E_{∞} RING THEORY

J.P. MAY

http://www.math.uchicago.edu/ may/RANT/

Date: March 13, 2008.

[figure: PERM CATS; B; BIPERM CATS; B; E_{∞} SPACES; E_{∞} RING SPACES; SPACES; C; Σ^{∞}; BLACK BOX; E_{∞} SPACES; C; Σ^{∞}; SPECTRA; Ω^{∞}; E_{∞} RING SPECTRA; Ω^{∞}]

E_{∞} RING THEORY 3

[figure: PERM CATS; B; BIPERM CATS; B; ℱ − SPACES; (ℱ ∫ ℱ) − SPACES; 𝒞̂ − SPACES; (𝒢̂ ∫ 𝒞̂) − SPACES; (𝒞̂, 𝒢̂) − SPACES; 𝒞 − SPACES; (𝒞 , 𝒢) − SPACES; E_{∞} SPACES; E_{∞} RING SPACES]

4 J.P. MAY

E_{∞} ring spaces

\underline{OPERAD}: Σ_{j}-spaces 𝒪(j), 𝒪(0) = \{∗\} (basepts), id ∈ 𝒪(1) (identity operation), γ: 𝒪(k)×𝒪(j_{1})×⋯×𝒪(j_{k}) ⟶ 𝒪(j_{1}+⋯+j_{k}). Associative, unital, equivariant. \underline{𝒪-space X} :

θ: 𝒪(j) × X^{j} ⟶ X.

\underline{OPERAD PAIR}: ‘Additive’, ‘multiplicative’ operads 𝒞, 𝒢, λ: 𝒢 (k)×𝒞 (j_{1})×⋯×𝒞 (j_{k}) ⟶ 𝒞 (j_{1} ⋯ j_{k}). Distributive, unital, equivariant, nullary. \underline{(𝒞 , 𝒢)-space} : 𝒞 -space and 𝒢 -space X

\xymatrix{𝒢 (k) × 𝒞 (j_{1}) × X^{j_{1}} × ⋯ × 𝒞 (j_{k}) × X^{j_{k}} \ar[r]^{id ×θ^{k}} \ar[d]_{ξ} & 𝒢 (k) × X^{k} \ar[d]^{ξ} \\ 𝒞 (j_{1} ⋯ j_{k}) × X^{j_{1}···j_{k}} \ar[r]_{θ} & X}

ξ on left induced from λ and ξ on X.

E_{∞} RING THEORY 5

Monadic reinterpretation Monad O in 𝒯 = based spaces 𝒪-spaces ≅ O-algebras.

OX = ∐ 𝒪(j) ×_{Σ_{j}} X^{j}/(∼) ∼= basepoint identifications

(𝒞 , 𝒢): G ‘acts’ on C; C induces a monad on G[𝒯 ], CG becomes a monad on 𝒯 . Isomorphic categories:

(𝒞 , 𝒢) − spaces C − algebras in G[𝒯 ] CG − algebras in 𝒯 .

6 J.P. MAY

Lewis May spectra; QX U = ℝ^{∞}; V ⊂ W ⊂ U fin dim. Prepectrum T: T V ⟶ Ω^{W −V} T W

Spectrum E: EV ⟶^{≅} Ω^{W −V} EW

Spectrification L, forgetful ℓ:

𝒮 (LT, E) ≅ 𝒫(T, ℓE)

𝒫({Σ^{V} X}, T) ≅ 𝒯 (X, T_{0})

Σ^{∞}X = L{Σ^{V} X} Ω^{∞}E = E_{0}.

𝒮 (Σ^{∞}X, E) ≅ 𝒯 (X, Ω^{∞}E)

QX = colim Ω^{V} Σ^{V} X = Ω^{∞}Σ^{∞}X

E_{∞} RING THEORY 7

Approximation Theorem There is a map of monads

α: C ⟶ Q

on 𝒯 such that

α: CX ⟶ QX

is a group completion for all X, hence a weak equivalence for all connected X.

Canonical operad pair

Linear isometries operad ℒ.

ℒ (j) = ℐ (U^{j}, U)

ℒ acts on the Steiner operad 𝒞, a variant of the infinite little cubes operad. 𝒞 acts on Ω^{∞}E; α is the composite

\xymatrix{CX \ar[r]^{Cη} & CQX \ar[r]|{Cθ} & QX.}

ℒ and 𝒞 are E_{∞} operads: their j^{th} spaces are Σ_{j}-free and contractible.

E_{∞} ring spectra

External smash products T ⊼ T′, E ⊼ E′:

(T ⊼ T′)(V, V′) = T V ∧ T′V′
(E ⊼ E′) = L(ℓE ⊼ ℓE′)

T^{[j]}: external j-fold smash power.

f: U ⟶ U′ induces f^{∗}: 𝒫(U′) ⟶ 𝒫(U).

(f^{∗}T′)(V) = T′(fV)

Restricts to f^{∗}: 𝒮 (U′) ⟶ 𝒮 (U). Left adjoints f_{∗} on 𝒫, f_{∗} = Lf_{∗}ℓ on 𝒮.

ℒ -prespectrum T: maps of prespectra

ξ_{j}(f): f_{∗}T^{[j]} ⟶ T

for all f ∈ ℒ (j); suitably continuous and compatible with operad structure on ℒ.

ℒ (j) ⋉ T^{[j]} ⟶ T.

ℒ (j) ⋉ E^{[j]} ⟶ E.

E_{∞} RING THEORY 9

Monadic reinterpretation; Ω^{∞}

Monad L_{+} on 𝒯 wrt ∧ rather than ×:

L_{+}X = ∨_{j≥0} ℒ (j)_{+} ∧_{Σ_{j}} X^{(j)}.

Monad L_{+} on 𝒮 analogously:

L_{+}E = ∨_{j≥0} ℒ (j) ⋉_{Σ_{j}} E^{[j]}.

L_{+}Σ^{∞}X ≅ Σ^{∞}L_{+}X

𝒮 (Σ^{∞}X, E) ≅ 𝒯 (X, Ω^{∞}E)

induces

L_{+}[𝒮](Σ^{∞}Y, R) ≅ L_{+}[𝒯 ](Y, Ω^{∞}R).

10 J.P. MAY

• The monad Q on 𝒯 induces a monad Q on L_{+}[𝒯 ]; for an ℒ -spectrum R, Ω^{∞}R is a Q-algebra, hence a C-algebra, in L_{+}[𝒯 ].

• Therefore Ω^{∞}R is an E_{∞} ring space.

• The 1-component and unit components SL_{1}R and GL_{1}R of Ω^{∞}R are ℒ -spaces.

• Therefore SL_{1}R and GL_{1}R are the 0^{th} spaces of spectra sl_{1}R and gl_{1}R.

E_{∞} RING THEORY 11

Orientation theory

F (X, Y): based maps X ⟶ Y

∘: F (Y, Z) × F (X, Y) ⟶ F (X, Z) E ∈ 𝒮, E_{0} ≅ Ω^{V} EV for V ⊂ U = ℝ^{∞}. ∘: F (S^{V} , EV)×F (S^{V} , S^{V}) ⟶ F (S^{V} , EV)

E_{0} × Ω^{V} Σ^{V} S^{0} ⟶ E_{0}

R a ring spectrum. 1970’s notations:

SF = SL_{1}S F = GL_{1}S SF R = SL_{1}R F R = GL_{1}R

12 J.P. MAY

Think of F as a functor ℐ ⟶ monoids:

F V = hAut(S^{V}) ⊂ F (S^{V} , S^{V}) G an ℐ -monoid (or group) mapping to F:

GV × GV ⟶ GV GV ⟶ F V SGV ×SGV ⟶ SGV SGV ⟶ SF V B(F R, GV, S^{V}) → B(F R, GV, ∗) ≡ B(GV; R) (or with F R, GV replaced by SF R, SGV) Classifying spaces for R-oriented V -sphere bundles, or with preassigned Hℤ-orientation. Pass to colimits. Get classifying spaces

B(G; R) B(SG; R)

for stable R-oriented sphere bundles.

[figure: SG; SF R; B(SG; R); BSG; G; F R; B(G; R); BG; G; π_{0}(R)^{×}; B(G; π_{0}R); BG]

PUNCH LINE: If R is an E_{∞} ring spectrum, this is a diagram of ℒ -spaces, hence has an associated diagram of connective spectra.

Universal obstruction to R-orientability:

w: BSG ⟶ BSF R = BSGL_{1}R

E.g: G = O or G = F, R = kO:

w = w_{2} : BSF ⟶ BO_{⊗}

14 J.P. MAY

Theorem 1. At p > 2,

BST OP ≃ B(SF; kO) as E_{∞}-spaces: equivalent spectra.

Theorem 2. At p > 2,

MST OP ≃ M(SF; kO) as E_{∞}-ring spectra.

BCokerJ, MCokerJ: Can compute!!!

E_{∞} RING THEORY 15

Algebraic and topological K-theory

KR ≡ EB𝒢 ℒ R

𝒢 ℒ R = ∐_{n≥0} GL(n, R)

E is the BLACK BOX functor Fix a prime q.

Brauer lift (Quillen, plus May-Tornehave):

Theorem 3. Completed away from q, K𝔽̄_{q} and kU are equivalent ring spectra. For r = q^{a}, Frobenius and Adams agree:

\xymatrix{K𝔽̄_{q} \ar[r]^{ϕ^{r}} \ar[d]_{≃} & K𝔽̄_{q} \ar[d]^{≃} \\ kU \ar[r]_{ψ^{r}} & kU}

Multiplicative Brauer lift (May-Tornehave):

Theorem 4. Completed away from q, sl_{1}K𝔽̄_{q} and sl_{1}KU are equivalent spectra.

sl_{1}KU_{p} ≃ K(ℤ_{p}, 2) × bsu_{p}

16 J.P. MAY

Complete at p > 2, p ≠ q. Let r = q^{a} be a unit mod p^{2}.

Write bR for the connected cover of KR.

\xymatrix{K𝔽_{r} \ar[r] \ar[d]_{≃} & K𝔽̄_{q} \ar[r]^{ϕ^{r}−1} \ar[d]^{≃} & bR \ar[d]^{≃} \\ j \ar[r] & kU \ar[r]_{ψ^{r}−1} & bU}

Theorem 5. There is a composite “exponential equivalence”

b𝔽_{r} ⟶ sl_{1}S ⟶ sl_{1}𝔽_{r}

So sl_{1}S splits. Infinite loop space splitting

SF ≃ J × Coker J.

As infinite loop spaces,

B Coker J ≃ B(SF; K𝔽_{r})

BO×B Coker J ≃ B(SF; kO) ≃ B(SF; K𝔽̄_{q}).

E_{∞} RING THEORY 17

Additive recognition principle

Theorem 6. For a 𝒞 -space X, define

EX = B(Σ^{∞}, C, X).

EX is connective. There is a diagram of maps of C-spaces

\xymatrix{X & B(C, C, X) \ar[l]|{ε} \ar[r]|{Bα} & B(Q, C, X) \ar[r]^{ζ} & Ω^{∞}EX.}

ε is a homotopy equivalence, inverse η; ζ is a weak equivalence; Bα is a group completion. Therefore the composite

η: X ⟶ Ω^{∞}EX

is a group completion. It is a weak equivalence if X is grouplike.

For a spectrum Y, there is a composite map of spectra

\xymatrix{ε: EΩ^{∞}Y \ar[r]|{Bα} & B(Σ^{∞}, Q, Ω^{∞}Y) \ar[r]|{ε} & Y.}

Apply Ω^{∞}. The maps of 𝒞 -spaces

\xymatrix{Ω^{∞}EΩ^{∞}Y \ar[r]^{Ω^{∞}Bα} & Ω^{∞}B(Σ^{∞}, Q, Ω^{∞}Y) \ar[r]^{Ω^{∞}ε} & Ω^{∞}Y}

are weak equivalences. Therefore ε is a weak equivalence if Y is connective.

\underline{Conclusion}: E and Ω^{∞} induce an equivalence between the homotopy category of grouplike E_{∞} spaces and the homotopy category of connective spectra.

[Can replace by 𝒞 by 𝒞 × ℒ, 𝒞 × 𝒟, etc.]

E_{∞} RING THEORY 19

Multiplicative recognition principle

Theorem 7. For a (𝒞 , ℒ)-space X,

EX = B(Σ^{∞}, C, X)

is a connective ℒ -spectrum. All maps in the diagram

\xymatrix{X & B(C, C, X) \ar[l]|{ε} \ar[r]|{Bα} & B(Q, C, X) \ar[r]^{ζ} & Ω^{∞}EX}

are maps of (𝒞 , ℒ)-spaces. Therefore the composite

η: X ⟶ Ω^{∞}EX

is a ring completion.

For an ℒ -spectrum Y, the maps

\xymatrix{ε: EΩ^{∞}Y \ar[r]|{Bα} & B(Σ^{∞}, Q, Ω^{∞}Y) \ar[r]|{ε} & Y}

are maps of ℒ -spectra and the maps

\xymatrix{Ω^{∞}EΩ^{∞}Y \ar[r]^{Ω^{∞}Bα} & Ω^{∞}B(Σ^{∞}, Q, Ω^{∞}Y) \ar[r]^{Ω^{∞}ε} & Ω^{∞}Y}

are maps of (𝒞 , ℒ)-spaces.

\underline{Conclusion}: E and Ω^{∞} induce an equivalence between the homotopy category of ringlike E_{∞} ring spaces and the homotopy category of connective E_{∞} ring spectra.

E_{∞} RING THEORY 21

Localizations of unit spectra

Let X be an E_{∞} ring space, e.g. Ω^{∞}R.

Let N = {0, 1, 2, ⋯ } ⊂ π_{0}X.

Let M ⊂ N be a multiplicative subset, for example {p^{i}} or {n|(n, p) = 1}.

Let X_{M} = ∐ X_{m}, union of components.

Mild homological convergence condition.

Theorem 8. As an E_{∞} space, the localization of SL_{1}E = Ω_{1}^{∞}E(X, θ) at M is the basepoint component Ω_{1}^{∞}E(X_{M}, ξ).

E(X, θ) is defined using X as a 𝒞 -space. The localizations of SL_{1}E(X) depend only on X as an ℒ -space.

Let X = Ω^{∞}R. E(X, θ) is the connective cover of R, SL_{1}E(X) is SL_{1}R.

Corollary 9. For an E_{∞} ring spectrum R, sl_{1}(R)[M^{−1}] is the connected cover of E((Ω^{∞}R)_{M}, ξ).

Lewis conundrum

Everything enriched over based spaces (or simplicial sets) in the following hypotheses.

Theorem 10 (Lewis). Let 𝒮 satisfy

(i) 𝒮 is closed symmetric monoidal. Have functors ∧ and F such that

𝒮 (E ∧ E′, E′′) ≅ 𝒮 (E, F (E′, E′′)).

(ii) Have functors Σ^{∞} and Ω^{∞} such that

𝒮 (Σ^{∞}X, E) ≅ 𝒯 (X, Ω^{∞}E).

(iii) The unit for ∧ is S ≡ Σ^{∞}S^{0}.

Let E be a commutative monoid in 𝒮 (e.g. S). Then SL_{1}(E) is a product of Eilenberg-MacLane spaces.

Corollary 11. No model structure on 𝒮 with S cofibrant can give the right Ho𝒮.

E_{∞} RING THEORY 23

EKMM S-modules

Quillen equivalences:

[figure: 𝒫; L; 𝒮; 𝕃; ℓ; 𝒮 [𝕃]; S∧_{ℒ} (−); ℓ; ℳ_{S}; F_{ℒ} (S,−)]

Monad 𝕃: 𝕃E = ℒ (1) ⋉ E. η : E ⟶^{≃} 𝕃E.

𝕃-spectrum: action 𝕃E ⟶ E; e.g. 𝕃E.

E ∧_{ℒ} E′ ≡ ℒ (2) ⋉_{ℒ (1)×ℒ (1)} E ⊼ E′

Σ^{∞}(X ∧ Y) ≅ Σ^{∞}X ∧_{ℒ} Σ^{∞}Y. Associative and commutative.

τ: E ∧_{ℒ} E′ ≅ E′ ∧_{ℒ} E.

24 J.P. MAY

Unit weak equivalence λ: S ∧_{ℒ} E ⟶^{≃} E.

E is an S-module if λ is an isomorphism.

Examples: Σ^{∞}X, S ∧_{ℒ} E.

𝒮 [𝕃] is the category of 𝕃-spectra.

ℳ_{S} is the category of S-modules.

Definition 12. A commutative ‘monoid’ in 𝒮 [𝕃] is an 𝕃-spectrum R with a unit map η: S ⟶ R and a commutative and associative product ϕ: R∧_{ℒ} R ⟶ R such that the following diagram commutes.

\xymatrix{S ∧_{ℒ} R \ar[r]^{η∧id} \ar[dr]_{λ} & R ∧_{ℒ} R \ar[d]^{ϕ} & R ∧_{ℒ} S \ar[l]_{id ∧η} \ar[dl]^{λτ} \\  & R}

Theorem 13. The category of commutative monoids in 𝒮 [𝕃] is isomorphic to the category of E_{∞} ring spectra.

E ∧_{S} E′ = E ∧_{ℒ} E′
Σ^{∞}(X ∧ Y) ≅ Σ^{∞}X ∧_{S} Σ^{∞}Y.

ℳ_{S} is closed symmetric monoidal, unit S.

Definition 14. A commutative S-algebra is a commutative monoid in ℳ_{S}. That is, it is an E_{∞} ring spectrum which is an S-module.

If R is an E_{∞} ring spectrum, then S∧_{ℒ} R is a weakly equivalent commutative S-algebra.

S = Σ^{∞}S^{0} is cofibrant in 𝒮; 𝕃S, S ∧_{ℒ} 𝕃S are cofibrant approximations in 𝒮 [𝕃], ℳ_{S}.

Adjunctions in 𝒮, 𝒮 [𝕃], ℳ_{S}:

(Σ^{∞}, Ω^{∞})
(𝕃Σ^{∞}, Ω^{∞}ℓ)
(S ∧_{ℒ} 𝕃Σ^{∞}, F_{ℒ} (S, Ω^{∞}ℓ))

Diagram spectra

[figure: 𝒫; ℙ; ℙ; Σ𝒮; ℙ; ℙ; 𝕌; ℐ 𝒮; 𝕌; ℙ; 𝕌; ℱ 𝒯; ℙ; 𝒲 𝒯; 𝕌; 𝕌; 𝕌]

Lexicon:

• 𝒫 is the category of 𝒩 -spectra, or (coordinatized) prespectra.

• Σ𝒮 is the category of Σ-spectra, or symmetric spectra.

• ℐ 𝒮 is the category of ℐ -spectra, or orthogonal spectra.

• ℱ 𝒯 is the category of ℱ -spaces, or Γ-spaces.

• 𝒲 𝒯 is the category of 𝒲 -spaces.

E_{∞} RING THEORY 27

𝒟-spectra for diagram of domains 𝒟:

[figure: 𝒩; Σ; ℐ; ℱ; 𝒲]

Start with 𝒟-spaces, 𝒟𝒯 . For a sphere functor S: 𝒟 ⟶ 𝒯 with smash products, S-modules are 𝒟-spectra. No distinction when 𝒟 = ℱ or 𝒟 = 𝒲 : 𝒟𝒯 = 𝒟𝒮.

Lexicon:

• 𝒩 is the category of natural numbers.

• Σ is the category of symmetric groups.

• ℐ is the category of linear isometric iso’s.

• ℱ is the category of finite based sets.

• 𝒲 is the category of based spaces that are homeomorphic to finite CW complexes.

Forgetful, prolongation functors 𝕌, ℙ. All (ℙ, 𝕌) are Quillen equivalences. (Connective spectra only for ℱ 𝒯 ).

For 𝒟-spaces T and T′, have external smash product T ⊼ T′, a 𝒟 × 𝒟-space.

(T ⊼ T′)(d, e) = T d ∧ T′e.

Given ⊕: 𝒟 ×𝒟 ⟶ 𝒟, left Kan extension gives a 𝒟-space T ∧ T′.

((𝒟×𝒟)𝒯 )(T ⊼T′, V ∘⊕) ≅ 𝒟𝒯 (T ∧T′, V)

For S-modules T and T′, get coequalizer

T ∧_{S} T′.

All but 𝒩 𝒮 above are symmetric monoidal. —————————————————- Quillen equivalence

\xymatrix{ℐ 𝒮 \ar[r]^{ℕ} & ℳ_{S} \ar[l]^{ℕ^{#}}}

from positive model structure on ℐ 𝒮.

E_{∞} RING THEORY 29

Theorem 15. The functors ℙ and ℕ induce Quillen equivalences from commutative symmetric ring spectra to commutative orthogonal ring spectra and from the latter to commutative S-algebras.

We have comparison functors

\xymatrix{Commutative symmetric ring spectra \ar[d]^{ℙ} \\ Commutative orthogonal ring spectra \ar[d]^{ℕ} \\ Commutative S-algebras \\ E_{∞} ring spectra \ar[u]_{S∧_{ℒ} (−)} \ar[d]^{Ω^{∞}} \\ E_{∞} ring spaces.}

But: no 0^{th} space information in diagram commutative ring spectra. The E_{∞} ring theory relating spaces and spectra is lost.

Naive E_{∞} ring spectra

𝒪(j)_{+} ∧ R^{(j)} ⟶ R.

Proposition 16. For a positive cofibrant symmetric or orthogonal spectrum or for a cofibrant S-module E,

π: (EΣ_{j})_{+} ∧_{Σ_{j}} E^{(j)} ⟶ E^{(j)}/Σ_{j} is a weak equivalence.

Proposition 17. The homotopy categories of naive 𝒪-spectra and commutative ring spectra (in any of Σ𝒮, ℐ 𝒮, or ℳ_{S}) are equivalent.