.

The main goal of this paper is generalizing Theorem 1.1 so that it applies to ‘hypercovers’, rather than just Čech covers. These are defined in detail in Section 4, but for now we will just give an intuitive definition. An open hypercover of a space X is a simplicial space U_{∗} such that

(1) Each U_{n} is a disjoint union of open subsets of X,

(2) The spaces appearing in U_{0} are an open cover of X,

(3) The spaces in U_{1} cover the double intersections of those in level 0,

(4) The spaces in U_{2} cover the triple intersections of those in level 1, and so on. Of course making sense of (4)—especially the ‘and so on’ part—requires a certain amount of bookkeeping, which is why we are postponing the formal definition. But the essence is that hypercovers are like Čech complexes except that instead of taking the double intersections at level 1 we may refine them further, and we may continue this refining process at each level. Our second main result concerns homotopy colimits of hypercovers. Theorem 1.3. If U_{∗} is an open hypercover of a space X, then the natural map hocolim U_{∗} → X is a weak equivalence.

This result could almost be considered folklore since everyone immediately agrees it’s true, but a proof seems to be missing from the literature. One might consider tackling it by appealing to the Whitehead theorem, proving an isomorphism on fundamental groupoids and homology with local coefficients. This is the approach taken in [F, Prop. 8.1] in the related context of étale hypercovers, but this is messy and obscures in computation the underlying geometric explanation of the theorem. In the case of topological spaces, the isomorphism on fundamental groupoids was the subject of the paper [RT] (although they only dealt with Čech complexes, not hypercovers). The approach we take here, on the other hand, is very elementary. The idea is to reduce to the case of Čech covers in a clever way.

Our interest in these results arose from attempts to understand topological realization functors in the 𝔸^{1}-homotopy theory of schemes [MV]. Given an algebraic variety X defined over ℂ, there is an associated topological space X(ℂ) obtained by giving X the analytic topology. Of course this should extend to a map of ‘homotopy theories’ from the Morel-Voevodsky category Spc(ℂ) to the category of topological spaces. In [MV] this extension is only provided at the level of homotopy categories, but we are interested in extending it to the model category level.

The first thing to note is that Spc(ℂ) must be replaced by a Quillen-equivalent model, denoted Spc′(ℂ) in section 5. Once this is done, the key fact needed to make things work is precisely Theorem 1.3. This is worked out in detail in Section 5, following the basic program of [D2, Rem. 8.2] or [I]. We also prove that taking analytic spaces for schemes defined over ℝ induces a Quillen map from Spc′(ℝ) to ℤ_{2}-equivariant topological spaces:

Theorem 1.4. The functor X ↦ X(ℂ) extends to left Quillen functors of the form Spc′(ℂ) → 𝒯op and Spc′(ℝ) → ℤ_{2}−𝒯op.

Finally, we give in this paper several interesting corollaries to Theorem 1.3. On the whole these seem too disparate to recount in the introduction, but as an example let us mention two of them. We refer the reader to Sections 3 and 4 for more results like these. Corollary 1.5. Let E → B be any map which is locally split (for example, a covering space), and form the associated Čech complex Č(E)_{∗} given by

Č(E)_{n} := E_{B}^{n+1} = E ×_{B} E ×_{B} ⋯ ×_{B} E.

Then the natural map hocolim Č(E)_{∗} → B is a weak equivalence. Corollary 1.6. Let 𝒰 be an open cover of a space X with the property that every finite intersection U_{a_{0}···a_{n}} is covered by other elements of 𝒰. Form the diagram consisting of all the U_{a}’s and all the inclusions between them. Then the homotopy colimit of this diagram is weakly equivalent to X.

The first corollary is an immediate consequence of Proposition 4.10, and the second is restated and proved as Proposition 4.6(c).

Using open covers to give homotopy decompositions for spaces, or to detect weak equivalences, is of course a classical topic. In addition to [S1] it is worthwhile to mention [Mc1], [Mc2], and [Dk]. Hypercovers were invented by Verdier in [SGA4, Expose V, Sec. 7], where they were used as a way of computing sheaf cohomology in arbitrary Grothendieck topologies. Finally, after writing this paper we learned of the unpublished preprint [Si], which deals with some overlapping issues from a different perspective.

We would like to express our thanks to Bill Dwyer, Michael Mandell, and Jeff Smith for several useful conversations about these results, and to Boris Chorny for pointing out an error in an earlier version. We thank Phil Hirschhorn for helpful conversations about the results in the appendix, in particular for his ideas on removing an annoying T_{1} separation condition. The final form of Lemmas A.2 and A.3 is something we owe to him.

1.7. Notation and terminology. We assume that the reader is familiar with homotopy colimits, and in a few places also with the theory of model categories. The original reference for the latter is [Q], but we generally follow [H] in notation and terminology ([Ho] is also a good reference). Regarding homotopy colimits,

[H] uses ‘hocolim D’ to denote the result of applying a certain explicit formula to any diagram D. This has the disadvantage that the resulting object has the correct homotopy type only when the diagram consists entirely of cofibrant objects. We instead adopt the position that ‘hocolim D’ should always denote the correct homotopy-invariant construction: it is obtained by first applying cofibrantreplacement to the objects in the diagram, and only then using the usual explicit formulas. In model-theoretic terms, homotopy colimit is the left derived functor of the ordinary colimit functor, when the category of diagrams is given the projective model structure (see below). Having made the previous point, we now get to say that for topological spaces it isn’t really necessary because of Appendix A.

We review one last piece of machinery, used often in the body of the paper. Given a small category I, recall that there is a model structure on the category s𝒮et^{I} of diagrams of simplicial sets such that a map is a weak equivalence (resp., fibration) if it is so in every spot of the diagram [H, Sec. 11.6]. We call this the projective model structure on s𝒮et^{I}, and the cofibrant diagrams have the property that the homotopy colimit and ordinary colimit are weakly equivalent.

Finally, some notation: Throughout this paper our open covers 𝒰 = {U_{a}} are always indexed by a set A. In particular, we are allowing the possibility that U_{a} = U_{a′} for different values a ≠ a′. For every finite set σ = {a_{0}, …, a_{n}} in A, we’ll write U_{σ} or U_{a_{0}···a_{n}} for U_{a_{0}} ∩ ⋯ ∩ U_{a_{n}}. Also, once and for all we fix our model for Δ^{n} as the subset of ℝ^{n+1} consisting of (n + 1)-tuples t = (t_{0}, …, t_{n}) such that 0 ≤ t_{i} ≤ 1 for all i and Σ_{i=0}^{n}t_{i} = 1. The symbol 𝒯op denotes the category of all topological spaces—we don’t assume any hypotheses like compactly-generated, although the results all work in that context as well. Also, the symbol s𝒮et denotes the category of simplicial sets.

2. Čech complexes

The purpose of this section is to prove the following: Theorem 2.1. For any open cover 𝒰 of a topological space X, the natural map π : |Č(𝒰)_{∗}| → X is a weak equivalence.

We start by recalling the following result and its corollary: Proposition 2.2 (Gray). Let f: X → Y be a map of spaces and let U and V form an open cover of Y. Suppose that the induced maps

f^{−1}U → U, f^{−1}V → V, and f^{−1}(U ∩ V) → U ∩ V

are all weak equivalences. Then X → Y is also a weak equivalence.

This is proven (in more generality) in [G, 16.24], using an elegant small-simplices argument. With enough technology it can also be done by a Whitehead-type theorem: it’s easy to see that X → Y is an isomorphism on π_{0}, a souped-up van Kampen theorem yields the isomorphism on π_{1}, and for homology with local coefficients one uses the Mayer-Vietoris exact sequence. Gray’s argument is much nicer, though. Corollary 2.3 (May). Let f: X → Y be a map of spaces and let 𝒰 = {U_{a}} be an open cover of Y. Suppose that f^{−1}U_{σ} → U_{σ} is a weak equivalence for every finite set σ of indices. Then X → Y is also a weak equivalence.

May deduces the generalization by a quick application of Zorn’s Lemma [M2, Cor. 1.4]: look at the set of all opens W such that f^{−1}(W ∩ U_{σ}) → W ∩ U_{σ} is a weak equivalence for all σ, including σ = ∅. This set has a maximal element, and Gray’s result shows it must be X. In an earlier paper McCord proved a more general version of this result [Mc1, Th. 6], but the proof is quite a bit more complicated.

Proof of Theorem 2.1. Let ϵ: E_{∗} → X be an augmented simplicial space and let p: |E_{∗}| → X be the induced map. For any V ⊆ X, there is a continuous bijection from the realization of [n] ↦ ϵ_{n}^{−1}V to the space p^{−1}V. If V is open in X, one checks that this is actually a homeomorphism.

In our particular example of Č(𝒰)_{∗} → X, we can now say that given any open set V in X, the space π^{−1}V is homeomorphic to the space |Č(𝒰′)_{∗}|, where 𝒰′ is the open cover {U_{a} ∩V } of the space V. We want to consider the maps π^{−1}(U_{σ}) → U_{σ}, but in this case the cover 𝒰′ of U_{σ} actually contains the whole space U_{σ} as one of its elements. Using Corollary 2.3 we therefore reduce the theorem to the case where the open cover contains the whole space as one of its elements, which is the following lemma. □

Lemma 2.4. Let 𝒰 be an open cover of X such that U_{b} = X for some index b. Then the natural map |Č(𝒰)_{∗}| → X is a weak equivalence (in fact, a homotopy equivalence).

Proof. There is a section χ: X → |Č(𝒰)_{∗}| obtained from the map U_{b} ⊗ Δ^{0} → |Č(𝒰)_{∗}| and the identification U_{b} = X. We only need to show that χπ is homotopic to the identity.

Let Č(𝒰)_{∗} ×I be the simplicial space obtained by crossing all the levels of Č(𝒰)_{∗} with the unit interval. Then |Č(𝒰)_{∗} × I| is the quotient

[ ∐_{a_{0}···a_{n}} U_{a_{0}···a_{n}} × Δ^{n} × I]/ ∼

where the relations are the usual ones, not affecting the I factor at all. Define a map |Č(𝒰)_{∗} × I| → |Č(𝒰)_{∗}| in the following way. Take an element (x, t_{0}, …, t_{n}, s) where x belongs to U_{a_{0}···a_{n}} and (t_{0}, …, t_{n}) belongs to Δ^{n}, and send it to the element (x, 1 − s, st_{0}, …, st_{n}) in the factor U_{ba_{0}...a_{n}} ⊗ Δ^{n+1}. This definition respects the various identifications.

Now, there is also an obvious map f: |Č(𝒰)_{∗} × I| → |Č(𝒰)_{∗}| × I induced by sending (x, t, s) to ((x, t), s). We claim that this is a homeomorphism, thereby giving us a homotopy |Č(𝒰)_{∗}| × I → |Č(𝒰)_{∗}| between χπ and the identity. The reason f is a homeomorphism is just because geometric realization and crossing with I are both left adjoints, and the right adjoints are easily seen to commute. It is important that I and Δ^{n} are locally compact Hausdorff so that the relevant mapping spaces with compact-open topologies have the correct adjointness properties. □

It’s possible to give a ‘slick’ proof of the above lemma by noticing that there is a contracting homotopy for the simplicial space Č(𝒰)_{∗}. One needs to check that homotopic maps of simplicial spaces give homotopic maps on realizations, but this is essentially the above argument. See the proof of Proposition 4.10 for a broader perspective.

2.5. Connection with Segal’s results. To close this section we make the connection between our Theorem 2.1 and the result proven in [S1]. Segal doesn’t explicitly deal with Čech complexes, but the objects he deals with turn out to be homeomorphic to them. This connection will be needed later on.

Let A be the indexing set for a cover 𝒰. We have already introduced the Čech complex Č(𝒰)_{∗}, but if A is given an ordering we may also consider the ordered Čech complex Č^{o}(𝒰)_{∗} which is often easier to work with. This is the simplicial space given by Č^{o}(𝒰)_{n} = ∐_{a_{0}···a_{n}} U_{a_{0}···a_{n}}, where the coproduct ranges over all ordered multi-indices in A. That is, we only consider multi-indices for which a_{0} ≤ a_{1} ≤ ⋯ ≤ a_{n}. Note that there is an inclusion of simplicial spaces Č^{o}(𝒰)_{∗} → Č(𝒰)_{∗}. Proposition 2.6. The map Č^{o}(𝒰)_{∗} → Č(𝒰)_{∗} induces a homotopy equivalence |Č^{o}(𝒰)_{∗}| → |Č(𝒰)_{∗}|.

Proof. For any (not necessarily ordered) multi-index a_{0} ⋯ a_{n}, there is a canonical reordering a_{σ0} ⋯ a_{σn} such that a_{σ0} ≤ ⋯ ≤ a_{σn}. If a_{i} = a_{j} for some i < j, then always choose σi < σj. This allows us to define an inverse map |Č(𝒰)_{∗}| → |Č^{o}(𝒰)_{∗}|. If (x, t) is an element of U_{a_{0}···a_{n}} ⊗ Δ^{n}, then send (x, t) to the element (x, σt) of U_{a_{σ0}···a_{σn}} ⊗ Δ^{n}, where σt is defined by (σt)_{i} = t_{σi}.

One composition is the equal to the identity. It remains to construct a homotopy H : |Č(𝒰)_{∗}| × I → |Č(𝒰)_{∗}| between the other composition and the identity. As in the proof of Lemma 2.4, we use the space |Č(𝒰)_{∗} × I| rather than |Č(𝒰)_{∗}| × I. We define H as follows: An element (x, t) of U_{a_{0}···a_{n}} ⊗ Δ^{n} is equivalent in |Č(𝒰)_{∗}| to the element (x, t_{0}, …, t_{n}, 0, …, 0) of U_{a_{0}···a_{n}a_{σ0}···a_{σn}} ⊗ Δ^{2n+1}. Also, (x, σt) is equivalent in |Č(𝒰)_{∗}| to the element (x, 0, …, 0, t_{σ0}, …, t_{σn}) of U_{a_{0}···a_{n}a_{σ0}···a_{σn}} ⊗ Δ^{2n+1}. Define H((x, t), s) to be the element

(x, st_{0}, …, st_{n}, (1 − s)t_{σ0}, …, (1 − s)t_{σn})

of U_{a_{0}···a_{n}a_{σ0}···a_{σn}} ⊗ Δ^{2n+1}. □

Proposition 2.7. Let 𝒰 be an open cover of a space X indexed by a set A. Consider the realization of the simplicial space

[n] ↦ ∐_{σ_{0}⊆···⊆σ_{n}} U_{σ_{n}},

where the coproduct is indexed by chains of nonempty, finite subsets of A. This realization is homeomorphic to the realization |Č^{o}(𝒰)_{∗}| of the ordered Čech complex and is homotopy equivalent to |Č(𝒰)_{∗}|.

The realization in the above proposition is the object considered in [S1]. The ordered Čech complex is another construction of the same space, which for us seems somewhat easier to work with. One disadvantage, of course, is that it is not natural: a total ordering on A must be chosen to begin with.

Proof. The second claim follows from the first claim and Proposition 2.6.

For the first claim, it is convenient to use a slightly unusual construction of |Č^{o}(𝒰)_{∗}|. When forming the geometric realization, instead of forming Cartesian products with Δ^{k} we instead form products with sd Δ^{k}; since they are homeomorphic it doesn’t matter which one we use. Given this, the key observation is that we can coordinatize sd Δ^{k} in the following way: assuming that the vertices of Δ^{k} are labelled by the numbers 0, …, k in the usual way, a point on sd Δ^{k} is represented uniquely by a chain of proper inclusions σ_{0} ⊂ ⋯ ⊂ σ_{j} of subsets of {0, …, k} together with an element t of Δ^{j}. Essentially, the chain of subsets determines in which sub-simplex the point lies, and then t gives local coordinates inside that sub-simplex.

Using this coordinate scheme, we can write down maps in both directions between the two realizations

[ ∐_{σ_{0}⊆···⊆σ_{n}} U_{σ_{n}} × Δ^{n}]/ ∼  and  [ ∐_{a_{0}≤···≤a_{k}} U_{a_{0}···a_{k}} × sd Δ^{k}]/ ∼ .

For instance, let’s give the map from left to right. Using degeneracy relations, a point p in the left space can be represented by a chain of proper inclusions σ_{0} ⊂ ⋯ ⊂ σ_{n}, a point x of U_{σ_{n}}, and an element t of Δ^{n}. Let a_{0}, a_{1}, …, a_{k} be the ordered list of elements of σ_{n}. The chain σ_{∗} together with t defines a point s in sd Δ^{k}, and so we map p to the pair (x, s). It is easy to see that this map is well-defined and continuous, and just as easy to write down its inverse. □

In the case that {U_{a}} admits a partition of unity {ψ_{a}} it is fairly easy to see that the map π: |Č^{o}(𝒰)_{∗}| → X admits a section: First, a point x of X has a neighborhood which intersects the support of ψ_{a} only for finitely many indices a = a_{0}, …, a_{n}. The section χ sends x to the point of |Č^{o}(𝒰)_{∗}| represented by (x, t) in U_{a_{0}···a_{n}} ⊗ Δ^{n} where t_{i} = ψ_{a_{i}}(x). One has to check that χ is continuous (use the local-finiteness of the partition of unity), and that χπ ≃ id via a straight-line homotopy. See Proposition 4.1 of [S1].

3. Passing to homotopy colimits

The results of the previous section all concerned geometric realizations. In this section we translate these into results about various homotopy colimits. In general, there is a ‘Reedy cofibrancy’ condition on simplicial spaces which guarantees that geometric realization and homotopy colimit agree. Unfortunately our Čech complexes are not Reedy cofibrant, due to the fact that the open sets appearing in them are not necessarily cofibrant spaces. However, Theorem 1.2 shows that in the category of topological spaces this cofibrancy issue is unimportant: homotopy colimits can be computed naively, without first making things cofibrant.

Theorem 3.1. If 𝒰 is an open cover of a space X, then the natural map hocolim Č(𝒰)_{∗} → X is a weak equivalence.

Proof. By Theorem 1.2, the map hocolim Č(𝒰)_{∗} → |Č(𝒰)_{∗}| is a weak equivalence because the Čech complex has free degeneracies in the sense of Definition A.4. Theorem 2.1 now gives the result. □

Here are several alternative formulations: Proposition 3.2. Let A be an indexing set for the cover 𝒰, and let 𝒫_{A} denote the partially ordered set consisting of all nonempty finite subsets of A. Let Γ denote the functor 𝒫_{A}^{op} → 𝒯op which sends σ to U_{σ}. Then the natural map hocolim Γ → X is a weak equivalence.

Proof. To construct hocolim Γ we can take the realization of the simplicial replacement for Γ (by Theorem A.7 we don’t need to first make the spaces cofibrant). That is, we take the realization of the simplicial space

[n] ↦ ∐_{σ_{0}⊆···⊆σ_{n}} U_{σ_{n}},

where the coproduct is indexed by chains of nonempty, finite subsets of A. Now Proposition 2.7 tells us that this realization is homotopy equivalent to |Č(𝒰)_{∗}|, so Theorem 2.1 finishes the proof. □

Corollary 3.3. Let 𝒫_{𝒰} denote the subcategory of 𝒯op whose objects are the open sets U_{a} belonging to 𝒰 together with their finite intersections; the morphisms are the inclusions of open subsets of X. Let Γ denote the inclusion functor 𝒫_{𝒰} → 𝒯op. Then the natural map hocolim Γ → X is a weak equivalence.

Proof. Consider the obvious functor F: 𝒫_{A}^{op} → 𝒫_{𝒰} sending σ to U_{σ}. We will show that it is homotopy cofinal, so for every object V in 𝒫_{𝒰} we prove that the undercategory (V ↓ F) is contractible. It suffices to show that any map K → N(V ↓ F) can be extended over the cone on K, as K ranges over all finite simplicial sets. Every n-simplex s in K maps to a chain of open sets V → U_{σ_{0}} → U_{σ_{1}} → ⋯ → U_{σ_{n}} in (V ↓ F). Since K has only finitely-many non-degenerate simplices, only finitelymany of the U_{σ} will ever appear. Define μ to be the union of all the σ_{i} arising from the map K → N(V ↓ F). To extend the map over CK, we send the cone on s to the (n+1)-simplex corresponding to the chain V → U_{μ} → U_{σ_{0}} → U_{σ_{1}} → ⋯ → U_{σ_{n}}. □

The following corollary was shown to us by Bill Dwyer. Let (𝒯op ↓ X)_{𝒰} denote the full subcategory of (𝒯op ↓ X) consisting of all maps Z → X that factor through the space E = ∐_{a} U_{a}. Let Γ: (𝒯op ↓ X)_{𝒰} → 𝒯op be the canonical functor sending Z → X to Z. We would like to claim that the homotopy colimit of the diagram Γ is weakly equivalent to X, but (𝒯op ↓ X)_{𝒰} is not a small category. So we choose an infinite cardinal κ larger than the size of E and restrict to the spaces Z that have at most κ elements. As the proof of the following corollary indicates, the weak homotopy type of hocolim Γ is independent of the choice of κ, as long as κ is sufficiently large. Corollary 3.4. For the functor Γ: (𝒯op ↓ X)_{𝒰} → 𝒯op defined above, the natural map hocolim Γ → X is a weak equivalence.

Proof. The nth level of the Čech complex is E_{X}^{n} := E ×_{X} E ×_{X} ⋯×_{X} E (n factors). Let’s write 𝒞 = (𝒯op ↓ X)_{𝒰} , for brevity. So we have the functor F: Δ^{op} → 𝒞 given by [n] ↦ E_{X}^{n}. The composition Δ^{op} → 𝒞 → 𝒯op is just Č(𝒰)_{∗}. Because of Theorem 3.1, it will be enough to show that F is homotopy cofinal.

For this we pick an object z : Z → X in 𝒞 and show that (z ↓ F) is contractible. This undercategory is isomorphic to the category of simplices of K [H, 15.1.16], where K is the simplicial set sending [n] to Hom_{𝒞} (z, E_{X}^{n}). But observe that Hom_{𝒞} (z, E_{X}^{n}) is equal to T^{n} where T = Hom_{𝒞} (z, E_{X}). So K is the simplicial set [n] ↦ T^{n}, which is contractible because T is nonempty (using the fact that z: Z → X factors through E). Thus (z ↓ F) is isomorphic to the category of simplices of a contractible simplicial set, and therefore has a contractible nerve. □

Corollary 3.5 (Small simplices theorem). Let Sing_{𝒰} X denote the simplicial set whose n-simplices are the maps Δ^{n} → X that factor through some U_{a}. Then Sing_{𝒰} X → Sing X is a weak equivalence.

Proof. Let 𝒫_{A} be the category defined in Proposition 3.2, where A is the indexing set for the cover. Consider the diagram Γ: 𝒫_{A}^{op} → s𝒮et defined by Γ(σ) = Sing (U_{σ}). By general nonsense |hocolim Γ| ≃ hocolim |Γ|. Also, there is a commutative diagram

\xymatrix{hocolim_{𝒫_{A}^{op}} |Sing U_{σ}| \ar[r] \ar[d] & |Sing X| \ar[d] \\ hocolim_{𝒫_{A}^{op}} U_{σ} \ar[r] & X}

in which the vertical maps are weak equivalences because the natural map |Sing Y | → Y is a weak equivalence for every space Y. We know from Proposition 3.2 that the bottom horizontal map is a weak equivalence, so the top horizontal map is also a weak equivalence. We conclude that the map hocolim Γ → Sing X is a weak equivalence of simplicial sets. Therefore, we shall compare hocolim Γ and Sing_{𝒰} X.

For the moment, assume that A is finite. In this case 𝒫_{A}^{op} is a Reedy category [Ho, Def. 5.2.1], where we think of all the maps as increasing degree. Since there are no non-identity downward maps, the fibrations are objectwise in the Reedy model structure on s𝒮et^{𝒫_{A}^{op}} (see [Ho, Th. 5.2.5]). So in this case the Reedy and projective model structures (cf. Section 1.7) are the same. In particular, a Reedy-cofibrant diagram is also projective-cofibrant, which guarantees that the homotopy colimit and the ordinary colimit are weakly equivalent.

The functor Γ may be checked to be Reedy cofibrant: at the spot indexed by σ = {a_{0}, …, a_{n}}, the latching object is the subobject of Sing U_{σ} consisting of all simplices which are contained in some other U_{b}. The fact that it is actually a subobject says that the latching map is a cofibration. So we know that hocolim Γ and colim Γ are weakly equivalent. It is easy to check that colim Γ ≅ Sing_{𝒰} X. We have shown that if 𝒰 is a finite cover, then Sing_{𝒰} X is weakly equivalent to Sing X.

Now let A be arbitrarily large. For any finite subcollection 𝒰′, let ∪𝒰′ denote the union of the open sets in 𝒰′. Then we know the map Sing_{𝒰′} (∪𝒰′) → Sing (∪𝒰′) is a weak equivalence. But Sing_{𝒰} X → Sing X is the filtered colimit of these maps, where the indexing category is the poset of all finite subcollections 𝒰′. This uses that each space Δ^{n} is compact. Our result now follows from the fact that filtered colimits of simplicial sets preserve weak equivalences. □

4. Hypercovering Theorems

In this section we define hypercovers, and then prove our main result, Theorem 1.3. We go on to deduce various corollaries.

Before giving a rigorous definition of hypercovers, we need to recall a few pieces of machinery related to simplicial objects. For any category 𝒞, let s𝒞 denote the category of simplicial objects in 𝒞. Likewise, let s_{≤n}𝒞 denote the category of truncated simplicial objects of dimension n. There is the obvious forgetful functor sk_{n}: s𝒞 → s_{≤n}𝒞, and if 𝒞 has all finite limits then sk_{n} has a right adjoint called cosk_{n}; these are the skeleton and coskeleton functors. If U_{∗} belongs to s𝒞 then we’ll often abbreviate cosk_{n}(sk_{n} U)_{∗} as just cosk_{n} U_{∗}. Finally, the nth matching object M_{n}U is defined to be the nth object of cosk_{n−1} U_{∗}. There is a canonical map of simplicial spaces U_{∗} → cosk_{n−1} U_{∗}, and in level n it gives U_{n} → M_{n}U. In levels less than n, this map is the identity. We write cosk_{n}^{X} for the nth coskeleton functor for s(𝒞 ↓ X).

These definitions have somewhat easier interpretations when 𝒞 is the category of topological spaces. To describe these, note that any simplicial set may be regarded as a simplicial space which is discrete in every dimension, and if U_{∗} and W_{∗} are simplicial spaces then the set of maps from U_{∗} to W_{∗} has a natural topology coming from the compact-open topology on function spaces. Using these observations, one checks that

(i) U_{n} ≅ Map(Δ^{n}, U_{∗}), (ii) [cosk_{n} U]_{k} ≅ Map(sk_{n}Δ^{k}, U_{∗}), and (iii) M_{n}U ≅ Map(∂Δ^{n}, U_{∗}). The first property is immediate from the Yoneda lemma. The second property follows from the first and the adjunction between sk_{n} and cosk_{n}. The third property is a special case of the second.

Finally, say that a map of spaces Z → X is an open covering map if it is isomorphic to a map of the form ∐_{a} U_{a} → X where {U_{a}} is an open cover of X. Definition 4.1. A hypercover of a space X is an augmented simplicial space U_{∗} → X such that the maps U_{n} → M_{n}^{X}U are open covering maps for all n ≥ 0. Here M_{n}^{X}U denotes the nth matching object of U_{∗} computed in the category s(𝒯op ↓ X) of simplicial spaces over X.

Note that M_{0}^{X}U ≅ X, so the condition for n = 0 says that U_{0} → X is an open covering map. Also M_{1}^{X}U ≅ U_{0} ×_{X} U_{0}, so when n = 1 we are requiring U_{1} → U_{0} ×_{X} U_{0} to be an open covering map. The reader should be aware that when n > 1 the objects M_{n}U and M_{n}^{X}U turn out to be isomorphic, so one can forget about the extra complication of the overcategory.

Using properties (i)–(iii) above, it can be checked that if U_{∗} → X is a hypercover and K → L is an inclusion of finite simplicial sets, then the map Map(L, U_{∗}) → Map(K, U_{∗}) is also an open covering map. From this, it follows that cosk_{n}^{X} U_{∗} → X is a hypercover whenever U_{∗} → X is a hypercover. Also, each map U_{k} → [cosk_{n}^{X} U]_{k} is an open covering map.

We leave it to the reader to check that in a hypercover each U_{n} must be a disjoint union of open subsets of X, and that Čech complexes are the hypercovers for which the maps U_{n} → M_{n}^{X}U are all isomorphisms. Generalizing this, a hypercover U_{∗} → X is called bounded if there exists an N such that the maps U_{n} → M_{n}^{X}U are isomorphisms for all n > N. The smallest such N for which this happens is called the dimension of the hypercover. Said intuitively, the bounded hypercovers of dimension N are the hypercovers for which the refinement process stops after the Nth level. A hypercover U_{∗} → X has dimension at most N if and only if U_{∗} ≅ cosk_{N}^{X} U_{∗}. Lemma 4.2. If U_{∗} → X is a bounded hypercover, then hocolim U_{∗} → X is a weak equivalence.

A more detailed version of the following proof, given in the context of an arbitrary Grothendieck topology, appears in [DHI].

Proof. We proceed by induction, starting from the fact that bounded hypercovers of dimension 0 are just Čech complexes and therefore are handled by Theorem 3.1.

Suppose that U_{∗} → X is a bounded hypercover of dimension n + 1. Define V_{∗} to be cosk_{n} U_{∗}, so V_{∗} is a bounded hypercover of dimension at most n. Therefore, we may assume by induction that hocolim V_{∗} → X is a weak equivalence. The canonical map U_{∗} → V_{∗} gives an open covering map U_{n+1} → V_{n+1}, by the very definition of what it means for U_{∗} to be a hypercover (since V_{n+1} = M_{n+1}U). In fact, one can check that U_{k} → V_{k} is an open covering map for all k.

Consider the following bisimplicial object, augmented horizontally by V_{∗}:

\xymatrix{V_{∗} & U_{∗} \ar[l] & U_{∗} ×_{V_{∗}} U_{∗} \ar[l] \ar[l] & ⋯ \ar[l] \ar[l] \ar[l]}

The kth row is the (augmented) Čech complex for the open covering map U_{k} → V_{k}. Note that for 0 ≤ k ≤ n the kth row is the constant simplicial object with value U_{k} because U_{k} → V_{k} is the identity. Call this bisimplicial object (without the horizontal augmentation) W_{∗∗}.

Let D_{∗} denote the diagonal of W_{∗∗}. Standard homotopy theory tells us that hocolim D_{∗} may be computed (up to weak equivalence) by first taking the homotopy colimits of the rows of W_{∗∗}, and then taking the homotopy colimits of the resulting simplicial object. But the homotopy colimit of the kth row is just V_{k} by Theorem 3.1. Since V_{∗} is a bounded hypercover of dimension at most n, we have assumed that hocolim V_{∗} is weakly equivalent to X. So hocolim D_{∗} → X is a weak equivalence.

We claim that U_{∗} is a retract, over X, of D_{∗}. Note first that one has, in complete generality, a map U_{∗} → D_{∗}; in dimension k it is the unique horizontal degeneracy W_{0k} → W_{kk} (which is the diagonal map in the Čech complex).

To produce a map D_{∗} → U_{∗} it is enough to give sk_{n+1} D_{∗} → sk_{n+1} U_{∗}, because U_{∗} = cosk_{n+1} U_{∗}. Notice that sk_{n} D_{∗} = sk_{n} U_{∗}. Choosing any face map [0] → [n+1] gives a map W_{n+1,n+1} → W_{0,n+1}, which is just D_{n+1} → U_{n+1}. This induces a corresponding map sk_{n+1} D_{∗} → sk_{n+1}U_{∗} as desired.

It is straightforward to check that U_{∗} → D_{∗} → U_{∗} is the identity (because U_{∗} = cosk_{n+1} U_{∗} one only has to check it on (n + 1)-skeleta), and all the maps commute with the augmentations down to X. We have already shown that hocolim D_{∗} → X is a weak equivalence. Since hocolim U_{∗} → X is a retract of hocolim D_{∗} → X, it must also be a weak equivalence. □

Theorem 4.3. If U_{∗} → X is a hypercover then the maps hocolim U_{∗} → |U_{∗}| → X are both weak equivalences.

Proof. The fact that hocolim U_{∗} → |U_{∗}| is a weak equivalence follows from Theorem 1.2: the simplicial object U_{∗} has free degeneracies (Definition A.4).

To show that hocolim U_{∗} → X is a weak equivalence, note first that we have an isomorphism π_{k}(hocolim U_{∗}) → π_{k}(hocolim [cosk_{k+1} U_{∗}]). This is true for any map of simplicial spaces X_{∗} → Y_{∗} which is an isomorphism on (k + 1)-skeleta—an easy proof is to apply the singular functor everywhere to get into bisimplicial sets, then use the diagonal in place of hocolim. But cosk_{k+1} U_{∗} is a bounded hypercover, so Lemma 4.2 gives us the isomorphism π_{k}(hocolim[cosk_{k+1} U_{∗}]) ⟶^{≅} π_{k}X. □

4.4. Complete covers. In this section we replace hypercovers with a related concept which captures the same phenomena. This second approach was suggested to us by Jeff Smith. Definition 4.5. An open cover 𝒰 = {U_{a}} of a space X is called complete if for all finite sets σ of indices, the intersection U_{σ} is covered by elements of 𝒰. It is called a Čech cover if every U_{σ} is again an element of the cover.

Complete covers appear in [DT, Satz 2.2], where they were used in the context of identifying quasi-fibrations. The paper [Mc1] then used them to detect weak equivalences.

We blur the distinction between a cover and the full subcategory that it spans inside the category of open sets of X. Given a cover 𝒰, we can construct an associated simplicial space in the following way: For any n ≥ 0, let P_{n} denote the category of nonempty subsets of {0, …, n}, where the maps are the inclusions. Note that the assignment [n] ↦ P_{n} defines a cosimplicial category in the obvious way. (Application of the nerve functor everywhere gives the cosimplicial space [n] → sd Δ^{n}.)

Define Ω_{∗} to be the simplicial space

[n] ↦ ∐_{F : P_{n}^{op}→𝒰} F ({0, …, n}),

where the coproduct runs over all functors P_{n}^{op} → 𝒰. The faces and degeneracies are induced by those in P in the expected way.

To give a point in Ω_{3}, for example, is to give the following data:

(1) A sequence of opens U_{0}, …, U_{3} in 𝒰, (2) 6 open subsets U_{01}, U_{02}, …, U_{23} in 𝒰 such that U_{ij} ⊆ U_{i} ∩ U_{j};

(3) 4 open subsets U_{012}, …, U_{123} in 𝒰 such that U_{ijk} ⊆ U_{ij} ∩ U_{jk} ∩ U_{ik};

(4) An open subset U_{0123} in 𝒰 which is contained in all the U_{ijk};

(5) A point on U_{0123}. It is usually helpful to think of these open sets as indexed by the faces of a 3-simplex.

In forming the Čech complex of a cover 𝒰 we are throwing in all the finite intersections U_{σ} into the higher levels of the simplicial object, and these are typically objects which are not in 𝒰 itself. The simplicial object Ω_{∗} is in some sense the closest thing we can get to a Čech complex while requiring all the open sets to belong to 𝒰. Proposition 4.6.

(a) If the cover 𝒰 is complete then Ω_{∗} is a hypercover of X.

(b) Regarding 𝒰 as a category, let Γ: 𝒰 → 𝒯op be the obvious inclusion. Then hocolim Γ ≃ |Ω_{∗}|.

(c) If the cover 𝒰 is complete then the natural map hocolim Γ → X is a weak equivalence.

Proof. For part (a), consider the full subcategory P̄_{n} of P_{n} consisting of all objects except for {0, 1, …, n}. Then the matching space M_{n}Ω is equal to

∐_{F̄: P̄_{n}^{op}→𝒰} [ ⋂_{σ∈P̄_{n}} F̄(σ)].

For example, a point in M_{3}Ω is determined by the data in (1)–(3) above, together with a point in U_{012} ∩ U_{013} ∩ U_{023} ∩ U_{123}.

Since the cover is complete, for each functor F̄: P̄_{n}^{op} → 𝒰 and each element x of ∩_{σ∈P̄_{n}} F̄(σ), there exists an extension F of F̄ to P_{n}^{op} such that x belongs to F ({0, …, n}). This shows that Ω_{n} → M_{n}Ω is an open covering map, which finishes part (a).

Now we proceed to part (b). To form hocolim Γ we can work in the Strom model structure on 𝒯op (see Appendix A), where we first take the simplicial replacement

[n] ↦ ∐_{U_{0}→···→U_{n}} U_{0}

and then form the realization. Here the coproduct is indexed over all functors Δ^{n} → 𝒰, where Δ^{n} denotes the category of n composable maps. Note that Ω_{∗} was formed in almost the same way as the simplicial replacement of Γ, except we indexed the coproduct by functors P_{n}^{op} → 𝒰. Each P_{n} is essentially just a subdivision of Δ^{n}, so it’s not surprising that |Ω_{∗}| is another model of the homotopy colimit.

In somewhat more detail: Let sd′ denote the ‘opposite’ of the usual subdivision functor on s𝒮et, in which the orientations of all the simplices have been changed so that they point away from the barycentres, rather than towards them. (We need this because we are using P_{n}^{op} rather than P_{n}.) The functor sd′ has a right adjoint Ex′. There is a natural ‘first vertex map’ sd′ K → K, inducing K → Ex′ K. Given our diagram Γ: 𝒰 → 𝒯op, the realization of the simplicial replacement is isomorphic to the coend Γ ⊗_{𝒰} B, where B: 𝒰 → s𝒮et sends U_{a} to the classifying space B(U_{a} ↓ 𝒰). Likewise, one checks that the realization of Ω_{∗} is isomorphic to the coend Γ ⊗_{𝒰} Ex′ B, where Ex′ B is the obvious composite functor. The natural map B → Ex′ B is an objectwise weak equivalence. The object B of s𝒮et^{𝒰} is cofibrant (see [H, Cor. 14.8.8]), where this diagram category has the projective model structure described in Section 1.7. The exact same arguments show that Ex′ B is also cofibrant in this structure. So we have an objectwise weak equivalence between two cofibrant diagrams. The diagram Γ: 𝒰 → 𝒯op is objectwise cofibrant (since we are working with the Strom model structure on 𝒯op), and so by [H, Cor. 18.4.5] it follows that Γ ⊗_{𝒰} B → Γ ⊗_{𝒰} Ex′ B is a weak equivalence.

Finally, part (c) is an immediate consequence of (a), (b), and Theorem 4.3. □

The following corollary was originally proven by McCord [Mc1, Th. 6], but is an easy consequence of our hypercovering theorem. It generalizes May’s result from Corollary 2.3, which handled the case of Čech covers. For the proof we will need the following observations: (1) If U → X is an open covering map and f: Y → X is any map, there is an induced open covering map Y ×_{X} U → Y. (2) If U_{∗} → X is a hypercover and f: Y → X is a map of spaces, one gets a hypercover f^{−1}U_{∗} → Y whose space in level n is Y ×_{X} U_{n}. Corollary 4.7. Let f: X → Y be a map of spaces. Suppose there is a complete cover 𝒰 = {U_{a}} of Y such that each f^{−1}(U_{a}) → U_{a} is a weak equivalence. Then f itself is a weak equivalence.

Proof. From 𝒰 form the associated hypercover Ω_{∗}^{Y} as described in the paragraph preceding Proposition 4.6. Pulling this back to X gives a hypercover Ω_{∗}^{X} := f^{−1}Ω_{∗}^{Y}, as described above (note that this is not the hypercover associated to the covering {f^{−1}U_{a}}). Now f induces a map Ω_{∗}^{X} → Ω_{∗}^{Y} compatible with the augmentations. This map of simplicial spaces is a levelwise weak equivalence, by assumption. Upon taking homotopy colimits we get

\xymatrix{hocolim Ω_{∗}^{X} \ar[r]^{∼} \ar[d]_{∼} & hocolim Ω_{∗}^{Y} \ar[d]^{∼} \\ X \ar[r] & Y,}

and so we conclude that X → Y is also a weak equivalence. □

4.8. Generalized hypercovers for topological spaces. Up until now we have only considered open covers, but now we turn to a broader notion. We’ll say that a map p: E → B of spaces is a generalized cover if it is locally split: that is, every element of B has a neighborhood U such that p^{−1}(U) → U admits a section. Observe that covering spaces, and in fact fibre bundles in general, are generalized covers. Definition 4.9. An augmented simplicial space U_{∗} → X is a generalized hypercover of X if the maps U_{n} → M_{n}^{X}U are generalized covers. Proposition 4.10. If U_{∗} is a generalized hypercover of X then hocolim U_{∗} → X is a weak equivalence.

Proof. The general results of [DHI] show that this is a formal consequence of Theorem 4.3, because open covers and generalized covers generate the same Grothendieck topology on topological spaces. To keep within the spirit of the present paper, however, we will also give an elementary proof.

One can almost repeat all of the results leading up to Theorem 4.3 for generalized covers verbatim, but there is the slight problem that for a generalized cover E → X the Čech complex Č(E) need not be Reedy cofibrant, even in the Strom model category. Here Č(E)_{∗} is the simplicial space whose nth level is E ×_{X} ⋯ ×_{X} E (with n + 1 factors), and for Reedy cofibrancy one needs to know that the inclusion of the ‘fat diagonals’ in each Č(E)_{n} are cofibrations. Although this will be satisfied in almost all real-life situations, we avoid the issue by working only with hocolims.

Using the same arguments as in Lemma 4.2 and Theorem 4.3, one reduces to showing that if E → X is a generalized cover then π: hocolim Č(E)_{∗} → X is a weak equivalence. Since E → X is a generalized cover, there is an open cover {U_{a}} of X such that the map has a section over each U_{a}. By Corollary 2.3, it suffices to prove that π^{−1}(U_{σ}) → U_{σ} is a weak equivalence for every σ. But for any open U ⊆ X, π^{−1}(U) is homeomorphic to hocolim Č(p^{−1}U)_{∗} (the argument is the same as for realizations—see the first paragraph of the proof of Theorem 2.1). So at this point we have further reduced to the case where E → X has a section.

But a section gives a contracting homotopy for the augmented simplicial space Č(E)_{∗} → X, or equivalently a homotopy equivalence between Č(E)_{∗} and the constant simplicial space with X in every dimension. So if Δ^{1} denotes the usual simplicial set, regarded as a discrete simplicial space, we have a homotopy Č(E)_{∗} × Δ^{1} → Č(E)_{∗}. One can check that if Z_{∗} is any simplicial space, then the two inclusions Z_{∗} ↪ Z_{∗} × Δ^{1} induce weak equivalences after taking hocolims (there are different ways to prove this, one of which is to quote [D1, Thm 6.1]). It follows that a homotopy equivalence between simplicial spaces induces a homotopy equivalence between their hocolims, and so hocolim Č(E)_{∗} → X is a homotopy equivalence. □

Corollary 1.5 is an immediate consequence of the above proposition. Example 4.11. Let G be a topological group and consider the usual G-fibre bundle ξ: EG → BG. Form the Čech complex Č(ξ)_{∗}, which is a generalized hypercover of BG. Using only the fact that EG has a free G-action, one can see that the nth level of Č(ξ)_{∗} is homeomorphic to G^{n} × EG, and the face and degeneracy maps are the familiar ones of the two-sided bar construction B(∗, G, EG). Now using that EG is contractible, we find that Č(ξ)_{∗} is levelwise weakly equivalent to the simplicial space

\xymatrix{∗ & G \ar[l] \ar[l] & G × G · · · . \ar[l] \ar[l] \ar[l]}

The above proposition tells us that |Č(ξ)_{∗}| ≃ BG, and so in this way we recover the usual bar construction for BG.

5. Topological realization functors for 𝔸^{1}-homotopy theory

Let k be a field. Morel and Voevodsky [MV] produced a model category Spc(k) which captures the ‘motivic homotopy theory’ of smooth schemes over k. Here Spc(k) stands for ‘spaces over k’. There are several possible choices for what model category to choose, all Quillen equivalent in the end, but we choose our reference point to be the category of simplicial presheaves on the Nisnevich site of smooth schemes over Spec k, with the model structure provided in [J, Appendix B].

When k comes with an embedding k ↪ ℂ, then any k-scheme X gives rise to a topological space X(ℂ) consisting of its ℂ-valued points with the analytic topology. A natural expectation is to use this functor to relate Spc(k) to the usual model category 𝒯op of topological spaces. Morel and Voevodsky showed how to extend this functor on the level of homotopy categories (by somewhat awkward methods), but they didn’t produce functors at the model category level. In this section we use Proposition 4.10 to produce such functors, with the small provision that we have to replace Spc(k) with a Quillen-equivalent variant. We also address the situation when k ↪ ℝ, in which case one can construct topological realization functors into ℤ_{2}-equivariant spaces.

As in [D2], a Quillen pair L: ℳ ⇄ 𝒩: R will be called a Quillen map ℳ → 𝒩.

5.1. The Complex case. Let 𝒯 denote either the Nisnevich or étale Grothendieck topology on the category Sm/k of smooth k-schemes. In the terminology of [D2], let Spc′(k)_{𝒯} denote the universal model category built from Sm/k subject to the following relations:

(1) X ⨿ Y ⟶^{∼} (X ∪ Y) (here ⨿ denotes the coproduct in our model category, whereas ∪ denotes disjoint union of schemes);

(2) hocolim U_{∗} ⟶^{∼} X for any 𝒯-hypercover U_{∗} of a smooth scheme X (called ‘basal hypercovers’ in [DHI]);

(3) X × 𝔸^{1} ⟶^{∼} X.

(Relation (1) is morally a special case of (2), but must be included separately for technical reasons—see [DHI]).

The model categories Spc(k)_{𝒯} and Spc′(k)_{𝒯} have the same underlying category and the same class of weak equivalences, but differ in their notions of cofibration and fibration. They are injective and projective versions of the same homotopy theory. Theorem 5.2. There are Quillen maps Spc′(k)_{et} → 𝒯op and Spc′(k)_{Nis} → 𝒯op sending a smooth k-scheme X to X(ℂ).

Proof. By general nonsense from [D2], to give a Quillen map Spc′(k)_{𝒯} → 𝒯op we just need to give a functor Sm/k → 𝒯op which respects the above relations. The functor we’re interested in is X ↦ X(ℂ), and this clearly preserves relations (1) and (3). In the case of the étale topology, the fact that it preserves relation (2) is just Proposition 4.10; the point is that if p: E → B is an étale cover, then p(ℂ): E(ℂ) → B(ℂ) satisfies the hypotheses of the inverse function theorem and hence is locally split.

Since the étale topology is finer than the Nisnevich topology, there is an obvious map Spc′(k)_{Nis} → Spc′(k)_{et} (in essence, there are more relations of type (2) for the étale topology). So one also gets a topological realization map Spc′(k)_{Nis} → 𝒯op by composition. □

It is possible to show that the functor X ↦ X(ℂ) takes elementary distinguished squares [MV, Sec. 3.1, Def. 1.3] to homotopy pushout squares of topological spaces. Together with results of [B], this can be used to give an alternative proof of the above theorem for the Nisnevich topology.

5.3. The Real case. If we have a Real field k ↪ ℝ, then the space X(ℂ) comes equipped with an action of the group Gal(ℂ/ℝ) = ℤ_{2}. So we might hope to compare Spc′(k) to a model category of ℤ_{2}-equivariant spaces.

Recall that if G is a finite group then there are two notions of weak equivalence for G-spaces, called the non-equivariant and G-equivariant equivalences. An equivariant map X → Y is a non-equivariant equivalence if it is a weak equivalence after forgetting the equivariant structure, and it is a G-equivariant equivalence if X^{H} → Y^{H} is a non-equivariant weak equivalence for every subgroup H ⊆ G. There are associated G-equivariant and non-equivariant model structures on the category of G-spaces, which we will denote 𝒯op(G) and 𝒯op(G)_{non}.

If p: E → B is an equivariant map which is also a covering space (nonequivariantly), the map hocolim Č(E)_{∗} → B is a non-equivariant equivalence but not necessarily a G-equivariant equivalence. For instance, if p is G → ∗ then the map hocolim Č(E)_{∗} → B is equal to EG → ∗. So when we have a subfield k ↪ ℝ the arguments given above show that the functor X ↦ X(ℂ) induces a Quillen map Spc′(k)_{et} → 𝒯op(ℤ_{2})_{non}, but not a Quillen map Spc′(k)_{et} → 𝒯op(ℤ_{2}). However, when we use the Nisnevich topology something special happens. Lemma 5.4. If E → B is a Nisnevich cover of k-schemes, then E(ℂ)^{ℤ_{2}} → B(ℂ)^{ℤ_{2}} is locally split. For a counterexample to this in the case of étale covers, try Spec ℂ → Spec ℝ.

Proof. First note that X(ℂ)^{ℤ_{2}} is homeomorphic to X(ℝ) for any scheme X over k. The map p(ℝ) : E(ℝ) → B(ℝ) is surjective by the defining property of Nisnevich covers; every ℝ-point in B lifts to E.

By definition of étale covers, p(ℝ) satisfies the hypothesis of the inverse function theorem. Since p(ℝ) is surjective, it is locally split. □

Theorem 5.5. There is a Quillen map Spc′(k)_{Nis} → 𝒯op(ℤ_{2}) sending a smooth k-scheme X to X(ℂ).

Proof. The argument exactly parallels the non-equivariant case in Theorem 5.2, so the only nontrivial part is to show that if U_{∗} → X is a Nisnevich hypercover then the map hocolim U_{∗}(ℂ) → X(ℂ) is a ℤ_{2}-equivariant weak equivalence of ℤ_{2}- spaces. The fact that it is a non-equivariant equivalence has already been discussed in Theorem 5.2, because U_{∗} → X is in particular an étale hypercover. So we must consider what happens when we take ℤ_{2}-fixed points.

It is a fact that for any diagram D of G-spaces (G any finite group) and any subgroup H of G, one has (hocolim D)^{H} ≃ (hocolim D^{H}) (see Remark 5.6 below). So we just need to convince ourselves that hocolim{U_{∗}(ℂ)^{ℤ_{2}}} → X(ℂ)^{ℤ_{2}} is a nonequivariant weak equivalence. But by the above lemma one sees that U_{∗}(ℂ)^{ℤ_{2}} is a generalized hypercover of X(ℂ)^{ℤ_{2}}, and so the result is an instance of Proposition 4.10. □

Remark 5.6. In the above proof we needed the fact that (hocolim D)^{H} is weakly equivalent to hocolim(D^{H}). This is well-known in equivariant topology, but it’s hard to find an actual reference. We give a brief sketch, for which we are grateful to Michael Mandell.

First of all, it clearly suffices to consider the case where all the D_{i} are cofibrant. This means in particular that they are Hausdorff. We form hocolim D by first writing down the simplicial replacement of the diagram, and then taking geometric realization. Taking H-fixed points obviously commutes with the simplicial replacement functor, so it suffices to worry about the geometric realization part. But one can check that if X_{∗} is a simplicial space in which all X_{n} are Hausdorff, then |X_{∗}|^{H} is homeomorphic to |X_{∗}^{H}|. To do this, use the skeletal filtration on |X_{∗}| and the fact that |Sk_{n} X_{∗}| is obtained from |Sk_{n−1} X_{∗}| by pushing out along a closed inclusion (this is one of the places where the Hausdorff condition is needed). Check that taking fixed-points commutes with filtered colimits, and for Hausdorff spaces it also commutes with pushouts along closed inclusions.

Appendix A. Homotopy colimits for diagrams of non-cofibrant spaces

Let 𝒯op denote the category of all topological spaces, with its usual model category structure. Given a diagram D: I → 𝒯op, the usual instructions for computing the homotopy colimit of D are (1) to apply a cofibrant-replacement functor to every object in the diagram, and (2) to then use an explicit formula like that of Bousfield-Kan [BK, Sec. XII.2]. This is the situation in an arbitrary model category. In this section we show that for the special case of 𝒯op, the first step of cofibrantreplacement is actually not needed. What we show is that no matter what formula one uses for computing homotopy colimits—whether it is the Bousfield-Kan formula or your favorite alternative—that formula always gives a homotopy invariant construction in 𝒯op, even without the cofibrant-replacement step. This fact seems to be folklore in certain circles, although not well-known in others.

The most useful way to formulate this result seems to be in model category terms, as a comparison between the usual model structure on 𝒯op and the Strom model structure, where everything is cofibrant. See Theorem A.7.

To begin with, we need the following Lemma A.1. Let A → B and X → Y be weak equivalences. Given a diagram

\xymatrix{A × D^{n} \ar[d] & A × S^{n−1} \ar[l] \ar[d] \ar[r] & X \ar[d] \\ B × D^{n} & B × S^{n−1} \ar[l] \ar[r] & Y,}

where the maps in the left-hand-square are the obvious ones, the induced map from the pushout of the top row to the pushout of the bottom row is also a weak equivalence.

Note that if A and B are cofibrant then this is an easy consequence of leftproperness for 𝒯op, but we claim the result in greater generality.

Proof. Let X_{A} and Y_{B} be the pushouts of the top and bottom rows respectively, and write f: X_{A} → Y_{B} for the map between them. We will produce a suitable cover of these spaces and use Proposition 2.2.

Let U_{B} be the pushout of

\xymatrix{B × (D^{n} − \{0\}) & B × S^{n−1} \ar[r] \ar[l] & Y.}

Write D_{ϵ} for {x ∈ D^{n}: |x| < ϵ} (where 0 < ϵ < 1), and let V_{B} = B × D_{ϵ}. The spaces U_{B} and V_{B} clearly form an open cover of Y_{B}, and notice that U_{B} deformationretracts down to Y. The intersection U_{B} ∩ V_{B} is equal to B × (D_{ϵ} − {0}).

The same definitions give us a cover {U_{A}, V_{A}} of X_{A}, and it is easy to check that f^{−1}(U_{B}) = U_{A} and f^{−1}(V_{B}) = V_{A}. So the map f^{−1}(V_{B}) → V_{B} is the map A × D_{ϵ} → B × D_{ϵ}, which is a weak equivalence. Similar reasoning shows that f^{−1}(U_{B} ∩ V_{B}) → U_{B} ∩ V_{B} is a weak equivalence. Finally, one argues that f^{−1}(U_{B}) → U_{B} is a weak equivalence because it deformation-retracts down to X → Y. Proposition 2.2 now shows that X_{A} → Y_{B} is a weak equivalence. □

We’ll say that an inclusion Y ↪ Z is relatively T_{1} if given any open set U in Y and any point z of Z\U, there is an open set W of Z such that U ⊆ W and z ∉ W (compare the similar definition from [Ho, p. 50]). It follows that if E is any finite subset of Z\U, one can find an open set W ⊆ Z which contains U and doesn’t intersect E. Note that a space X is T_{1} precisely if all the inclusions {x} ↪ X are relatively T_{1}. Lemma A.2. Given a pushout diagram of the form

[figure: A × S^{n}; Y; A × D^{n+1}; Z,]

the inclusion Y ↪ Z is relatively T_{1}.

Proof. Suppose given a point z in Z and an open U in Y. Either z is in Y or else it is represented by a pair (a, t) where t is in the interior of D^{n+1}. The argument works the same for the two cases, and so for convenience we’ll assume the latter.

Pull back U to A × S^{n} and express it as a union of rectangles V_{i} × W_{i}, where V_{i} is open in A and W_{i} is open in S^{n}. Each W_{i} can be fattened into an open subset W′_{i} of D^{n+1} with the properties that W′_{i} ∩ S^{n} = W_{i} and W′_{i} does not contain t.

Let M be the union of the V_{i} × W′_{i}; it is an open subset of A × D^{n+1}. Let N be the union of the images of M and U in Z. One checks that N ∩ Y = U, and the pullback of N to A × D^{n+1} is M. So N is open in Z and N contains U, but N does not contain z. □

The following lemma is well-known for closed inclusions of T_{1}-spaces (see also [Ho, Prop. 2.4.2]). The usual proof still works in our case. Lemma A.3. Suppose that Y_{1} ↪ Y_{2} ↪ ⋯ is a sequence of relatively T_{1} inclusions and that K is a compact space. Then any map f: K → colim Y factors through some Y_{k}.

Proof. Suppose the map does not factor through any Y_{k}. By taking a subsequence of Y if necessary, we can find a sequence of points k_{1}, k_{2}, … in K with the property that f(k_{i}) lies in Y_{i}\Y_{i−1}.

Pick an n and set V_{n} = Y_{n}. Next, choose an open set V_{n+1} in Y_{n+1} which contains V_{n} but doesn’t contain f(k_{n+1}). Then pick an open set V_{n+2} in Y_{n+2} which contains V_{n+1} but neither f(k_{n+1}) nor f(k_{n+2}). Continuing this process gives an infinite sequence of opens, so their colimit W_{n} is an open subset of colim Y.

As n varies, the open subspaces W_{n} form a cover of colim Y. But f(K) is a compact subspace of colim Y, and it is not covered by any finite subcover. This is a contradiction. □

We now need some machinery related to simplicial spaces. Definition A.4. A simplicial space X_{∗} is said to be split, or to have free degeneracies, if there exist subspaces N_{k} ↪ X_{k} such that the canonical map

∐_{σ} N_{σ} → X_{k}

is an isomorphism. Here the variable σ ranges over all surjective maps in Δ of the form [k] → [n], N_{σ} denotes a copy of N_{n}, and the map N_{σ} → X_{k} is the one induced by σ^{∗}: X_{n} → X_{k} (see [AM, Def. 8.1]).

The idea is that the spaces N_{k} represent the ‘non-degenerate’ part of X_{k}, sitting inside of X_{k} as a direct summand. It is an easy exercise to check that if X_{∗} has free degeneracies and all the N_{k} are cofibrant spaces, then X_{∗} is Reedy cofibrant in s𝒯op.

If X_{∗} is any simplicial space, let Sk_{n} X_{∗} be the simplicial space equaling X_{∗} through dimension n and equaling the degenerate subspaces of X_{∗} in larger dimensions. This is slightly different than the n-truncated simplicial space sk_{n} X_{∗}. There are maps Sk_{0} X_{∗} → Sk_{1} X_{∗} → ⋯ and the colimit is X_{∗}. It follows that |X_{∗}| is equal to colim_{n} |Sk_{n} X_{∗}|, using that geometric realization is a left adjoint (and this doesn’t require any assumptions on X, only hinging upon the fact that the spaces Δ^{n} are locally compact Hausdorff). An important point is that when X_{∗} has free degeneracies the space |Sk_{n} X_{∗}| is obtained from |Sk_{n−1} X_{∗}| via the pushout diagram

\xymatrix{N_{n} × ∂Δ^{n} \ar[d] \ar[r] & |Sk_{n−1} X_{∗}| \ar@{.>}[d] \\ N_{n} × Δ^{n} \ar@{.>}[r] & |Sk_{n} X_{∗}|.} (A.1)

Proposition A.5. Let X_{∗} be a simplicial space with free degeneracies. If K is a compact space then any map K → |X_{∗}| factors through some |Sk_{n} X_{∗}|.

Proof. This is a direct application of Lemmas A.2 and A.3, using the skeletal filtration of |X_{∗}| and the pushout square (A.1). □

The following corollary is the crucial ingredient for Theorem A.7. It is very similar to things in the literature, notably [M1, Th. 11.13] and [S2, Lem. A.5]. May’s result assumes the spaces are compactly-generated and Hausdorff, and also that the realizations are simply-connected. Segal’s result is more similar to ours, and the proofs follow the same pattern, but he works with homotopy equivalences rather than weak equivalences. Corollary A.6. If X_{∗} → Y_{∗} is a map of simplicial spaces with free degeneracies such that X_{n} → Y_{n} is a weak equivalence for each n, then |X_{∗}| → |Y_{∗}| is also a weak equivalence.

Proof. For every k and every basepoint ∗ of X_{0}, there is an isomorphism

colim_{n} π_{k}(| Sk_{n} X_{∗}|, ∗) → π_{k}(|X_{∗}|, ∗)

(and the same statement holds with X_{∗} replaced by Y_{∗}). This follows from Proposition A.5, taking K to be a sphere. Therefore, it suffices to show that |Sk_{n} X_{∗}| → |Sk_{n} Y_{∗}| is a weak equivalence. Using induction, this follows from the pushout square (A.1) and Lemma A.1. □

Recall that the Strom model category is a model structure for topological spaces, denoted 𝒯op^{S}, in which the weak equivalences are homotopy equivalences and the cofibrations (resp., fibrations) are the Hurewicz cofibrations (resp., fibrations). Note that all objects are cofibrant in this structure. Theorem A.7. Let D: I → 𝒯op be a diagram of spaces. Then the homotopy colimits of D as computed in 𝒯op and 𝒯op^{S} have the same weak homotopy type.

Proof. We don’t know if the Strom model category is simplicial—the usual definitions for the simplicial action run into adjointness problems because 𝒯op is the category of all topological spaces. However, one may check directly that the homotopy colimit of a diagram in 𝒯op^{S} can still be computed by first taking the simplicial replacement of D and then applying the geometric realization functor, as usual. No cofibrant-replacement is necessary, since all objects are cofibrant.

In 𝒯op we first apply a cofibrant-replacement functor to all the objects in the diagram, and only then do we take simplicial replacement and realize. Simplicial replacements always have free degeneracies (see [D2, Proof of Lem. 2.7]), hence Corollary A.6 applies. □

Remark A.8. Theorem A.7 also holds if one uses the category of compactlygenerated, weak Hausdorff spaces with its usual model structure. The same proofs work, with some extra caution that the various colimits are what they’re supposed to be.

Finally, we give a short proof of Theorem 1.2.

Proof of Theorem 1.2. By Theorem A.7, we can compute the homotopy colimit in the Strom model category. In this model structure U_{∗} is Reedy cofibrant because it has free degeneracies. So the realization already has the correct homotopy type. □

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Department of Mathematics, University of Oregon, Eugene, OR 97403

Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556

   E-mail address: ddugger@math.uoregon.edu
   E-mail address: isaksen.1@nd.edu