Derived Algebraic Geometry XIII: Rational and p-adic Homotopy Theory

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Derived Algebraic Geometry XIII: Rational and p-adic Homotopy Theory

December 15, 2011

Introduction

The principal goal of algebraic topology is to study topological spacesX by means of invariants associated toX, such as the homology and cohomology groups H∗(X;Z) and H^∗(X;Z). Ideally, we would like to devise invariantsX7→F(X) which satisfy the following requirements:

(a) The invariantF is algebraic in nature: that is, it assigns to each spaceX something like a group or a vector space, which is amenable to study using methods of abstract algebra.

(b) The invariantF is powerful: that is, F(X) contains a great deal of useful information aboutX.

There is a natural tension between these requirements: the simpler an objectF(X) is, the less information we should expect it to contain. Nevertheless, there are some invariants which do a good job of meeting both objectives. One example is provided by Sullivan’s approach to rational homotopy theory. To every topological spaceX, Sullivan associates apolynomial deRham complexA(X), which is a commutative differential graded algebra over the fieldQof rational numbers. This is an object of a reasonably algebraic nature (albeit not quite so simple as a group or a vector space), which at the same time captures the entire rational homotopy type of X: for example, if X is a simply connected space whose homotopy groups {πnX}_n≥2 are finite- dimensional vector spaces over Q, then we can functorially recoverX (up to homotopy equivalence) from A(X).

As a chain complex, Sullivan’s polynomial deRham complexA(X) is quasi-isomorphic with the complex of singular cochainsC^∗(X;Q). The advantage of A(X) of C^∗(X;Q) is thatA(X) is equipped with a multiplication which is commutative at the level of cochains, whereas the multiplication on C^∗(X;Q) (given by the Alexander-Whitney construction) is only commutative at the level of cohomology. We can therefore think ofA(X) as a remedy for the failure of the multiplication onC^∗(X;Q) to be commutative at the level of cochains. This is specific to the case of rational coefficients. Ifpis a prime number andFpis the finite field withpelements, then there is no way to replace the chain complexC^∗(X;Fp) by a commutative differential graded algebra overF_p, which is functorially quasi-isomorphic to C^∗(X;F_p). However, a different remedy is available overF_p: although the multiplication onC^∗(X;F_p) is not commutative, it is commutative up to coherent homotopy. More precisely,C^∗(X;F_p) has the structure of anE∞-algebra overF_p. Moreover, Man- dell has used this observation to develop a “p-adic” counterpart of rational homotopy theory. For example, he has shown that ifX is a simply connected space whose homotopy groups are finitely generated modules overZp, thenX can be functorially recovered fromC^∗(X;Fp), together with its E∞-algebra structure (see [54]).

Our goal in this paper is to give an exposition of rational and p-adic homotopy theory from the ∞- categorical point of view, emphasizing connections with the earlier papers in this series. We will begin with the case of rational homotopy theory. Sullivan’s work on the subject was preceded by Quillen, who showed that the homotopy theory of rational spaces can be described in terms of the homotopy theory of differential graded Lie algebras overQ. This result of Quillen was the impetus for later work of many authors, relating differential graded Lie algebras to the study of deformation problems in algebraic geometry. In [49], we made this relationship explicit by constructing an equivalence of ∞-categories Liek 'Modulik, where k is any field of characteristic zero. Here Liek denotes the ∞-category of differential graded Lie algebras over k, and Moduli_k the ∞-category of formal moduli problems overk. In §1, we will apply this result (in the special casek=Q) to recover Quillen’s results. Along the way, we will discuss Sullivan’s approach to rational homotopy theory and its relationship with the theory of coaffine stacks developed in [47].

In §2 we will turn our attention to the case of p-adic homotopy theory. We will say that a space X is p-finite if it has finitely many connected components and finitely many nonzero homotopy groups, each of which is a finite p-group (Definition 2.4.1). If X is ap-finite space, then Mandell shows thatX can be recovered as the mapping space

Map_CAlg

Fp(C^∗(X;Fp),Fp)'Map_CAlg

Fp

(C^∗(X;Fp),Fp)

where CAlg_kdenotes the∞-category ofE∞-algebras overkandFpdenotes the algebraic closure ofFp. Here it is important to work overFp rather thanFp: the functorX 7→C^∗(X;Fp) is not fully faithful (even when

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restricted top-finite spaces). We can explain this point as follows: ifkis an arbitrary field of characteristic p, then the homotopy theory of E∞-algebras over k is most naturally related not to the homotopy theory of (p-finite) spaces, but to the theory of (p-finite) spaces equipped with a continuous action of the absolute Galois group Gal(k/k). More generally, we will show that ifkis a commutative ring in whichpis nilpotent, then there is a fully faithful embedding fromShv^p−fc_k into CAlg^op_k (Corollary 2.6.12); hereShv^p−fc_k denotes the∞-category ofp-constructible´etale sheaves (of spaces) on SpecR(see Definition 2.4.1).

In order to apply the results of§2 to the study of an arbitrary spaceX, we need to study the problem of approximatingX byp-finite spaces. For this, it is convenient to introduce the notion of ap-profinite space.

By definition, ap-profinite space is a Pro-object in the∞-category S^p−fc ofp-finite spaces. The collection of p-profinite spaces can be organized into an ∞-category S^Pro(p) = Pro(S^p−fc). In §3 we will study the

∞-category S^Pro(p), and show that it behaves in many respects like the usual∞-category of spaces. Using Corollary 2.6.12, we will construct a fully faithful embedding

S^Pro(p)→CAlg^op_k X 7→C^∗(X;k),

where k is any separably closed field (Proposition 3.1.16). Moreover, if k is algebraically closed, we can explicitly describe the essential image of this functor (Theorem 3.5.8). We then recover some results of [54]

by restricting our attention top-profintie spaces of finite type (Corollary 3.5.15).

Notation and Terminology

We will use the language of ∞-categories freely throughout this paper. We refer the reader to [43] for a general introduction to the theory, and to [44] for a development of the theory of structured ring spectra from the∞-categorical point of view. We will also assume that the reader is familiar with the formalism of spectral algebraic geometry developed in the earlier papers in this series. For convenience, we will adopt the following reference conventions:

(T) We will indicate references to [43] using the letter T.

(A) We will indicate references to [44] using the letter A.

(V) We will indicate references to [45] using the Roman numeral V.

(V II) We will indicate references to [46] using the Roman numeral VII.

(V III) We will indicate references to [47] using the Roman numeral VIII.

(IX) We will indicate references to [48] using the Roman numeral IX.

(X) We will indicate references to [49] using the Roman numeral X.

(XI) We will indicate references to [50] using the Roman numeral XI.

(XII) We will indicate references to [51] using the Roman numeral XII.

For example, Theorem T.6.1.0.6 refers to Theorem 6.1.0.6 of [43].

If C is an ∞-category, we let C^' denote the largest Kan complex contained in C: that is, the ∞- category obtained fromCby discarding all non-invertible morphisms. We will say that a map of simplicial sets f : S → T is left cofinal if, for every right fibration X → T, the induced map of simplicial sets FunT(T, X)→FunT(S, X) is a homotopy equivalence of Kan complexes (in [43], we referred to a map with this property as cofinal). We will say that f isright cofinal if the induced mapS^op →T^op is left cofinal:

that is, iff induces a homotopy equivalence FunT(T, X)→FunT(S, X) for everyleftfibrationX →T. IfS

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andT are∞-categories, thenf is left cofinal if and only if for every objectt∈T, the fiber productS×TT_t/

is weakly contractible (Theorem T.4.1.3.1).

Throughout this paper, we let CAlg denote the ∞-category of E∞-rings. If R is an E∞-ring, we let CAlg_R = CAlg(ModR) denote the ∞-category of E∞-algebras over R. We let Spec^´êtR denote the affine spectral Deligne-Mumford stack associated to R. This can be identified with the pair (Shv^´êt_R,O), where Shv^´êt_R ⊆Fun(CAlg^´êt_R,S) is the full subcategory spanned by those functors which are sheaves with respect to the

´

etale topology, andOis the sheaf ofE∞-rings onShv^´^et_R determined by the forgetful functor CAlg^´^et_R →CAlg.

1 Rational Homotopy Theory

Definition 1.0.1. Let Y be a simply connected space. We say that Y is rational if, for each n≥2, the homotopy groupπnY is a vector space over the field Qof rational numbers (since Y is simply connected, the homotopy groupsπ_n(Y, y) are canonically independent of the choice of base pointy ∈Y). We let S^rat denote the full subcategory ofSspanned by the simply connected rational spaces, andS^rat∗ the∞-category of pointed objects ofS^rat (that is, the∞-category of pointed simply connected rational spaces).

Definition 1.0.2. Letkbe a field of characteristic zero, and let Liek denote the∞-category of differential graded Lie algebas over k (Definition X.2.1.14). We will say that a differential graded Lie algebra g_∗ is connectedif the graded Lie algebra H_∗(g_∗) is concentrated in positive degrees: that is, if the homology groups Hn(g_∗) vanish forn≤0 (here our notation indicates passage to the homology groups of the underlying chain complex ofg_∗, rather than the Lie algebra homology ofg_∗). We let Lie^≥1_k denote the full subcategory of Lie_k spanned by the connected differential graded Lie algebras.

Quillen’s work on rational homotopy theory establishes a close connection between rational spaces and differential graded Lie algebras. We can formulate his main result as follows:

Theorem 1.0.3 (Quillen). The ∞-category S^rat∗ of rational pointed spaces is equivalent to the ∞-category Lie^≥1_Q of connected differential graded Lie algebras over the fieldQof rational numbers.

In§X.2, we studied a different interpretation of the homotopy theory of differential graded Lie algebras:

for any fieldk of characteristic zero, there is a canonical equivalence of ∞-categories Ψ : Liek →Modulik, where Modulikdenotes the∞-category of formal moduli problems overk(Theorem X.2.0.2). In this section, we will explore the relationship between this statement and Quillen’s work. To this end, we will associate to every fieldkof characteristic zero an∞-category RType(k), which we refer to as the∞-category ofk-rational homotopy types. By definition, RType(k) is a full subcategory of the∞-category Fun(CAlg^cn_k ,S). Our main results can be summarized as follows:

(a) Ifk=Qis the field of rational numbers, then the evaluation mapX 7→X(Q) induces an equivalence of ∞-categories RType(Q) → S^rat (Theorem 1.3.6). Restricting to pointed objects, we obtain an equivalence RType(Q)_∗→S^rat∗ .

(b) For any fieldkof characteristic zero, and k-rational homotopy typeX, and any base pointη∈X(k), we can associate a formal moduli problemX^∨∈Modulik, which we call theformal completionofX at the pointk. The construction (X, η)7→X^∨induces a fully faithful embedding RType(k)_∗→Moduli^≥2_k , where Moduli^≥2_k denotes the full subcategory of Modulik spanned by those formal moduli problems having a 2-connective tangent complex (Theorem 1.5.3).

(c) For every fieldkof characteristic zero, the equivalence Ψ : Liek →Modulik restricts to an equivalence Lie^≥1_k →Moduli^≥2_k .

Takingkto be the field of rational numbers, assertions (a), (b), and (c) give a diagram of equivalences S^rat∗

←α RType(Q)_∗→^β Moduli^≥2_Q ←^γ Lie^≥1_Q ,

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thereby giving a proof of Theorem 1.0.3.

Let us now outline the contents of this section. We will begin in §1.1 by reviewing a convergence criterion for the cohomological Eilenberg-Moore spectral sequence, which plays an important role in our study of rational and p-adic homotopy theory. In §1.2, we define the ∞-category RType(k) of k-rational homotopy types and establish some of its formal properties. In§1.3, we review the Sullivan model for rational homotopy theory, and use it to construct the equivalence RType(Q)→S^rat described in (a). The functor β: RType(k)_∗→Moduli^≥2_k of (b) will be constructed in§1.5. The proof thatβ is an equivalence will require some basic facts about the homotopy theory of differential graded Lie algebras, which we review in§1.4.

1.1 Cohomological Eilenberg-Moore Spectral Sequences

Notation 1.1.1. LetC be an ∞-category. For every Kan complex X, we let C^X denote the ∞-category Fun(X,C) of functors from X to C. If f : X → Y is a map of Kan complexes, then composition with f induces a functorC^Y →C^X, which we will denote byf_∗.

Assume that, for each vertex y ∈ Y, the ∞-category C admits colimits indexed by the Kan complex X×Y Y_/y. Then the functorf^∗ :C^Y →C^X admits a left adjoint, given by left Kan extension alongf; we will denote this functor byf!. If for eachy∈Y the∞-categoryCadmits limits indexed byX×Y Y_y/, then f^∗ admits a right adjoint (given by right Kan extension alongf), which we will denote byf_∗.

Let k be a field and let Modk denote the∞-category of k-module spectra. For every Kan complexX, we let Mod^X_k denote the∞-category Fun(X,Modk). The symmetric monoidal structure on Modk induces a symmetric monoidal structure on Mod^X_k. We will denote the unit object of Mod^X_k bykX(that is,kX ∈Mod^X_k denotes the constant functorX →Modk taking the valuek).

Remark 1.1.2. In the situation of Notation 1.1.1, the functor f_∗ : Mod^X_k → Mod^Y_k is lax symmetric monoidal. In particular,A=f_∗(k_X) is a commutative algebra object of Mod^Y_k, andf_∗ determines a functor Mod^X_k →Mod_A(Mod^Y_k).

Notation 1.1.3. LetX be a Kan complex and letf :X→∆⁰be the projection map. For eachF∈Mod^X_k , we set

C_∗(X;F) =f!F C^∗(X;F) =f_∗F.

In the special caseF=kX, we will denoteC_∗(X;kX) andC^∗(X;kX) byC_∗(X;k) andC^∗(X;k), respectively.

We regardC^∗(X;k) as a commutative algebra object of Modk.

Remark 1.1.4. If we identify Mod_kwith the underlying∞-category of the model category Vect^dg₍ k) of chain complexes of k-vector spaces, then C_∗(X;k) and C^∗(X;k) can be represented by the objects of Vect^dg₍ k) given by the usualk-valued chain and cochain complexes associated to the simplicial setX. In particular, we have canonical isomorphisms ofk-vector spaces

πnC_∗(X;k)'Hn(X;k) πnC^∗(X;k)'H⁻ⁿ(X;k).

Lemma 1.1.5. Suppose we are given a homotopy pullback diagram of Kan complexesσ: X⁰ ^f

0 //

g

Y⁰

g⁰

X ^f //Y.

LetCbe an∞-category, and assume that for each y∈Y the∞-categoryCadmits limits indexed by the Kan complex X×Y Yy/. Then the diagram of∞-categories

C^Y //

C^X

C^Y⁰ //C^X⁰

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is right adjointable.

Proof. Let F : X → C be a functor; we wish to show that the canonical map g^0∗f_∗F → f⁰_∗g^∗F is an equivalence inC^Y⁰. Unwinding the definition, we must show that for each vertexy⁰ ∈Y⁰, the map

lim←−(F|X×Y Yg⁰(y)/)→lim←−(F|X⁰×Y⁰Y_y/⁰ )

is an equivalence in C. This follows from the fact that the mapX⁰×Y⁰Y_y/⁰ →X×Y Y_g⁰_(y)/ is a homotopy equivalence, since we have assumed thatσis a homotopy pullback diagram.

Remark 1.1.6. LetX be a Kan complex. The stable∞-category Mod^X_k admits an accessible t-structure ((Mod^X_k)≥0,(Mod^X_k)≤0), where (Mod^X_k)≥0 is the full subcategory of Mod^X_k spanned by those functors F : X → Modk such that F(x) ∈ (Modk)≥0 for all x∈ X, and (Modk)≤0 is defined similarly. The heard of Mod^X_k can be identified with the category of functors from the fundamental groupoid of X to the category of vector spaces overk. IfX is connected and contains a vertexx, then we can identify the heart of Mod^X_k with the category of representations (ink-vector spaces) of the fundamental groupπ1(X, x).

If f : X →Y is a map of Kan complexes, then the functor f^∗ : Mod^Y_k →Mod^X_k is t-exact. It follows that the functorf_∗: Mod^X_k →Mod^Y_k is left t-exact and the functor f!: Mod^X_k →Mod^Y_k is right t-exact.

Definition 1.1.7. LetX be a connected Kan complex containing a vertexx. We will say that an object F∈Mod^X_k isnilpotentif the following conditions are satisfied:

(1) For each integern, regardV =πnFas a representation of the fundamental groupG=π1(X, x). Then V has a finite filtration by subrepresentations

0 =V₀⊆V₁⊆ · · · ⊆V_m=V

such that each quotientVi/V_i−1 is isomorphic tok(endowed with the trivialG-action).

(2) The homotopy groupsπnFvanish forn0.

Remark 1.1.8. In the situation of Definition 1.1.7, the condition that Fbe nilpotent does not depend on the choice of vertexx∈X.

Proposition 1.1.9. Let f : X → Y be a map of Kan complexes. Assume that Y is connected and let F∈Mod^Y_k be nilpotent. Then the canonical map

θ_F :C^∗(Y;F)⊗_C∗(Y;k)C^∗(X;k)→C^∗(X;f^∗F) is an equivalence inModk.

Proof. Let C ⊆ Mod^Y_k be the full subcategory spanned by those objects F ∈ Mod^Y_k for which θ_F is an equivalence. It is clear thatCis a stable subcategory of Mod^Y_k which containskY.

Let F∈ Mod^Y_k be nilpotent; we wish to show that F∈C. Replacing F by a shift if necessary, we can assume thatF∈(Mod^Y_k)_≤0. LetK_F be the cofiber of the mapθ_F. We will show thatK_F ∈(Modk)_≤−n for every integern, so thatK_F '0. To prove this, we observe that the fiber sequence

τ_≥−nF→F→τ_≤−n−1F in Mod^Y_k determines a fiber sequence

Kτ_≥−nF→K_F →Kτ_≤−n−1F

in Mod_k. For each n≥0, the truncation τ_≥−nFis a successive extension of (shifted) copies ofk_Y, so that τ_≥−nF∈ C and thereforeK_τ_≥−n_F '0. It follows thatK_F 'K_τ_≤−n−1_F. We may therefore replace F by τ_≤−n−1Fand thereby assume thatπ_iF'0 for i≥ −n.

To prove thatK_F∈(Modk)_≤−n, it will suffice to show thatC^∗(X;f^∗F) andC^∗(Y;F)⊗_C∗(Y;k)C^∗(X;k) belong to (Modk)_≤−n−1. In the first case, this follows from the left t-exactness of the functorC^∗(X;•). In the second case, it follows from the left t-exactness of the functorC^∗(Y;•) together with Corollary VIII.4.1.11 (note thatC^∗(Y;k) is a coconnectivek-algebra, since we have assumed thatY is connected).

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Corollary 1.1.10. Letk be a field, and let f :X →Y be a Kan fibration between Kan complexes. Assume that:

(a) The set of componentsπ0Y is finite.

(b) For each point y∈Y and each n≥0, the vector space V = Hⁿ(Xy;k)admits a finite filtration 0 =V0⊆V1⊆ · · · ⊆Vm⊆0

byπ₁(Y, y)-invariant subspaces such that each quotientV_i/V_i−1is a one-dimensional vector space over k endowed with the trivial action ofπ1(Y, y).

Then, for any mapg:Y⁰→Y, the canonical map

θ:C^∗(Y⁰;k)⊗C^∗(Y;k)C^∗(X;k)→C^∗(Y⁰×Y X;k) is an equivalence.

Proof. Using (a), we can immediately reduce to the case whereY is connected. Using Proposition 1.1.9, we can identify θ with the canonical map C^∗(Y⁰;g^∗f_∗k_X)→C^∗(Y⁰;f_∗⁰g^0∗k_X), wheref⁰ :Y⁰×Y X →Y⁰ and g⁰:Y⁰×Y X →X denote the projection maps. It follows from Lemma 1.1.5 thatθ is an equivalence.

Remark 1.1.11. Under the hypotheses of Corollary 1.1.10, Proposition A.7.2.1.19 gives a spectral sequence {E^p,q_r }r≥2 with E₂^p,∗ 'Tor^H_p^∗^(Y^;k)(H^∗(X;k),H^∗(Y⁰;k)) which converges to H^∗(Y⁰×Y X;k). This spectral sequence is called thecohomological Eilenberg-Moore spectral sequence.

Corollary 1.1.12(K¨unnethFormula). Letkbe a field, letX andZ be Kan complexes, and assume that the cohomology groupsHⁿ(X;k)are finite dimensional forn≥0. Then the canonical map

C^∗(X;k)⊗kC^∗(Z;k)→C^∗(X×Z;k) is an equivalence of E∞-algebras overk.

Proof. Apply Corollary 1.1.10 in the caseY = ∆⁰,Y⁰=Z.

Remark 1.1.13. In the situation of Corollary 1.1.12, the spectral sequence of Remark 1.1.11 degenerates, and we obtain the usual K¨unnethisomorphisms

Hⁿ(X×Z;k)' M

i+j=n

Hⁱ(X;k)⊗kH^j(Z;k).

Forn≥1 andA an abelian group, we letK(A, n) denote the associated Eilenberg-MacLane space: it is characterized up to equivalence by the requirements thatK(A, n) be connected and have homotopy groups

πiK(A, n)'

(A ifi=n 0 otherwise.

WhenAis a fieldk, we have a tautological cohomology class ηn ∈Hⁿ(K(A, n);k),

which classifies a mapk[−n]→C^∗(K(A, n), k) in Mod_k and therefore a map Sym^∗(k[−n])→C^∗(K(A, n);k) in CAlg_k.

Proposition 1.1.14. For each n≥1, the map

φ: Sym^∗(Q[−n])→C^∗(K(Q, n);Q) is an equivalence of E∞-rings.

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Lemma 1.1.15. Letkbe a field, letAbe a coconnectiveE1-algebra overk. Letf :M →M⁰ be a morphism in RModA and let N ∈ LModA. Assume thatN is nonzero, that πiM 'πiM⁰ 'πiN '0 fori >0, and that the induced mapM ⊗AN →M⁰⊗AN is an equivalence. Thenf is an equivalence.

Proof. LetKbe the fiber off, and assume (for a contradiction) thatK6= 0. Then there exists some largest integernsuch thatπ_nK6= 0. SinceN 6= 0, there exists some largest integermsuch thatπ_mN 6= 0. Corollary VIII.4.1.11 implies that the map

(πn)K⊗k(πmN)→πm+n(K⊗AN)'0 is injective, which is a contradiction.

Proof of Proposition 1.1.14. We proceed by induction onn. Consider first the casen= 1. The spaceK(Q,1) is given by the filtered limit of the diagram

K(Z,1)→K(1

2Z,1)→K(1

6Z,1)→K(1

24Z,1)→ · · ·

Each space appearing in this diagram can be identified with the circle S¹, and each map in the diagram induces an isomorphism on rational cohomology. It follows that

Hⁱ(K(Q,1);Q)'Hⁱ(S¹;Q)'

(Q ifi∈ {0,1}

0 otherwise.

Form≥0, we can identify Sym^m(Q[−1]) withV[−m], whereV is themth exterior power ofQ(as a vector space over Q). It follows that Sym^m(Q[−1]) '0 for m ≥ 2, so that the map ψ : L

i≤1Symⁱ(Q[−1]) → Sym^∗(Q[−1]) is an equivalence. It is easy to see thatφ◦ψis an equivalence, so thatφis an equivalence as well.

Now suppose thatn≥2. We have a homotopy pullback diagram of spaces K(Q, n−1) //

∗

∗ //K(Q, n).

The inductive hypothesis guarantees that each cohomology group Hⁱ(K(Q, n−1);Q) is finite dimensional, so that Corollary 1.1.10 gives an equivalence

C^∗(K(Q, n−1);Q)'Q⊗_C∗(K(Q,n);Q)Q.

LetV =C^∗(K(Q, n);Q)⊗Sym^∗(Q[−n])Q. It follows from Corollary VIII.4.1.11 thatπiV '0 fori >0. Using the inductive hypothesis, we deduce that

C^∗(K(Q, n−1);Q)'Sym^∗(Q[1−n])'Q⊗_Sym^∗_(Q[−n])Q'Q⊗_C^∗_(K(Q,n);Q)V.

SinceK(Q, n) is connected,C^∗(K(Q, n);Q) is coconnective and Lemma 1.1.14 impliesφinduces an equiv- alenceV 'Q. Using Lemma 1.1.14 again, we deduce thatφis itself an equivalence.

Corollary 1.1.16. Let V be a finite-dimensional vector space over Qand let V^∨ denote the dual space of V. Then forn≥1, the canonical map ofE∞-algebras overQ

Sym^∗(V^∨[−n])→C^∗(K(V, n);Q) is an equivalence.

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1.2 k-Rational Homotopy Types

Letkbe a field of characteristic zero, which we regard as fixed throughout this section. We let CAlg_k denote the ∞-category of E∞-algebras over k, CAlg^cn_k the full subcategory of CAlg_k spanned by the connective E∞-algebras overk, and CAlg⁰_k the full subcategory of CAlg^cn_k spanned by the discrete E∞-algebras over k (so that CAlg⁰_k is equivalent to the nerve of the ordinary category of commutativek-algebras).

Definition 1.2.1. LetX : CAlg^cn_k →S be a functor. We will say thatX is a k-rational homotopy typeif the following conditions are satisfied:

(a) The functorX is a left Kan extension of its restrictionX0=X|CAlg⁰_k to discrete E∞-algebras.

(b) For everyR∈CAlg⁰_k, the spaceX₀(R) is simply connected.

(c) For every integer n≥2, there exists a vector space V over ksuch that the functor R 7→πnX0(R) is given byR7→V ⊗kR.

We let RType(k) denote the full subcategory of Fun(CAlg^cn_k ,S) spanned by thek-rational homotopy types.

We will refer to RType(k) as the∞-category ofk-rational homotopy types.

Example 1.2.2. IfV is a vector space over k, we define a functorK(V, n) : CAlg^cn_k →S by the formula K(V, n)(R) = Ω^∞−n(V ⊗kR).

Ifn≥2, thenK(V, n) is ak-rational homotopy type. IfV is finite-dimensional, thenK(V,0) is the functor corepresented by the discreteE∞-algebra Sym^∗V^∨.

Our goal in this section is to study the∞-category RType(k) and establish some of its properties.

Remark 1.2.3. Let Vect_kdenote the category of vector spaces overk, letAbdenote the category of abelian groups, and define a functorT : Vect_k →Fun(CAlg⁰_k,Ab) by the formula

T(V)(R) =V ⊗kR.

If V is finite dimensional, then T carriesV to the functor represented by the group scheme SpecV^∨. In particular, T carries finite-dimensional vector spaces to compact objects of Fun(CAlg⁰_k,Ab). Since k has characteristic zero, the functorT is fully faithful when restricted to finite dimensional vector spaces. Since T commutes with filtered colimits, we conclude thatT is fully faithful in general.

Now suppose that X : CAlg^cn_k → S is a k-rational homotopy type and n ≥2 an integer. Then there exists a vector space V over k such that the functor R 7→X(R) is given by V ⊗_kR for R ∈CAlg⁰_k. The above argument shows that the vector space V is determined by X (up to unique isomorphism). We will emphasize this dependence by writingV =πnX.

Definition 1.2.4. LetX be a k-rational homotopy type. We will say that X is of finite typeif the vector spacesπ_nX are finite-dimensional for eachn≥2.

Remark 1.2.5. Recall that a functorX: CAlg^cn_k →Sis said to be acoaffine stackif it is corepresentable by a coconnectiveE∞-algebraA∈CAlg_k. We will say that a coaffine stackX issimply connectedifπ−1A'0.

Using Propositions VIII.4.4.8 and VIII.4.4.6, we see that an arbitrary functorX : CAlg^cn_k →S is a simply connected coaffine stack if and only if it satisfies conditions (a) and (b) of Definition 1.2.1, together with the following version of (c):

(c⁰) For every integern≥2, there exists a vector space W overk such that the functorR7→πnX0(R) is given byR7→Homk(W, R). HereX0 denotes the restrictionX|CAlg⁰_k.

We say that a simply connected coaffine stackX is offinite typeif the vector spacesW appearing in (c⁰) are finite-dimensional. The following conditions on a functorX: CAlg^cn_k →S are equivalent:

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(i) The functorX is ak-rational homotopy type of finite type.

(ii) The functorX is a coaffine stack which is simply connected and of finite type.

In particular, ifX is a k-rational homotopy type of finite type, thenX is a coaffine stack.

We now establish some basic formal properties ofk-rational homotopy types.

Proposition 1.2.6. Letkbe a field of characteristic zero and letX : CAlg^cn_k →Sbe ak-rational homotopy type. ThenX commutes with sifted colimits.

Lemma 1.2.7. LetA_• be a simplicial abelian group, and suppose that the unnormalized chain complex

· · · →A2→A1→A0

is an acyclic resolution of some abelian groupA. Then for each integern≥0, the canonical map

|K(A_•, n)| →K(A, n)

is a homotopy equivalence.

Proof. We proceed by induction on n. If n = 0, then |K(A•, n)| can be identified with the geometric realization|A•|, and its homotopy groups are computed by the chain complex

· · · →A₂→A₁→A₀.

Now suppose thatn >0, and consider the mapθ_n:|K(A_•, n)| →K(A, n). Using Corollary A.5.1.3.7, we see that the map Ω(θ_n) : Ω|K(A_•, n)| →ΩK(A, n) can be identified withθ_n−1, which is a homotopy equivalence by the inductive hypothesis. Since the spaces|K(A_•, n)|andK(A, n) are both connected, we conclude that θn is a homotopy equivalence.

Proof of Proposition 1.2.6. Let Poly_k denote the full subcategory of CAlg^cn_k spanned by those k-algebras of the form k[x1, . . . , xn] for some n ≥0. Then Poly_k is a subcategory of compact projective generators for CAlg^cn_k : that is, the inclusion Poly_k ,→ CAlg^cn_k extends to an equivalence PΣ(Poly_k) ' CAlg^cn_k (see Proposition A.7.1.4.20). Let X0 =X|Poly_k and let X⁰ : CAlg^cn_k → S be a functor which extendsX0 and commutes with sifted colimits. ThenX⁰ is a left Kan extension of X0, so that the identity transformation fromX⁰ to itself extends to a natural transformationα:X⁰→X. We wish to show thatαis an equivalence, or equivalently that X is a left Kan extension of X0. Since X is a left Kan extension of X|CAlg⁰_k, it will suffice to show that X|CAlg⁰_k is a left Kan extension ofX0 (Proposition T.4.3.2.8). For this, it suffices to show that for every objectR∈CAlg⁰_k, the canonical mapX⁰(R)→X(R) is an equivalence.

We first treat the case where R= Sym^∗V for somek-vector spaceV. ThenR'lim

−→Sym^∗V_α, where the colimit is taken over the filtered partially ordered set of all finite-dimensional subspacesV_α⊆V. We then have a commutative diagram

lim−→X⁰(Sym^∗Vα) //

X⁰(R)

lim−→X(Sym^∗Vα) //X(R).

Since each Sym^∗Vαbelongs to Poly_k, the left vertical map is a homotopy equivalence. The upper horizontal map is a homotopy equivalence because the functor X⁰ commutes with filtered colimits. It will therefore suffice to show that the lower horizontal map is a homotopy equivalence. Since the domain and codomain of this map are both simply connected, we are reduced to proving that the map

lim−→π_nX(Sym^∗V_α)→π_nX(R)

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is an isomorphism for eachn≥2. This is clear, since the functorW 7→(πnX)⊗k(Sym^∗W) commutes with filtered colimits.

Now suppose thatR∈CAlg⁰_k is arbitrary. Choose a representation ofRas the geometric realization of a simplicial objectR•of CAlg^cn_k , where eachRna polynomial algebra (on a possibly infinite set of generators) overk. We then have a commutative diagram

|X⁰(R_•)| //

X⁰(R)

|X(R_•)| //X(R).

The upper horizontal map is a homotopy equivalence because X⁰ commutes with sifted colimits, and the left vertical map is a homotopy equivalence by the previous step in the proof. We are therefore reduced to proving that the mapθ:|X(R_•)| →X(R) is a homotopy equivalence.

θ_m:|τ_≤mX(R_•)| →τ_≤mX(R)

is a homotopy equivalence. We proceed by induction onm, the casem≤1 being trivial. We have a simplicial diagram of fiber sequences

τ≤mX(R•)→τ≤m−1X(R•)→K((πmX)⊗kR•, m+ 1).

Since each of the spacesK((π_mX)⊗_kR_•, m+ 1) is connected, Lemma A.5.3.6.17 gives a fiber sequence of geometric realizations

|τ_≤mX(R_•)| → |τ_≤m−1X(R_•)| → |K((πmX)⊗kR_•, m+ 1)|.

Consequently, to show thatθmis an equivalence, it will suffice to show that the vertical maps in the diagram

|τ_≤m−1X(R_•)| //

|K((π_mX)⊗ −kR_•, m+ 1)|

τ≤m−1X(R) //K((πmX)⊗kR•, m+ 1)

are homotopy equivalences. For the left vertical map this follows from the inductive hypothesis, and for the right vertical map it follows from Lemma 1.2.7.

Corollary 1.2.8. Let X be a k-rational homotopy type. For each R ∈CAlg^cn_k , the space X(R) is simply connected.

Proof. WriteR as a geometric realization |R_•|, where each R_n is discrete. Proposition 1.2.6 implies that X(R)' |X(R_•)|. Since each of the spacesX(R_n) is simply connected, we conclude that X(R) is simply connected.

Corollary 1.2.9. Let X bek-rational homotopy type, and suppose we are given a pullback diagramσ: A⁰ //

A

B⁰ //B

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inCAlg^cn_k . Suppose that the right vertical map induces a surjection π0A→π0B. Then the diagramX(σ) X(A⁰) //

X(A)

X(B⁰) //X(B) is also a pullback square.

Proof. Choose a base point η ∈ X(k). Using η, we can liftX to a functor CAlg^cn_k →S∗ taking values in the∞-category of pointed spaces. Choose a simplicial object B_• of CAlg^cn_k whose geometric realization is equivalent toB, where eachBn is discrete. Set

A_•=A×BB_• B⁰_•=B⁰×BB_• A⁰_•=A⁰×BB_•. Thenσcan be obtained as the geometric realization of a simplicial diagramσ_•:

A⁰_• //

A_•

B⁰_• //B_•.

Proposition 1.2.6 implies thatX commutes with sifted colimits. It follows thatX(σ) can be identified with the geometric realization|X(σ•)|. We wish to show that|X(σ•)|is a pullback diagram. Each of the spaces X(B_•) is connected (and equipped with a base point determined by our chosen pointη∈X(k)). Applying Lemma A.5.3.6.17, we are reduced to proving that each of the diagramsX(σ_n) is a pullback square. We may therefore replaceBbyB_n and thereby reduce to the case whereB is discrete. Using a similar argument, we may suppose thatA andB⁰ are also discrete (so thatA⁰ 'A×_BB⁰ is likewise discrete). LetY denote the fiber productX(A)×_X(B)X(B⁰). The homotopy groups ofY fit into a long exact sequence

πn+1X(A)×πn+1X(B⁰)→πn+1X(B)→πnY →πnX(A)×πnX(B⁰)→πnX(B).

SinceX is ak-rational homotopy type, we can rewrite this sequence as

(πn+1X)⊗k(A⊕B⁰)→^α (πn+1X)⊗kB →πnY →(πnX)⊗k(A⊕B⁰)→^β (πnX)⊗kB.

Since the mapA→B is surjective,αis a surjection for eachn≥1. It follows thatY is simply connected and that for each n ≥ 2 we have a canonical isormophismπ_nY 'ker(β) ' (π_nX)⊗_k A⁰. From this, we immediately deduce that the mapX(A⁰)→Y is a homotopy equivalence.

Corollary 1.2.10. Let X be a k-rational homotopy type and letf :A→B be a morphism inCAlg^cn_k . Iff isn-connective for some n≥0, then the induced map X(A)→X(B)is(n+ 2)-connective.

Proof. We proceed by induction onn. Ifn >0, then the diagonal mapδ:A→A×BAis (n−1)-connective.

The inductive hypothesis then implies thatX(δ) :X(A)→X(A×BA) is (n+1)-connective. Using Corollary 1.2.9 we can identifyX(δ) with the diagonal mapX(A)→X(A)×X(B)X(A). SinceX(f) :X(A)→X(B) is surjective on connected components (both spaces are simply connected, by Corollary 1.2.8), the (n+ 1)- connectivity ofX(A)→X(A)×_X(B)X(A) implies that (n+ 2)-connectivity ofX(f).

It remains to treat the case n= 0. Since X(A) andX(B) are simply connected, it suffices to show that the mapX(f) induces a surjectionπ₂X(A)→π₂X(B). Choose an equivalenceB' |B_•|for some simplicial objectB_•of CAlg^cn_k , where eachBn is a polynomial algebra overk(possibly on an infinite set of generators).

Proposition 1.2.6 implies thatX(B) is equivalent to the geometric realization of the simplicial spaceX(B_•), where eachX(Bn) is simply connected. It follows that the mapπ2X(B0)→π2X(B) is surjective. Sincef is connective, the mapB0→B factors throughA, so that the mapπ2X(A)→π2X(B) is also surjective.

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Corollary 1.2.11. LetX be ak-rational homotopy type. For everyA∈CAlg^cn_k , the canonical mapX(A)→ lim←−X(τ_≤nA)is a homotopy equivalence.

We now turn our attention to the compactness properties ofk-rational homotopy types of finite type.

Lemma 1.2.12. Let X be ak-rational homotopy type of finite type. Then, for each n≥0, the functor V 7→Map_Fun(CAlgcn

k,S)(X, K(V, n)) commutes with filtered colimits,

Proof. Since X is a coaffine stack (Remark 1.2.5), Proposition VIII.4.4.4 implies that X is given by the geometric realization of a simplicial objectX_•of Fun(CAlg^conn_k ,S), where eachXn is the set-valued functor corepresented by some discretek-algebraRⁿ. SinceK(V, n) isn-truncated, the mapping space

Map_Fun(CAlg0

k,S)(X, K(V, n)) is equivalent to the finite limit

lim←−

[m]∈∆≤n+1

Map_Fun(CAlgcn

k,S)(Xm, K(V, n)).

It will therefore suffice to show that each of the functors V 7→Map_Fun(CAlgcn

k,S)(X_m, K(V, n))'K(R^m⊗_kV, n) commutes with filtered colimits, which is clear.

Definition 1.2.13. LetX be ak-rational homotopy type and letn≥0 be an integer. We will say thatX is n-truncated if, for eachR ∈CAlg⁰_k, the space X(R) is n-truncated. We let RType(k)_≤n denote the full subcategory of RType(k) spanned by the then-truncatedk-algebraic homotopy types.

Remark 1.2.14. The inclusion functor RType(k)_≤n,→RType(k) admits a left adjoint X 7→τ_≤nX, where τ_≤nX denotes a left Kan extension of the functor CAlg⁰_k →Sgiven byR7→τ_≤nX(R).

Remark 1.2.15. LetX be ann-truncatedk-rational homotopy type and letV =π_nX. To eachR∈CAlg⁰_k we can associate a fiber sequence

X(R)→τ_≤n−1X(R)→K(V ⊗kR, n+ 1),

depending functorially on R. Taking left Kan extensions, we obtain a commutative diagram of k-rational homotopy typesσ

X //

τ≤n−1X

∗ //K(V, n+ 1).

Let C ⊆ CAlg^cn_k denote the full subcategory spanned by those connective E∞-algebras A for which the diagram

X(A) //

(τ_≤n−1X)(A)

∗ //K(V, n+ 1)(A).

is a pullback square. Using Lemma A.5.3.6.17, we see thatCis closed under sifted colimits. SinceCcontains CAlg⁰_k, it follows thatC= CAlg^cn_k . That is, we have a fiber sequence of functors

X →τ_≤n−1X →K(V, n+ 1).

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Lemma 1.2.16. Let k be a field of characteristic zero. Then the ∞-category RType(k) is closed under filtered colimits in Fun(CAlg^cn_k ,S). Moreover, for eachn≥2, the functorX 7→πnX commutes with filtered colimits. In particular, of each of the subcategoriesRType(k)≤n is also closed under filtered colimits.

Proof. Let {Xα} be a filtered diagram of k-rational homotopy types, and let X denote its colimit in Fun(CAlg^cn_k ,S). It is clear thatXsatisfies conditions (a) and (b) of Definition 1.2.1. Moreover, forR∈CAlg⁰_k, we have a canonical isomorphism

π_nX(R)'lim

−→π_n(X_α(R))'(π_nX_α)⊗kR'V ⊗kR, where V denotes the direct limit lim

−→π_nX_α (see Remark 1.2.3). It follows thatX is ak-rational homotopy type and that the canonical map

lim−→πnXα=V →πnX is an isomorphism ofk-vector spaces.

Lemma 1.2.17. Let k be a field of characteristic zero, let n≥0, and let X be ann-truncated k-algebraic homotopy type. IfX is of finite type, thenX is a compact object of RType(k)_≤n.

Proof. Letθ : RType(k)→ S denote the functor corepresented byX. We will prove that for each integer m, the restrictionθ|RType(k)≤mcommutes with filtered colimits. We proceed by induction onm, the case m ≤ 1 being trivial. To carry out the inductive step, suppose we are given a filtered diagram {Yα} in RType(k)≤mhaving colimitY, and setVα=πnYα. We then have a fiber sequence of functors

Y_α→τ_≤n−1Y_α→K(V_α, n+ 1) depending functorially on α. Set V = lim

−→V_α so that V ' π_nY. We then have another fiber sequence Y →τ_≤n−1Y →K(V, n+ 1). Sinceθcommutes with limits, we have a map of fiber sequences

lim−→θ(Yα) //

α

lim−→θ(τ≤n−1Yα) //

β

lim−→θ(K(Vα, n+ 1))

γ

θ(Y) //θ(τ_≤n−1Y) //θ(K(V, n+ 1)).

The map β is a homotopy equivalence by the inductive hypothesis, and γ is a homotopy equivalence by Lemma 1.2.12. It follows thatαis a homotopy equivalence, as desired.

For each integer n≥0, we let RType(k)^ft_≤n denote the intersection RType(k)^ft∩RType(k)_≤n: that is, the∞-category ofk-rational homotopy types whcih aren-truncated and of finite type.

Proposition 1.2.18. Letkbe a field of characteristic zero. For each integern, the inclusionRType(k)^ft_≤n,→ RType(k)_≤n induces an equivalence of ∞-categoriesθ: Ind(RType(k)^ft_≤n)'RType(k)_≤n.

Remark 1.2.19. Since RType(k)^ft_≤n is closed under retracts in RType(k)_≤n, it follows from Proposition 1.2.18 and Lemma 1.2.17 that an objectX ∈RType(k)_≤n is compact if and only if it is finite type.

Remark 1.2.20. One can show that the∞-category RType(k) itself is compactly generated, but we will not need this.

Proof. Using Lemma 1.2.17 and Proposition T.5.1.3.1, we deduce thatθis fully faithful. LetC⊆RType(k)_≤n denote the essential image of θ. We wish to show that Ccontains everyn-truncatedk-algebraic homotopy typeX. The proof proceeds by induction on n, the casen≤1 being trivial. SetV =πnX so that we have a fiber sequence of functors

X →τ≤n−1X →K(V, n+ 1).

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Using the inductive hypothesis, we can writeτ_≤n−1X as a filtered colimit lim

−→Yα, where eachYα is (n−1)- truncated and of finite type. For eachα, letXαdenote the fiber of the composite map

Y_α→τ_≤n−1X →K(V, n+ 1).

ThenX 'lim

−→Xα. Since Cis closed under filtered colimits, it will suffice to prove that eachXα belongs to C. Using Lemma 1.2.12, we see that there exists a finite-dimensional subspaceW ⊆V such that the map Yα → K(V, n+ 1) factors through K(W, n+ 1). Write V ' W ⊕W⁰, so that we have an equivalence of functors

Xα'Zα×K(W⁰, n)

where Z_α denotes the fiber of the map Y_α → K(W, n+ 1). Then Z_α is of finite type. Since C is closed under products, we are reduced to proving that K(W⁰, n) belongs to C. This is clear, since K(W⁰, n) ' lim−→K(W_β⁰, n), whereW_β⁰ ranges over all finite-dimensional subspaces ofW⁰.

Proposition 1.2.21. Let kbe a field of characteristic zero. Then the canonical map fromRType(k)to the homotopy limit of the tower

· · · →RType(k)_≤3^τ→^≤2RType(k)_≤2→RType(k)_≤1' ∗ is an equivalence.

Proof. Let RType⁰(k) denote the full subcategory of Fun(CAlg⁰_k,S) spanned by those functors which satisfy conditions (b) and (c) of Definition 1.2.1, and for each integer n ≥0 define RType⁰(k)_≤n similarly. Using Proposition T.4.3.2.15, we see that the restriction maps

RType(k)→RType⁰(k) RType(k)_≤n →RType⁰(k)_≤n

are equivalences. It will therefore suffice to show that RType⁰(k) is equivalent to the homotopy inverse limit of the tower of∞-categories

· · · →RType⁰(k)_≤3^τ→^≤2RType⁰(k)_≤2→RType⁰(k)_≤1' ∗.

This follows from the observation that Fun(CAlg⁰_k,S) is given by the homotopy inverse limit of the tower {Fun(CAlg⁰_k, τ_≤nS).

1.3 Rational Homotopy Theory and E

^∞

-Algebras

LetY be a simply connected rational space. We will say thatX is offinite typeif the homotopy groupπnX is a finite-dimensional vector space overQfor eachn≥2. Sullivan has shown that the homotopy theory of rational spaces of finite type can be described algebraically, using commutative differential graded algebras overQ. His main result can be formulated as follows:

Theorem 1.3.1 (Sullivan). Let Qbe the field of rational numbers. Let S^ratft denote the full subcategory of Sspanned by those spacesX which are simply connected and such that each homotopy group πiX is a finite dimensional vector space over Q. Then the construction X 7→C^∗(X;Q) determines a fully faithful embedding S^ratft →CAlg_Q, whose essential image is the collection of those Q-algebras A satisfying the following conditions:

(a) The Q-vector spacesπ_iAare finite dimensional for all i.

(b) We have π_iA'







0 if i >0 Q if i= 0 0 if i=−1.

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In this section, we will review the proof of Theorem 1.3.1, and deduce from it an analogous description of theentire∞-categoryS^rat of rational spaces (Theorem 1.3.6).

Remark 1.3.2. Letk be a field. The constructionX 7→C^∗(X;k) determines a functor from Sto CAlg^op_k . This functor admits a right adjoint, given by the formula A 7→ Map_CAlg

k(A, k). In particular, for every space X we have a canonical unit map X →Map_CAlg

k(C^∗(X;k), k), which carries a vertex x∈ X to the evaluation mapC^∗(X;k)→C^∗({x};k)'k.

Proposition 1.3.3. Let X be a simply connected rational space of finite type. Then the unit map u_X:X →Map_CAlg

Q(C^∗(X;Q),Q) is a homotopy equivalence.

Proof. The mapX →τ_≤nX induces an isomorphism on homotopy groups in degrees≤n, hence an isomorphism Hⁱ(X;Q)→Hⁱ(τ_≤nX;Q) fori≤n. It follows thatC^∗(X;Q)'lim

−→C^∗(τ_≤nX;Q), so thatuX can be identified with the limit of the maps {uτ≤nX}n≥0. It will therefore suffice to prove that each of the maps uτ≤nX is a homotopy equivalence. We proceed by induction onn, the casen= 1 being obvious. Let n >1 and set V =πnX, so that V is a finite-dimensional vector space over Q. We have a pullback diagram of spaces

τ_≤nX //

∗

τ_≤n−1X //K(V, n+ 1) so that Corollary 1.1.10 yields a pushout diagram

C^∗(K(V, n+ 1),Q) //

Q

C^∗(τ≤n−1X;Q) //C^∗(τ≤nX;Q).

Consequently, to show thatu_τ_≤n_Xis a homotopy equivalence, it suffices to show thatu_τ_≤n−1_X andu_K(V,n+1) are homotopy equivalences. In the first case, this follows from the inductive hypothesis. In the second, it follows from Corollary 1.1.16.

Corollary 1.3.4. Let X be a simply connected rational space of finite type, and let Y be an arbitrary space.

Then the functorC^∗(•;Q) induces a homotopy equivalence Map_S(Y, X)→Map_CAlg

Q(C^∗(X;Q), C^∗(Y;Q)).

Proof of Theorem 1.3.1. Corollary 1.3.4 implies thatC^∗|S^ratft is fully faithful. We next show that ifX ∈S^ratft , then C^∗(X;Q) satisfies conditions (a) and (b). Since X is simply connected, condition (b) is obvious. We wish to prove that H⁻ⁱ(X;Q) is finite dimensional for every integer i. ReplacingX byτ_≤nX forn≥i, we can assume thatX isn-truncated, and we proceed by induction onn. Ifn= 1, then X ' ∗and the result is obvious. Otherwise, setV =πnX so that we have a pullback diagram

X //

∗

τ≤n−1X //K(V, n+ 1)

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and therefore (by Corollaries 1.1.10 and 1.1.16) a pushout diagram ofQ-algebras Sym^∗(V[−n−1]) //

Q

C^∗(τ_≤n−1X;Q) //C^∗(X;Q).

The desired result now follows from the inductive hypothesis and Lemma X.4.1.16.

We now show that if A ∈ CAlg_Q satisfies conditions (a) and (b), then A belongs to the essential of C^∗|S^ratft . Let X = cSpecA be the coaffine stack determined by A. Proposition VIII.4.4.18 implies thatX has finite type, so that each homotopy group πiX is representable by a commutative unipotent algebraic group over Q and therefore πiX(Q) is a finite dimensional vector space over Q for each i. Proposition VIII.4.4.12 implies that X(Q) is simply connected, so that X(Q) ∈ S⁰. Let A⁰ = C^∗(X(Q);Q), and let X⁰ be the coaffine stack determined by A⁰. The canonical map A → A⁰ induces a map of coaffine stacks v : X⁰ → X. Since C^∗|S⁰ is fully faithful, the induced map X⁰(Q) → X(Q) is a homotopy equivalence.

For eachi≥1, we have a mapπ_i(v) :π_iX⁰ →π_iX of unipotent algebraic groups overQ which induces an isomorphism on Q-points. It follows thatπ_i(v) is an isomorphism of algebraic groups and therefore thatv is an equivalence of coaffine stacks, so thatA'A⁰=C^∗(X(Q);Q) lies in the essential image ofC^∗|S^ratft as desired.

In the language of coaffine stacks, Theorem 1.3.1 has the following interpretation:

Theorem 1.3.5. LetC⊆Fun(CAlg^cn_Q,S)denote the full subcategory spanned by the simply connected coaffine stacks of finite type. The evaluation map X7→X(Q)determines an equivalence of ∞-categoriesC→S^ratft . Proof. It is clear that ifX ∈ C, the X(Q)∈ S^ratft . Let D denote the full subcategory of CAlg_Q spanned by those E∞-algebras A ∈CAlg_Q satisfying conditions (a) and (b) of Theorem 1.3.1. Using Propositions VIII.4.4.12 and VIII.4.4.18, we see that the constructionA7→cSpecAinduces an equivalence of∞-categories D^op → C. It will therefore suffice to prove that the composite functor D^op →C→ C⁰ is an equivalence of

∞-categories. Unwinding the definitions, we see that this composite functor is given by A7→Map_CAlg

Q(A,Q).

We now observe that this construction is right adjoint to the equivalence of ∞-categories appearing in Theorem 1.3.1.

We now use the theory ofk-rational homotopy types developed in§1.2 to remove the finiteness restrictions from Theorem 1.3.5.

Theorem 1.3.6. LetQdenote the field of rational numbers. Then the constructionX 7→X(Q)induces an equivalence of∞-categoriese: RType(Q)→S^rat.

Proof. We first show that the functorX 7→X(Q) is fully faithful. Fix Q-rational homotopy types X and Y; we wish to show that the canonical map

uX,Y : Map_RType(Q)(X, Y)→Map_S(X(Q), Y(Q))

is a homotopy equivalence. Note thatuX,Y is given by the homotopy limit of the tower of maps{uX,τ_≤nY}_n≥0. It will therefore suffice to show that each of the maps uX,τ_≤nY is a homotopy equivalence. We proceed by induction onn, the casen≤1 being trivial. To carry out the inductive step, write πnY(R)'V ⊗QR for someQ-vector spaceV. We then have a fiber sequence of functors

τ_≤nY →τ_≤n−1Y →K(V, n+ 1).

Derived Algebraic Geometry XIII: Rational and p-adic Homotopy Theory