FROM THE BEGINNING (ROSENDAL TALK)

J.P. MAY

First let me thank John for the opportunity to give a jet-lagged first talk. He wrote “I think it would be very nice if you could start it off with a talk about where this all began, plus whatever else you prefer to cover”. I’ll give a four part talk, starting with where it began, probably ending before getting to the interesting part.

The real beginning is the Freudenthal suspension theorem, but we can jump ahead to Boardman’s definitive definition of the stable homotopy category in 1964. All people doing foundational work back then thought in terms of CW complexes, especially finite ones. That precluded taking a point-set category too seriously.

I first became convinced that spectra should be well-behaved point-set level objects for very naive reasons connected with homology operations on infinite loop spaces. I wanted the infinite little cubes operad to act naturally on the zeroth spaces of spectra. To make this true, I defined a spectrum to be a sequence of spaces E_{i} and homeomorphisms σ̃: E_{i} ⟶ ΩE_{i+1}. That was in 1968. I did have in mind a redevelopment of the stable homotopy category from that starting point, but nothing serious was done for some years. I was into calculations and not foundations back then.

In September of 1972, Frank Quinn posed the question of defining operad actions on that kind of point-set level spectrum. I will come back to his original motivation. In November of 1972, Nigel Ray gave a talk about Thom spectra in Chicago’s topology seminar, and we talked about how to get a good construction of them. Something clicked, and by the end of the following week I had defined E_{∞} ring spectra. They didn’t look like they do now, because I only defined twisted half-smash products and extended powers a few years later, but the definition was precisely equivalent to the one that is now standard.

The crucial idea was to model the definition on Thom spectra as they come in nature, starting from the Grassmannians of inner product spaces. In particular, they are coordinate-free spectra, given by spaces T V and structure maps σ: Σ^{W} T V ⟶ T (V ⊕ W) for inner products V and W. Actually, in retrospect, Thom spectra as we understood them in 1972 were FSP’s, functors with smash product, in a form that is the external smash product version of an orthogonal ring spectra with its internalized smash product.

Spectrifying to make the adjoint structure maps σ̃ into homeomorphisms, the result is an E_{∞} ring spectra. Up to language, I asked myself then whether or not every E_{∞} ring spectrum arose as the spectrification of an orthogonal ring spectrum. I thought that the answer was, surely not, but Mandell, Schwede, Shipley, and I proved in 1999 that the answer is yes. Technology had moved on, and we proved that the two definitions, and their symmetric ring spectrum variant, are equivalent via Quillen equivalences of model categories. That is, any good definition of highly structured ring spectra gives a Quillen model category that is Quillen equivalent to the original category of E_{∞} ring spectra.

There was a conference at Northwestern in 1974 at which I proposed 18 problems and 3 conjectures in this area. Let me list a few of them to give the flavor of where we were then, compared with where we are now. Many of them, such as the very first one, are still open.

Problem 1. Does the Brown-Peterson spectrum admit a model as an E_{∞} ring spectrum?

We will hear two talks about that still open question this afternoon.

Problem 2. Are localizations and completions of E_{∞} ring spectra again E_{∞} ring spectra?

EKMM proves that Bousfield localization of an R-algebra A at an Rmodule E is a again an R-algebra, and similarly in the commutative case. For example, with R = ko, this gives a one line proof that KO is a koalgebra, hence an E_{∞}-ring spectrum by neglect of structure.

Conjecture 1. The complex Adams onjecture holds on the infinite loop space level.

There were two proofs announced in a joint paper of Seymour and Friedlander. Seymour’s is wrong and Friedlander’s is right.

Conjecture 2. The Atiyah-Bott-Shapiro orientation g: BSpin ⟶ B(Spin; ko) is an infinite loop map.

Conjecture 3. The Sullivan orientation ḡ: BST op ⟶ B(ST op; ko[1/2]) is an infinite loop map.

It was questions like the last one that motivated Quinn.

Problem 4. Is ψ^{r}: ko[1/r] ⟶ ko[1/r] a map of E_{∞} ring spectra?

Problem 5. Is Brauer lift λ̂ : ko^{δ} ⟶ k̂o[1/q] a map of E_{∞} ring spectra?

I will say more about these last two problems shortly. For this conference, the most relevant of these 1974 problems is

Problem 10. Develop a theory of E_{∞} pairings of E_{∞} modules over an E_{∞} ring spectrum.

FROM THE BEGINNING (ROSENDAL TALK) 3

Ancient history; a book; Thom Thom spectra

1968-73: ∞ LOOP SPACE THEORY 1968: Spectra: E = {E_{i} | E_{i} ≅ ΩE_{i+1}}. 1972-73: E_{∞} ring spaces and ring spectra. Coordinate-free spectra:

{EV | EV ≅ Ω^{W}E_{V ⊕W}}

ℐ = category of fin dim inner product spaces.

𝒯 = category of based spaces

ℐ -functor: continuous functor

T: ℐ ⟶ 𝒯

with commutative and associative nat. trans.

ω: T × T ⟶ T ∘ ⊕.

Inclusions on restriction to basepoints. ℐ -functors ⟹ ℒ -spaces,

ℒ the linear isometries operad.

ℐ -monoid G: values in monoids.

G = O, U, Spin, String, T op, F, special variants SG.

Orthogonal spectrum T: (ℐ_{∗}-prespectrum in 1980)

Σ^{W}T (V) ⟶ T (V ⊕ W)

ℐ -FSP (ℐ_{∗}-prefunctors in 1973):

T (V) ⊕ T (W) ⟶ T (V ⊕ W)

Thom spectra the original examples.

ℐ -FSP’s ⟹ E_{∞}-ring spectra

Modern route: Left Kan extension shows ℐ -FSP’s are the same as orthogonal ring spectra, and they are (Quillen) equivalent to E_{∞} ring spectra, EKMM S-algebras, and symmetric ring spectra. ALL NECESSARY.

Motivation: calculations, e.g. π_{∗}(MST OP).

Formalization of work of Sullivan; localize at some prime p ≠ 2.

[figure: SF; F/T op; ≃; BST op; ≃; BSF; SF; BO_{⊗}; B(SF; kO); BSF.]

Here B(SF; kO) is the classifying space for kO-oriented stable spherical fibrations, and

BO_{⊗} = SL_{1}(kO) = SF (kO).

Theorem 1. The diagram is a commutative diagram of infinite loop spaces and infinite loop maps.

∞ loop spaces J and J_{⊗}:

\xymatrix{J \ar[r] & BO \ar[r]^{ψ^{r}−id} & BO}

and

\xymatrix{J_{⊗} \ar[r] & BO_{⊗} \ar[r]^{ψ^{r}/ id} & BO_{⊗},}

(r mod p^{2} generates the group of units).

∞ loop space BCokerJ:

\xymatrix{BCokerJ \ar[r] & B(SF; kO) \ar[r]^{c(ψ^{r})} & BO_{⊗}.}

c(ψ^{r}) is the “universal cannibalistic class”. CokerJ = ΩBCokerJ.

Theorem 2. There is an equivalence of infinite loop spaces

SF ≃ J × CokerJ.

Theorem 3. There is an equivalence of infinite loop spaces

B(SF; kO) ≃ BCokerJ × BO.

These three theorems distill a huge amount of work by many people: Boardman and Vogt; Adams; Peterson; Adams and Priddy; Madsen, Snaith, and Tornehave; Friedlander; Hodgkin and Snaith.

This reduces calculation of H^{∗}(BST op) to calculation of H^{∗}(BCokerJ).

Why is BCokerJ an ∞ loop space? Is ψ^{r} a map of E_{∞} ring spectra? Discrete models from algebraic K-theory.

MULTIPLICATIVE INFINITE LOOP SPACE THEORY

0th space of an E_{∞} ring spectrum is an E_{∞} ring space. Converse up to completion of semi-ring to ring. (1975, 1983).

E_{∞} ring space: (𝒦 , ℒ) an operad pair. 𝒦 : Steiner’s version of infinite little cubes operad. Monad 𝕂 associated to 𝒦 restricts to a monad on the category of ℒ -spaces with zero. An E_{∞} ring space is an algebra over this monad. Has two operad actions.

A bipermutative category 𝒞 is a category with products ⊕ and ⊗; strictest version of symmetric bimonoidal. B𝒞 is equivalent to an E_{∞} ring space and so gives rise to an E_{∞} ring spectrum R𝒞.

Example 4. kO from O(n; ℝ), n ≥ 0.

8 J.P. MAY

Let r = q^{a}, q prime, where r mod p^{2} generates the group of units. Allow p = 2.

Let 𝔽_{r} be the field with r elements, 𝔽̄_{q} its algebraic closure. Bipermutative category 𝒪(R) of orthogonal groups O(n, R) of a commutative ring R gives E_{∞} ring spectrum kO(R). ∞ loop commutative diagram:

[figure: SF; e; BO(𝔽̄_{q})_{⊗}; λ_{⊗}; ≃; B(SF; kO(𝔽̄_{q})); ω; ≃; π; BSF; SF; e; BO_{⊗}; B(SF; kO); π; BSF.]

λ_{⊗} is Brauer lift. Why an ∞ loop map? Key: machine built property of SL_{1}(R𝒞).

R an E_{∞} ring spectrum. Unit components and component of identity (1973 notations)

GL_{1}(R) = F R and SL_{1}(R) = SF R. Infinite loop spaces; spectra now denoted

gl_{1}(R) and sl_{1}(R).

∞ loop maps X ⟶ GL_{1}R
⟷
E_{∞} ring maps Σ^{∞}X_{+} ⟶ R.

FROM THE BEGINNING (ROSENDAL TALK) 9

X an E_{∞} space with π_{0}(X) = {n|n ≥ 0}, associated E_{∞} ring spectrum R. Given a multiplicative set M ⊂ {n|n ≥ 0}, let X_{M} = ⨿X_{m}. It is an E_{∞} space.

Theorem 5. As an ∞ loop space, the localization of SL_{1}(R) at M is the basepoint component of the 0th space of the spectrum associated to X_{M}.

SL_{1}(R) comes from the additive structure, and is then given a multiplicative structure, but its localizations can be computed just from space level multiplicative structure.

Example of how this is used.

Example 6. Regard elements of O(n, 𝔽_{r}) as permutations of the set of r^{n} elements. Get a map of permutative categories

𝒪(𝔽_{r}, ⊕) ⟶ (ℰ , ×),

ℰ =finite sets, hence ∞ loop map J ≃ kO(𝔽_{r})_{0} ⟶ SF [1/q].

SF [1/q] computed from ℰ under ×, not ⨿.

Frobenius automorphisms

ϕ^{r} : O(n, 𝔽̄_{q}) ⟶ O(n, 𝔽̄_{q}),

with fixed points O(n, 𝔽_{r}). These give a map of bipermutative categories and so give an E_{∞} ring map

ϕ^{r}: kO(𝔽̄_{q}) ⟶ kO(𝔽̄_{q}).

ϕ^{r} corresponds to ψ^{r} under Brauer lift, and c(ϕ^{r}) corresponds to c(ψ^{r}):

\xymatrix{BCokerJ \ar[r] & B(SF; kO(𝔽̄_{q}) \ar[r]^{c(ϕ^{r})} & SL_{1}(kO(𝔽̄_{q}))}

c(ϕ^{r}) is an ∞ loop map, and

BCokerJ ≃ B(SF; kO(𝔽_{r})).

Question 7. Does the chromatic level 2 picture tell us anything about BCokerJ?

FROM THE BEGINNING (ROSENDAL TALK) 11

Coming soon to your local book seller: Parametrized Homotopy Theory by J.P. May and J. Sigurdsson

Everything to follow works equivariantly for any compact Lie group G of equivariance. Spaces and spectra can mean G-spaces and G-spectra. Will describe things model theoretically, although model structures are insufficient for the proofs.

BASIC STRUCTURE

Theorem 8. For any space B, there is a proper stable topological model category 𝒮_{B} of spectra over B. For b ∈ B and X ∈ 𝒮_{B}, the fiber X_{b} is an orthogonal spectrum. X is fibrant if each projection X(V) ⟶ B is a fibration and each adjoint structure map

σ̃: X_{b}(V) ⟶ Ω^{W}X_{b}(V ⊕ W)

is a weak equivalence.

12 J.P. MAY

Theorem 9. The stable homotopy category Ho𝒮_{B} is closed symmetric monoidal under the internal smash product and function spectrum functors ∧_{B} and F_{B}.

BASE CHANGE FUNCTORS

Theorem 10. For f: A ⟶ B, there is a pullback functor f^{∗}: 𝒮_{B} ⟶ 𝒮_{A} with left and right adjoints f_{!} and f^{∗}.

Theorem 11. (f_{!}, f^{∗}) is a Quillen pair, and it is a Quillen equivalence if f is an equivalence.

Theorem 12. (f^{∗}, f_{∗}) is a Quillen pair if f is a bundle with CW fibers, but not in general otherwise.

Example 13. f: A → B an inclusion.

f^{∗}Y = Y |_{A}
(f_{!}X)(V) = X(V) ∪_{sA} B

f_{∗}(X) = “skyscraper” over A.

Example 14. r: B → ∗. X a based space,

E = (E, p, s) an ex-space over B.

r^{∗}X = B × X

r_{!}E = E/s(B) = Thom space

r_{∗}E = Sec(B, E).

Example 15. H ⊂ G. H𝒮 ≅ G𝒮_{G/H}.

14 J.P. MAY

Thom Thom spectra

Let G be an ℐ -monoid. Bar construction

EG(V) = B(∗, G(V), S^{V}) Bundle (or quasifibration) over

BG(V) = B(∗, G(V), ∗) with section s at ∞. Classical Thom spectra T G(V) = EG(V)/sBG(V) = r_{!}EG(V). Reinterpretation. BG = colim BG(V).

i(V): BG(V) ⟶ BG i(V)_{!}EG(V) = EG(V) ∪_{sBG(V)} BG

These give a spectrum UG over BG, and T G = r_{!}UG.

B(∗, G(V), S^{V ⊕W}) ⟶ B(∗, G(V ⊕W), S^{V ⊕W}) induces required structural maps

σ: Σ_{BG}^{W}U(V) ⟶ U(V ⊕ W)

Definition 16. For f: B ⟶ BG,

T f = r_{!}f^{∗}UG.

Here r: B ⟶ ∗ is r ∘ f, so get

T f = r_{!}f_{!}f^{∗}UG ⟶ r_{!}UG = T G.

Reinterpretation of Mahowald and Gaunce Lewis construction. Properties easily proven.

UG = U(G; S), S the sphere spectrum. Let X be an ℐ -G orthogonal spectrum: maps

ξ: G(V)_{+} ∧ X(V) ⟶ X(V)

compatible with σ and G(V) → G(W).

Definition 17. For an ℐ -G spectrum X,

U(G; X)(V) = i(V)_{!}B(∗, G(V), X(V))

gives a spectrum over BG, and

T (G; X) = r_{!}U(G; X).

If X is an ℐ -FSP with product compatible with its G-action, called an ℐ -G-FSP, then T (G; X) is an ℐ -FSP.

Example 18. If G is group-valued, then T G is an ℐ -G-FSP. Action induced by

G(V) × G(V)^{q} × S^{V} ⟶ G(V)^{q} × S^{V}

(h, g_{1}, …, g_{q}, v) ↦ (hg_{1}h^{−1}, …, hg_{q}h^{−1}, hv)

Definition 19. T (G; T G) is the \underline{Thom} \underline{Thom} spectrum of G. Since it is an ℐ -FSP, it is an orthogonal ring spectrum.

Two-sided variant starting from B(Y, G, X).

The classifying space B(SF; R) for R-oriented spherical fibrations discussed earlier is

B(SF; R) = B(SL_{1}(R), SF, ∗).

SF acts on SL_{1}(R) since R is a coordinatefree spectrum, not an orthogonal spectrum. Need precise 0th space functor.

HOMOLOGY and COHOMOLOGY

Definition 20. Let E and X be spectra over B. For integers n, define the nth Ehomology and E-cohomology groups of X by

E_{n}(X) = π_{n}(r_{!}(E ∧_{B} X))

and

E^{n}(X) = π_{−n}(r_{∗}F_{B}(X, E)).

Then

E^{n}X ≅ [S_{B}^{−n}, F_{B}(X, E)]_{B}.

Applied to Σ_{B}^{∞}K, gives theories on ex-spaces K over B. Axiomatization, representability as usual, although Adams’ variant of Brown representability does not apply.

• Twisted theories

• Bundle construction

• Twisted K -theory

• Fundamental groupoids visible

• Twisted K -theory

• Eilenberg-Moore spectral sequence is a parametrized Künneth theorem Proposition 21. Fix f: A ⟶ B. For E ∈ 𝒮_{A} and X ∈ 𝒮_{B},

(f_{!}E)_{n}(X) ≅ E_{n}(f^{∗}X)

and

(f_{∗}E)^{n}(X) ≅ E^{n}(f^{∗}X).

For X ∈ 𝒮_{A} and E ∈ 𝒮_{B},

(f^{∗}E)_{n}(X) ≅ E_{n}(f_{!}X)

and

(f^{∗}E)^{n}(X) ≅ E^{n}(f_{!}X).

Corollary 22. Let E = r^{∗}k, where k is a spectrum. Let X be a spectrum over B. Then

E_{n}(X) ≅ k_{n}(r_{!}X)

and

E^{n}(X) ≅ k^{n}(r_{!}X).

Parametrized Spanier-Whitehead duality? New Costenoble-Waner duality needed!

FROM THE BEGINNING (ROSENDAL TALK) 19

BACK TO BASICS: COMPATIBILITIES Theorem 23. f^{∗}: 𝒮_{B} → 𝒮_{A} is “closed symmetric monoidal”: f^{∗}S_{B} ≅ S_{A} and

(1) f^{∗}(Y ∧_{B} Z) ≅ f^{∗}Y ∧_{A} f^{∗}Z

(2) f_{!}(f^{∗}Y ∧_{A} X) ≅ Y ∧_{B} f_{!}X

(3) F_{B}(Y, f_{∗}X) ≅ f_{∗}F_{A}(f^{∗}Y, X)

(4) f^{∗}F_{B}(Y, Z) ≅ F_{A}(f^{∗}Y, f^{∗}Z)

(5) F_{B}(f_{!}X, Y) ≅ f_{∗}F_{A}(X, f^{∗}Y) Theorem 24. Given a pullback

\xymatrix{C \ar[r]^{g} \ar[d]_{i} & D \ar[d]^{j} \\ A \ar[r]_{f} & B}

with f or j a fibration, j^{∗}f_{!} ≅ g_{!}i^{∗} f^{∗}j_{∗} ≅ i_{∗}g^{∗}

f^{∗}j_{!} ≅ i_{!}g^{∗} j^{∗}f_{∗} ≅ g_{∗}i^{∗}.

Not true without fibration hypothesis.

20 J.P. MAY

EXTERNAL OPERATIONS X, Y, Z spectra over A, B, A × B. Get X ⊼ Y over A × B, F̄(Y, Z) over A:

𝒮_{A×B}(X ⊼ Y, Z) ≅ 𝒮_{A}(X, F̄(Y, Z))

X ⊼ Y ≅ π_{A}^{∗}X ∧_{A×B} π_{B}^{∗}Y

F̄(Y, Z) ≅ π_{A ∗}F_{A×B}(π_{B}^{∗}Y, Z) X, Y ex-spectra over B, Δ : B → B × B

X ∧_{B} Y ≅ Δ^{∗}(X ⊼ Y)

F_{B}(X, Y) ≅ F̄(X, Δ_{∗}Y)

Quillen pairs

(− ⊼ Y, F̄(Y, −))

Compatibilities with base change, e.g. (f^{∗}Y ⊼ g^{∗}Z) ≅ (f × g)^{∗}(Y ⊼ Z)

FROM THE BEGINNING (ROSENDAL TALK) 21

𝒦_{B} category of ex-spaces over B. Adjoints Σ_{B}^{∞}: 𝒦_{B} → 𝒮_{B} and Ω_{B}^{∞}: 𝒮_{B} → 𝒦_{B} Case V = 0 of adjoints F_{V} : 𝒦_{B} → 𝒮_{B} and ev_{V} : 𝒮_{B} → 𝒦_{B} where ev_{V} = X(V). Generating cofibrations are F_{V} i, where

i: S_{+}^{n−1} ⟶ D_{+}^{n}

is a fiberwise cofibration over B. Nota bene: S_{B} = Σ_{B}^{∞}B × S^{0} is not among the generating cofibrations. Theorem 25. HoG𝒮_{B} is triangulated with compatible symmetric monoidal structure.

FIBERWISE DUALITY

Theorem 26 (Fiberwise duality theorem). (X fibrant). X is dualizable or invertible if and only if each fiber X_{b} is dualizable or invertible.

Dualizable in symmetric monoidal category:

η: S_{B} ⟶ X ∧_{B} Y, ε: Y ∧_{B} X ⟶ S_{B}

such that

(id ∧ε)(η∧id) = Id_{X}, (ε∧id)(id ∧η) = Id_{Y} .

D_{B}(X) ≡ F_{B}(X, S_{B}), Y ≃ D_{B}(X)

Alternatively, X is dualizable if

ν: D_{B}X ∧_{B} X → F_{B}(X, X)

is an isomorphism. Fiberwise, the map is

ν: D(X_{b}) ∧ X_{b} → F (X_{b}, X_{b}),

which gives the proof of the theorem.

Identifies dualizable and invertible guys.

Theorem 27 (Lewis–May, Greenlees). A G-spectrum is dualizable if and only if it is isomorphic in HoG𝒮 to a homotopy retract of a finite G-CW spectrum.

Theorem 28 (Fausk, Lewis, May). A G-spectrum is invertible if and only if it is isomorphic in HoG𝒮 to a “stable homotopy representation” Σ^{−V} Σ^{∞}T, where V is a representation of G and T is a finitely dominated based G-CW complex such that T^{H} ≃ S^{n(H)} for H ⊂ G.

TRACE and TRANSFER

Definition 29. Δ_{X}: X → X ∧_{B} C_{X} for some C_{X}. The trace τ(f) of f: X → X is

\xymatrix{S_{B} \ar[r]^{η} \ar[d]_{τ(f)} & X ∧_{B} D_{B}X \ar[r]^{γ} & D_{B}X ∧ X \ar[d]^{D_{B}f∧Δ_{X}} \\ C_{X} & S_{B} ∧ C_{X} \ar[l]^{≅} & D_{B}X ∧_{B} X ∧_{B} C_{X} \ar[l]^{ϵ∧1}}

C_{X} = S_{B}, Δ_{X} = id:

τ(f) is the Lefschetz constant χ(f); f = id: Euler characteristic χ(X).

C_{X} = X, Δ_{X} a diagonal map:

τ_{X} = τ(id) is the transfer map of X.

Compatible triangulation implies additivity:

\xymatrix{X \ar[r]^{f} & Y \ar[r]^{g} & Z \ar[r]^{h} & ΣX}

χ(Y) = χ(X) + χ(Z).

Nonparametrized transfers by applying r_{!}.

Definition 30 (TRANSFER MAPS). Let p: E → B be a G-fibration over B such that each fiber E_{b} is a homotopy retract of a finite G_{b}-CW-complex. Then Σ_{B}^{∞}(E ⨿ B) is dualizable, and we have a transfer

τ_{E⨿B}: S_{B} → Σ_{B}^{∞}(E ⨿ B)

of G-spectra over B. The usual transfer is

τ_{E} = r_{!}τ_{E⨿B}: Σ^{∞}B_{+} → Σ^{∞}E_{+}.

“Bundle construction” leads to a fiberwise variant: insert pretransfer maps of spectra fiberwise into bundles. The two agree where both are defined.

SYMMETRIC BICATEGORIES

Bicategory 𝒞: monoidal category with many objects. 0-cells = objects Category 𝒞 (A, B) for each pair of 0-cells. 1-cell X: A ⟶ B is an object of 𝒞 (A, B). 2-cell X ⟶ Y is a morphism of 𝒞 (A, B).

⊙: 𝒞 (B, C) × 𝒞 (A, B) ⟶ 𝒞 (A, C) for each triple of 0-cells, unit functor

U_{A}: ∗ ⟶ 𝒞 (A, A)

for each 0-cell, written A = U_{A} by abuse.

Associative and left and right unital up to coherent natural isomorphism. That’s all.

𝒞 is closed if it has left and right internal hom functors: ◁: 𝒞 (A, C) × 𝒞 (B, C)^{op} ⟶ 𝒞 (A, B)

▷: 𝒞 (A, B)^{op} × 𝒞 (A, C) ⟶ 𝒞 (B, C)

𝒞 (X, Z◁Y) ≅ 𝒞 (Y ⊙X, Z) ≅ 𝒞 (Y, X▷Z).

Definition 31. 𝒞 is symmetric if it has • A bijection t on the 0-cells of 𝒞 such that

ttA = A.

• Equivalences of categories

t: 𝒞 (A, B) ⟶ 𝒞 (tB, tA)

• Natural isomorphism 2-cells

ι: tU_{A} ⟶ U_{tA}

γ: tX ⊙ tY ⟶ t(Y ⊙ X)

for 1-cells X: A ⟶ B and Y : B ⟶ C such that appropriate coherence laws hold.

If 𝒞 is closed and symmetric, then

X ▷ Z ≃ t(tZ ◁ tX): B ⟶ C

for X: A ⟶ B and Z: A ⟶ C

28 J.P. MAY

Example 32. ℬ_{R}, R a commutative ring:

• 0-cells are R-algebras.

• ℬ_{R}(A, B) is the category of (B, A)-bimodules.

• ⊗_{B} defines the composition

⊙: ℬ_{R}(B, C) × ℬ_{R}(A, B) ⟶ ℬ_{R}(A, C)

• The left and right homs are

Z ◁ Y = Hom_{C}(Y, Z)
X ▷ Z = Hom_{A}(X, Z).

• tA is the opposite R-algebra of A

• For a (B, A)-bimodule M, tM is M regarded as a (tA, tB)-bimodule.

• γ is the usual interchange isomorphism.

Example 33. DGA variant, using derived categories of bimodules.

Example 34. Structured ring spectrum variant, using stable homotopy categories.

Right home for Morita theory and variants.

DUALITY IN SYMMETRIC BICATEGORIES

Definition 35. Let

X: B ⟶ A and Y : A ⟶ B

be 1-cells. (X, Y) is a dual pair if there are 2-cells

η: A ⟶ X ⊙ Y
ε: Y ⊙ X ⟶ B

such that the following diagrams commute in 𝒞 (B, A) and 𝒞 (A, B).

\xymatrix{X \ar[d]_{id} & A ⊙ X \ar[l]_{≅} \ar[r]^{η⊙id} & (X ⊙ Y) ⊙ X \ar[d]^{≅} \\ X & X ⊙ B \ar[l]^{≅} & X ⊙ (Y ⊙ X) \ar[l]^{id ⊙ϵ}}

\xymatrix{Y \ar[d]_{id} & Y ⊙ A \ar[l]_{≅} \ar[r]^{id ⊙η} & Y ⊙ (X ⊙ Y) \ar[d]^{≅} \\ Y & B ⊙ Y \ar[l]^{≅} & (Y ⊙ X) ⊙ Y \ar[l]^{ϵ⊙id}}

We say that X is left dual to Y and Y is right dual to X. Right dualizability is not equivalent to left dualizability.

30 J.P. MAY

Definition 36. For a 1-cell X: B ⟶ A,

D_{r}X = X ▷ B: A ⟶ B. For a 1-cell Y : A ⟶ B,

D_{ℓ}Y = B ◁ Y : B ⟶ A.

ε: Y ⊙ X ⟶ B has left and right adjoints

ε̃: X ⟶ D_{ℓ}Y and ε̃: Y ⟶ D_{r}X such that the following diagram commutes:

\xymatrix{Y ⊙ D_{ℓ}Y \ar[dr]_{ε} & Y ⊙ X \ar[l]_{ε̃⊙id} \ar[d]^{ε} \ar[r]^{id ⊙ε̃} & D_{r}X ⊙ X \ar[dl]^{ε} \\  & B}

Proposition 37. If ε: Y ⊙ X ⟶ B is the evaluation map of a dual pair (X, Y), ε̃: X ⟶ D_{ℓ}Y and ε̃: Y ⟶ D_{r}X are isomorphisms.

FROM THE BEGINNING (ROSENDAL TALK) 31

Example 38. Let A be a commutative R-algebra: tA = A. Write U_{A}: A ⟶ A; A as an (A, A)-bimodule

S_{A}: A ⟶ R; A as a left A-module tS_{A}: R ⟶ A; A as a right A-module

tS_{A} ⊙ S_{A} = A ⊗_{A} A ≅ A and

S_{A} ⊙ tS_{A} = A ⊗_{R} A.

Let

η: U_{R} = R ⟶ A ≅ tS_{A} ⊙ S_{A}

and

ε: tS_{A} ⊙ S_{A} ≅ A ⊗_{R} A ⟶ A = U_{A} be the unit and product on A. Left and right unit laws say (tS_{A}, S_{A}) is a dual pair.

For (S_{A}, tS_{A}) to be a dual pair, need maps

η: A ⟶ A ⊗_{R} A and ε: A ⟶ R of (A, A) and (R, R)-bimodules such that left and right counit laws hold.

We have natural maps

μ: Z ⊙ D_{r}X ⟶ X ▷ Z

and

ν: D_{ℓ}Y ⊙ W ⟶ W ◁ Y

Proposition 39. A 1-cell X: B ⟶ A is right dualizable iff μ is an isomorphism when Z = X, and then μ is an isomorphism for all Z and X ≅ D_{ℓ}D_{r}X.

Proposition 40. A 1-cell Y : A ⟶ B is left dualizable iff ν is an isomorphism when W = Y, and then ν is an isomorphim for all W and Y ≅ D_{r}D_{ℓ}Y.

THE BICATEGORY ℰ x

• 0-cells are spaces, with tB = B.

• ℰ x(A, B) = Ho𝒮_{B×A} with t = t^{∗},

t: B × A ≅ A × B.

• Composition

⊙: Ho𝒮_{C×B} × Ho𝒮_{B×A} ⟶ Ho𝒮_{C×A}

Y ⊙ X = θ_{B}(Y ⊼ X)

Y ⊼ X is a spectrum over C × B × B × A.

θ_{B}: Ho𝒮_{C×B×B×A} ⟶ Ho𝒮_{C×A}

is

π_{C×A}_{!}(id ×Δ_{B} × id)^{∗}.

• The unit U_{B} is Δ_{!}S_{B}.

• θ_{B} has right adjoint θ_{B}^{∗} = (id ×Δ_{B} × id)_{∗}π_{C×A}^{∗}.

Z ◁ Y = F̄(Y, θ_{B}^{∗}Z): A ⟶ B

X ▷ Z = F̄(X, θ_{B}^{∗}Z): B ⟶ C

X: A ⟶ B, Y : B ⟶ C, Z: A ⟶ C.

34 J.P. MAY

TILTING SPECTRA

Theorem 41. For f: B ⟶ A, there are

S_{f}: B ⟶ A and tS_{f}: A ⟶ B such that (tS_{f}, S_{f}) is a dual pair and:

f_{!}Y ≃ Y ⊙ tS_{f}

f_{!}tY ≃ S_{f} ⊙ tY
X ⊙ S_{f} ≃ f^{∗}X ≃ tS_{f} ▷ X
tS_{f} ⊙ tX ≃ f^{∗}tX ≃ X ◁ S_{f}
f_{∗}Y ≃ S_{f} ▷ tY
f_{∗}tX ≃ X ◁ S_{f}.

For r: B ⟶ ∗,

S_{r} ≅ S_{B} ≅ tS_{r}.

FROM THE BEGINNING (ROSENDAL TALK) 35

Theorem 42. Let (X, Y) be a dual pair, X: B ⟶ A and Y : A ⟶ B, and let f: B ⟶ C be a map. Then

((id ×f)_{!}X, (f × id)_{!}Y)

is a dual pair. With C = ∗ and f = r,

(r_{!}X, r_{!}Y)

is a dual pair. (A = ∗: duality in 𝒮).

Proof. This says that

(X ⊙ tS_{f}, S_{f} ⊙ Y)

is a dual pair. Since (tS_{f}, S_{f}) is a dual pair, this is formal. □

36 J.P. MAY

Have two copies ℰ x(∗, B) and ℰ x(B, ∗) of HoG𝒮_{B} in ℰ x and the copy ℰ x(∗, ∗) of HoG𝒮. Also have ℰ x(B, B) in play.

Notations 43. View a spectrum X over B as a 1-cell B ⟶ ∗. Write tX for X viewed as a 1-cell ∗ ⟶ B. For X and Y over B,

X ⊙ tY : ∗ ⟶ ∗

is a spectrum, but

tY ⊙ X: B ⟶ B

is a spectrum over B × B.

Proposition 44. Let X and E be spectra over B. As spectra,

E ⊙ tX ≃ r_{!}(E ∧_{B} X)

and tE ◁ tX ≃ r_{∗}F_{B}(X, E) ≃ X ▷ E.

Therefore

E_{∗}(X) ≅ π_{∗}(E ⊙ tX)

and

E^{∗}(X) ≅ π_{−∗}(X ▷ E).

FROM THE BEGINNING (ROSENDAL TALK) 37

Duals with respect to ∗:

D_{ℓ}X = S ◁ X: ∗ ⟶ B

D_{r}tX = tX ▷ S: B ⟶ ∗,

Both are the spectrum D_{B}(X) over B. Proposition 45. A spectrum X over B is fiberwise dualizable with dual Y if and only if (tX, Y) is a dual pair in ℰ x. Duals with respect to B; write B for Δ_{!}S_{B}.

D_{ℓ}tX = B ◁ tX: B ⟶ ∗

D_{r}X = X ▷ B: ∗ ⟶ B.

They are the same spectrum over B. Definition 46. The Costenoble-Waner dual of X in HoG𝒮_{B} is

D_{B}^{CW}X = F̄(X, B).

Viewed as a 1-cell in ℰ x,

D_{B}^{CW}X ≃ B ◁ tX = D_{ℓ}X: B ⟶ ∗ and

tD_{B}^{CW}X ≃ X ▷ B = D_{r}X: ∗ ⟶ B.

38 J.P. MAY

Definition 47. X is Costenoble-Waner [CW] dualizable with dual Y if (X, tY) is a dual pair in ℰ x. Then Y ≃ D_{B}^{CW}X.

Proposition 48. X is CW dualizable with dual Y if and only if Y is CW dualizable with dual X, and then

X ≃ D_{B}^{CW}D_{B}^{CW}X.

Proposition 49. If X is CW dualizable and E is any spectrum over B, then

r_{!}(E ∧_{B} D_{B}^{CW}X) ≃ E ⊙ tD_{B}^{CW}X
μ
r_{∗}F_{B}(X, E) ≃ X ▷ E

is an equivalence of spectra. Therefore

E_{∗}(D_{B}^{CW}X) ≅ E^{−∗}(X).

FROM THE BEGINNING (ROSENDAL TALK) 39

When A = B = ∗, duality theory in ℰ x reduces to Spanier-Whitehead [SW] duality theory in Ho𝒮.

Corollary 50. If S_{B} is CW dualizable, then Σ^{∞}(B_{+}) is SW dualizable.

Proof. r_{!}S_{B} ≃ Σ^{∞}(B_{+}). □

In general S_{B} is fiberwise but not CW dualizable, whereas finite cell spectra are CW but not fiberwise dualizable.

Theorem 51. If a spectrum X over B is a retract in HoG𝒮_{B} of a finite cell spectrum, then X is CW dualizable.

Will state steps of proof. Say that a space T is CW dualizable if S_{T} is CW dualizable.

Theorem 52 (Parametrized Atiyah duality). Any smooth compact G-manifold M is CW dualizable.

Applying r_{!}, this reproves Atiyah duality.

Corollary 53. Any G-space G/H × S^{n} is CW dualizable.

Lemma 54. (K, p) a space over B. Then (K, p)_{+} = K ⨿ B is isomorphic to p_{!}S_{K}^{0}.

Proposition 55. If K is CW dualizable and (K, p) is a space over B, then Σ_{B}^{∞}(K, p)_{+} is CW dualizable.

Proof. p_{!}S_{K} = Σ_{B}^{∞}(K, p)_{+}, so done. □

Implies one cell case. Induction:

Lemma 56. The cofiber of a map between CW dualizable spectra is CW dualizable.

Lemma 57. A retract of a CW dualizable spectrum is CW dualizable.

Converse? Don’t know.

Question 58. Is S_{M} equivalent in HoG𝒮_{M} to a retract of a finite cell spectrum over M?

FROM THE BEGINNING (ROSENDAL TALK) 41

Let (F, C_{F}) be a dual pair, so that F is CW dualizable with right dual C_{F}. Then

μ: X ⊙ C_{F} ⟶ S_{F} ▷ X is an equivalence for X over F. This is

r_{!}(X ∧_{F} C_{F}) ≃ r_{∗}X.

F = G/H: the Wirthmüller isomorphism.

Fiberwise generalization:

Let G = Γ/Π, where Π◁Γ. Let p: E ⟶ B be a (Π; Γ)-bundle. Fiber a CW dualizable Γ-space F with right dual C_{F}. Associated principal (Π; Γ)-bundle P ⟶ B.

Theorem 59. Set

C_{p} = P ×_{Π} C_{F}.

Then

(S_{p}, Δ_{!}C_{p} ⊙ tS_{p})

is a dual pair and, via a duality map μ,

p_{!}(X ∧_{E} C_{p}) ≃ p_{∗}X

for all spectra X over E.

The Adams isomorphism is a special case.