DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS

J.P. MAY

In October, 1999, I organized a conference here in honor of MacLane’s 90th birthday. I’ll repeat how I started my talk then. “A great deal of modern mathematics would quite literally be unthinkable without the language of categories, functors, and natural transformations introduced by Eilenberg and MacLane in 1945. It was perhaps inevitable that some such language would have appeared eventually. It was certainly not inevitable that such an early systematization would have proven so remarkably durable and appropriate.

I talked then about another piece of language that Saunders introduced, that of symmetric monoidal category, which comes from a 1963 paper entitled “Natural associativity and commutativity”. It gave the first clear articulation of the general categorical problem of coherence. The necessary isomorphisms are called “constraints” in the later literature of algebraic geometry. To quote from Mathematical Reviews, “A very pleasant feature of the paper is the skillful, economical and rigorous manner in which the problem is formulated”.

More precisely, what I talked about then was duality theory in symmetric monoidal categories. As Mac Lane so well understood, category theory turns analogies into mathematical theory that turns the things being compared into examples. The category theory then allows one to understand new things about the examples, and to compare them rigorously. Duality theory gives a strikingly beautiful and important illustration of this kind of mathematics. The purpose of this talk is to explain duality theory in symmetric bicategories. This is a new theory whose basic definitions are less than a year old. It is joint work with Johann Sigurdsson, but its starting point was a key insight in work of Steven Costenoble and Stefan Waner.

Jean Benabou, who I think was here at the time, already introduced bicategories in 1967. There is a notion of a symmetric monoidal bicategory, but I only mention that for the well-informed category theorists. The notion of symmetric bicategories is different. The new theory is not difficult, and it could well have been developed immediately after Benabou’s work. However, the need for it only became apparent with the development of parametrized stable homotopy theory. Niles Johnson is beginning to verify my belief that it also gives the definitively right framework for understanding Morita duality in derived categories. Surprisingly, Kate Ponto discovered that it also gives the definitively right framework for understanding topological fixed point theory. Here even the analogies are not immediately obvious.

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OUTLINE

(1) Duality in symmetric monoidal categories (2) Closed symmetric bicategories

(3) Duality in closed symmetric bicategories (4) Composites of dualities

(5) Parametrized spectra

(6) Dictionaries

(7) Parametrized duality theory

(8) Parametrized homology & cohomology 1. Duality in symmetric monoidal

categories

(𝒞 , ⊗, Hom, I, γ) ⊗: 𝒞 × 𝒞 ⟶ 𝒞

Unital, associative, and commutative up to coherent natural isomorphism; associativity and unit isomorphisms are implicit, and γ is the commutativity isomorphism.

I ⊗ X ≅ X X ≅ Hom(I, X)

Hom(X ⊗ Y, Z) ≅ Hom(X, Hom(Y, Z))

Hom(Y ⊗ X, Z) ≅ Hom(X, Hom(Y, Z))

Hom(X, Y) ⊗ Hom(X′, Y′)
⊗
Hom(X ⊗ X′, Y ⊗ Y′)

ν : Hom(X, Y) ⊗ Z ⟶ Hom(X, Y ⊗ Z)

DX ≡ Hom(X, I)

Evaluation map: ε : DX ⊗ X ⟶ I.

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Definition 1. X is dualizable if

ν : DX ⊗ X ⟶ Hom(X, X) is an isomorphism. Coevalutation map:

\xymatrix{I \ar[d]_{ι} \ar[r]^{η} & X ⊗ DX \\ Hom(X, X) & DX ⊗ X \ar[l]^{ν} \ar[u]_{γ}}

Here ι is adjoint to id: X ⟶ X.

ℳ_{R}, 𝒟_{R}, Ho𝒮 sh(X) Finitely projective R-module Perfect chain complex Finite CW-spectrum

Theorem 2. Fix X and Y. TFAE.

(i) X is dualizable and Y ≅ DX. (ii) Triangle identities: There are η : I ⟶ X⊗Y and ε : Y ⊗X ⟶ I s.t.

\xymatrix{X ≅ I ⊗ X \ar[r]^{η⊗id} & X ⊗ Y ⊗ X \ar[r]^{id ⊗ε} & X ⊗ I ≅ X}
\xymatrix{Y ≅ Y ⊗ I \ar[r]^{id ⊗η} & Y ⊗ X ⊗ Y \ar[r]^{ε⊗id} & I ⊗ Y ≅ Y}

are identity maps.

(iii) There is a map η : I ⟶ X ⊗ Y such that the composite

𝒞 (W ⊗ X, Z)
(−)⊗Y
𝒞 (W ⊗ X ⊗ Y, Z ⊗ Y)
𝒞 (id ⊗η,id)
𝒞 (W, Z ⊗ Y)

is a bijection for all W and Z. (iv) There is a map ε : Y ⊗ X ⟶ I such

that the composite

𝒞 (W, Z ⊗ Y)
(−)⊗X
𝒞 (W ⊗ X, Z ⊗ Y ⊗ X)
𝒞 (id,id ⊗ε)
𝒞 (W ⊗ X, Z)

is a bijection for all W and Z.

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The adjoint ε̃ : Y ⟶ DX of ε satisfying (ii) or (iv) is an isomorphism under which the given map ε corresponds to the canonical evaluation map ε : DX ⊗ X ⟶ I. Terminology: (X, Y) is a dual pair.

Proposition 3. If X and Y are dualizable, then DX and X ⊗ Y are dualizable and the canonical map ρ : X ⟶ DDX is an isomorphism.

Proposition 4. If X or Z is dualizable, then the map ν above is an isomorphism.

Proposition 5. If X and X′ are dualizable or if X is dualizable and Y = I, then the map ⊗ above is an isomorphism.

Definition 6. X is invertible if there is a Y and an isomorphism X ⊗ Y ≅ I. Then (X, Y) is a dual pair. Pic(𝒞) = group of isomorphism classes of invertible X.

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2. Closed symmetric bicategories

A bicategory 𝒞 is a monoidal category with many objects, called 0-cells. For each pair of 0-cells A, B, it has a category 𝒞 (A, B) of object 1-cells A → B and morphism 2-cells X → Y between 1-cells X, Y : A → B. There are “horizontal” compositions

⊙: 𝒞 (B, C) × 𝒞 (A, B) ⟶ 𝒞 (A, C)

and unit 1-cells I_{A} [abbreviated A by abuse] such that ⊗ is associative and unital up to coherent natural isomorphism. If 𝒞 has one object ∗, then 𝒞 (∗, ∗) is a monoidal category.

𝒞 at: categories, functors, nat. trans. ℬ_{R}: R-algebras, (B, A)-bimodules, maps ℰ x: spaces, spectra over B × A, maps

Symmetric version? Directionality of 1-cells? Turns out that symmetric bicategories with one object are not just symmetric monoidal categories.

𝒞^{op}: same 0-cells,

𝒞^{op}(A, B) = 𝒞 (B, A).

Definition 7. An involution on 𝒞 consists of the following data subject to coherence diagrams.

(1) A bijection t on 0-cells with ttA = A.

(2) Equivalences of categories

t: 𝒞 (A, B) ⟶ 𝒞 (tB, tA) = 𝒞^{op}(tA, tB),

with isomorphism 2-cells ξ: id ≅ tt.

(3) Isomorphism 2-cells

ι: I_{tA} ⟶ tI_{A} and

γ: tY ⊙^{op} tX ≡ tX ⊙ tY ⟶ t(Y ⊙ X).

A symmetric bicategory 𝒞 is a bicategory with an involution. A 0-cell A of 𝒞 is said to be commutative if tA = A.

t: 𝒞 (A, A) ⟶ 𝒞 (A, A) need not be the identity functor, so a one object symmetric bicategory need not be a symmetric monoidal category.

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Example 8. ℬ_{R}. tA is the opposite R-algebra of A. The commutative 0-cells are the commutative R-algebras. The 1-cells X: A → B are the (B, A)- bimodules, and tX: tB → tA is the same R-module X viewed as a (tA, tB)-bimodule. The 2-cells α: X → Y are the maps of (B, A)-bimodules, and t(α) is the same Rmap viewed as a map of (tA, tB)-bimodules.

ℬ_{R}(B, C) × ℬ_{R}(A, B)
⊙=⊗_{B}
ℬ_{R}(A, C).

I_{A} is A regarded as an (A, A)-bimodule,

γ: tX ⊗_{tB} tY ⟶ t(Y ⊗_{B} X)

is given by γ(x ⊗ y) = y ⊗ x.

ℬ_{R}(A, R) = right A-modules
ℬ_{R}(R, A) = left A-modules

tA = A ⇒ ℬ_{R}(A, A) = (A, A)-bimodules.

Here t ≠ id: ℬ_{R}(A, A) ⟶ ℬ_{R}(A, A). The symmetric monoidal full subcategory

ℳ_{A} ⊂ ℬ_{R}(A, A)

consists of the central (A, A)-bimodules X, those for which ax = xa for all a and x.

Example 9. Graded version of ℬ_{R} has γ defined with a sign. Leads to differential graded version. Topological version in brave new algebra.

ℬ_{R} and other examples arise from anchored bicategories by neglect of structure. Maps R ⟶ S of commutative rings and maps A ⟶ B of algebras have been ignored. LOGIC: ℬ_{R} is EASY. Use it as a toy model to understand deeper examples, such as those above and parametrized spectra.

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 11

Definition 10. A bicategory 𝒞 is right and left closed if there are right and left internal hom functors ▷: 𝒞 (A, B)^{op} × 𝒞 (A, C) ⟶ 𝒞 (B, C) and ◁: 𝒞 (A, C) × 𝒞 (B, C)^{op} ⟶ 𝒞 (A, B) such that

𝒞 (Y ⊙ X, Z) ≅ 𝒞 (Y, X ▷ Z)

𝒞 (Y ⊙ X, Z) ≅ 𝒞 (X, Z ◁ Y)

for X: A ⟶ B, Y : B ⟶ C, Z: A ⟶ C. Can’t write X ⊙ Y, not defined!

Unit and counit 2-cells ε: (X▷Z)⊙X ⟶ Z, η: Y ⟶ X▷(Y ⊙X) ε: Y ⊙(Z◁Y) ⟶ Z, η: X ⟶ (Y ⊙X)◁Y. Definition 11. A symmetric bicategory is closed if it is left closed. It is then also right closed with

X ▷ Z ≅ t(tZ ◁ tX).

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In symmetric monoidal categories, use of γ collapses ◁ and ▷ into Hom; replaced by involutive relation in symmetric bicategories.

Remark 12. X ▷ Z is the 1-cell of “maps pointing right from X to Z”; Z ◁ Y is the 1-cell of “maps pointing left from Y to Z”. X ▷ Z is a 1-cell from the target of X to the target of Z (with X and Z having the same source) and Z ◁ Y is a 1-cell from the source of Z to the source of Y (with Y and Z having the same target).

Example 13. ℬ_{R} is closed. For a (B, A)-bimodule X,

(C, B)-bimodule Y,
(C, A)-bimodule Z,

Z ◁ Y = Hom_{C}(Y, Z), a (B, A)-bimodule and X▷Z = Hom_{A}(X, Z), a (C, B)-bimodule.

ℬ_{R}(Y ⊗_{B} X, Z) ≅ ℬ_{R}(Y, Hom_{A}(X, Z))

ℬ_{R}(Y ⊗_{B} X, Z) ≅ ℬ_{R}(X, Hom_{C}(Y, Z))

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Example 14. Base change bimodules. For an R-algebra map f: A′ ⟶ A

A_{f}: A′ ⟶ A, an (A, A′)-bimodule

fA = tA_{f}: A ⟶ A′ an(A′, A)-bimodule. For an R-algebra B, pullback action functor

f^{∗}: ℬ_{R}(A, B) ⟶ ℬ_{R}(A′, B)

has a left adjoint f_{!} and a right adjoint f_{∗}, extension and coextension of scalars.

f^{∗}M ≅ M ⊙ A_{f}
f_{!}M′ ≅ M′ ⊙_{f}A
f_{∗}M′ ≅ A_{f} ▷ M′.

For g: B′ ⟶ B and any A,

g^{∗}: ℬ_{R}(A, B) ⟶ ℬ_{R}(A, B′)

has a left adjoint g_{!} and a right adjoint g_{∗},

g^{∗}N ≅ _{g}B ⊙ N,
g_{!}N′ ≅ B_{g} ⊙ N′,
g_{∗}N′ ≅ _{g}B ◁ N′.

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Formal isomorphisms:

(Y ⊙ X) ▷ Z ≅ Y ▷ (X ▷ Z)

Z ◁ (Y ⊙ X) ≅ (Z ◁ Y) ◁ X
(X ▷ Z) ◁ Y ≅ X ▷ (Z ◁ Y)

Formal maps: μ: Z ⊙ (X ▷ Y) ⟶ X ▷ (Z ⊙ Y) ν: (Z ◁ Y) ⊙ W ⟶ (Z ⊙ W) ◁ Y

ω: (Z ◁Y)⊙(X ▷W) ⟶ X ▷(Z ⊙W)◁Y (Two ways of parenthesizing are isomorphic) DUALITY THEORY: WHEN ARE THESE ISOMORPHISMS?

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3. Duality in closed symmetric bicategories

Write B for I_{B} [So B is a 0-cell and 1-cell].

Definition 15. (X, Y), X: B ⟶ A and Y : A ⟶ B, is a dual pair of 1-cells if there are 2-cells

η: A ⟶ X ⊙ Y and ε: Y ⊙ X ⟶ B,

coevaluation and evaluation maps, such that the following diagrams commute in 𝒞 (B, A) and 𝒞 (A, B), respectively.

\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}}

X is right dualizable with right dual Y. Y is left dualizable with left dual X.

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Example 16. Let f: B ⟶ A be a map of R-algebras. We have

A_{f}: B ⟶ A and _{f}A: A ⟶ B. fA ⊙ A_{f}: B ⟶ B is A, regarded as a (B, B)-bimodule. Let

η = f: B ⟶ _{f}A ⊙ A_{f}. A_{f} ⊙_{f}A = A ⊗_{B} A. Let

ε: A_{f} ⊙_{f}A ⟶ A

be given by the product on A. Then (η, ε) display (_{f}A, A_{f}) as a dual pair; the left and right unit laws for A imply the diagrams. (A_{f},_{f}A) is not a dual pair in general.

Example 17. Specialize. Take f to be the unit ι: R ⟶ A of an R-algebra with product ϕ: A ⊗_{R} A ⟶ A. (ι, ϕ) display (_{ι}A, A_{ι}) as a dual pair. For (A_{ι},_{ι}A) to be a dual pair, we need a coproduct and counit

η: A ⟶ A ⊗_{R} A

ε: A ⟶ R

such that the left and right counit laws hold.

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 17

In topology, the situation is reversed: spaces have a coproduct and a counit, but not a product and a unit.

Proposition 18. If 𝒞 is symmetric, then η and ε exhibit (X, Y) as a dual pair iff

γ^{−1}t(η) : tA ⟶ tY ⊙ tX

and

t(ε)γ: tX ⊙ tY ⟶ tB

exhibit (tY, tX) as a dual pair.

Analogous to symmetric monoidal duality, given 1-cells X: B ⟶ A and Y : A ⟶ B and a 2-cell ε: Y ⊙ X ⟶ B, we have

ε_{#} : 𝒞 (W, Z ⊙ Y) ⟶ 𝒞 (W ⊙ X, Z).

For a 2-cell η: A ⟶ X ⊙ Y, we have

η_{#} : 𝒞 (W ⊙ X, Z) ⟶ 𝒞 (W, Z ⊙ Y)

(Both for W: A ⟶ C, Z: B ⟶ C.) Duality (η, ε): these are inverse isomorphisms.

Proposition 19. Let

X: B ⟶ A and Y : A ⟶ B

be 1-cells. TFAE for a given 2-cell

ε: Y ⊙ X ⟶ B.

(1) (X, Y) is a dual pair with evaluation map ε.

(2) ε_{#} is a bijection for all W and Z.

(3) ε_{#} is a bijection when W = A and Z = X and when W = Y and Z = B.

Dually, TFAE for a given 2-cell

η: A ⟶ X ⊙ Y.

(1) (X, Y) is a dual pair with coevaluation map η.

(2) η_{#} is a bijection for all W and Z.

(3) η_{#} is a bijection when W = A and Z = X and when W = Y and Z = B.

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 19

Proposition 20. Let X, X′: B ⟶ A and Y, Y′: A ⟶ B be 1-cells such that (X, Y) and (X′, Y′) are dual pairs and let α: X ⟶ X′ and β: Y ⟶ Y′ be 2-cells.

(1) There is a unique 2-cell α^{∗}: Y′ ⟶ Y that makes either of the following diagrams commute, and then the other diagram also commutes.

\xymatrix{A \ar[r]^{η} \ar[d]_{η} & X′ ⊙ Y′ \ar[d]^{id ⊙α^{∗}} & Y′ ⊙ X \ar[r]|{id ⊙α} \ar[d]_{α^{∗}⊙id} & Y′ ⊙ X′ \ar[d]^{ε} \\ X ⊙ Y \ar[r]_{α⊙id} & X′ ⊙ Y & Y ⊙ X \ar[r]_{ε} & B}

(2) There is a unique 2-cell β^{∗}: X′ → X that makes either of the following diagrams commute, and then the other diagram also commutes.

\xymatrix{A \ar[r]^{η} \ar[d]_{η} & X′ ⊙ Y′ \ar[d]^{β^{∗}⊙id} & Y ⊙ X′ \ar[r]^{β⊙id} \ar[d]_{id ⊙β^{∗}} & Y′ ⊙ X′ \ar[d]^{ε} \\ X ⊙ Y \ar[r]_{id ⊙β} & X ⊙ Y′ & Y ⊙ X \ar[r]_{ε} & B}

Now assume that 𝒞 is closed.

Definition 21. For a 1-cell X: B ⟶ A, define

D_{r}X = X ▷ B: A ⟶ B.

For a 1-cell Y : A ⟶ B, define

D_{ℓ}Y = B ◁ Y : B ⟶ A.

A 2-cell ε: Y ⊙ X ⟶ B has adjoints

ε̃: X ⟶ D_{ℓ}Y and ε̃: Y ⟶ D_{r}X.

\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 22. If ε: Y ⊙ X ⟶ B is the evaluation map of a dual pair (X, Y), then the adjoint 2-cells

ε̃: X ⟶ D_{ℓ}Y and ε̃: Y ⟶ D_{r}X

are isomorphisms.

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 21

The maps μ and ν specialize to

(1)  μ: Z ⊙ D_{r}X ⟶ X ▷ Z

and

(2)  ν: D_{ℓ}Y ⊙ W ⟶ W ◁ Y.

Proposition 23. The following are equivalent for a 1-cell X: B ⟶ A, and then (X, D_{r}X) is canonically a dual pair.

(1) X is right dualizable.

(2) μ is an isomorphism when Z = X.

(3) μ is an isomorphism for all Z.

When these hold, the adjoint of ε is an isomorphism X ⟶ D_{ℓ}D_{r}X. Dually, the following are equivalent for a 1-cell Y : A ⟶ B, and then (D_{ℓ}Y, Y) is canonically a dual pair.

(1) Y is left dualizable.

(2) ν is an isomorphism when W = Y.

(3) ν is an isomorphism for all W.

When these hold, the adjoint of ε is an isomorphism Y ⟶ D_{r}D_{ℓ}Y.

Left and right analogues of X ≅ DDX. Other analogues.

Proposition 24. Consider μ, ν, and ω.

(1) If X or Z is right dualizable, then

μ: Z ⊙ (X ▷ Y) ⟶ X ▷ (Z ⊙ Y)

is an isomorphism.

(2) If W or Y is left dualizable, then

ν: (Z ◁ Y) ⊙ W ⟶ (Z ⊙ W) ◁ Y

is an isomorphism.

(3) If X is right and Y is left dualizable, then

ω: (Z ◁Y)⊙(X ▷W) ⟶ X ▷(Z ⊙W)◁Y

is an isomorphism.

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 23

4. Composites of dualities

Theorem 25. Consider 1-cells W, X, Y, Z as in the diagram

\xymatrix{C \ar[r]^{W} & B \ar[l]^{Z} \ar[r]^{X} & A. \ar[l]^{Y}}

If (η, ε) exhibits (X, Y) as a dual pair and (ζ, σ) exhibits (W, Z) as a dual pair, then

((id ⊙ζ ⊙ id) ∘ η, σ ∘ (id ⊙ϵ ⊙ id)) exhibits (X ⊙ W, Z ⊙ Y) as a dual pair.

Theorem 26. Let F: ℬ ⟶ 𝒞 be a lax functor between symmetric bicategories. Let (X, Y) be a dual pair in ℬ,

X: B ⟶ A and Y : A ⟶ B. Assume that the unit and composition coherence 2-cells

I_{F B} ⟶ F I_{B}

and

F X ⊙ F Y ⟶ F (X ⊙ Y)

are isomorphisms. Then (F X, F Y) is a dual pair in 𝒞. There is a dual result for oplax functors.

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5. Parametrized spectra

Ho𝒮_{B}: The homotopy category of spectra X parametrized over B. For each b ∈ B, there is a “fiber spectrum” X_{b}, and these are nicely glued together. Base change functors for f: A ⟶ B:

f_{!} : Ho𝒮_{A} ⟶ Ho𝒮_{B},

f^{∗}: Ho𝒮_{B} ⟶ Ho𝒮_{A},
f_{∗}: Ho𝒮_{A} ⟶ Ho𝒮_{B}.

The closed symmetric bicategory ℰ x has as 0-cells spaces B, with tB = B, and it has

ℰ x(A, B) = Ho𝒮_{B×A}.

t: B×A ≅ A×B induces involution t = t^{∗}. For X over B × A and Y over C × B, we have a fiberwise smash product

Y ⊼ X over C × B × B × A.

Y ⊙X = (id_{C} ×r×id_{A})_{!}(id_{C} ×Δ×id_{A})^{∗}(Y ⊼X) where Δ: B ⟶ B ×B is the diagonal and r: B ⟶ ∗ is the unique map to a point. The unit I_{B} over B × B is Δ_{!}S_{B}, where S_{B} is the sphere spectrum over B.

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Ho𝒮_{B} is closed symmetric monoidal with unit S_{B} under the fiberwise smash product

X ∧_{B} Y = Δ^{∗}(X ⊼ Y).

DUALITY IN Ho𝒮_{B} VERSUS DUALITY IN ℰ x

Embed Ho𝒮_{B} in ℰ x as

Ho𝒮_{∗×B} = ℰ x(B, ∗)

and

Ho𝒮_{B×∗} = ℰ x(∗, B)

1-cells X: B ⟶ ∗ and tX: ∗ ⟶ B. ℰ x informs on Ho𝒮_{B} by relating it with

Ho𝒮 = ℰ (∗, ∗)

and

Ho𝒮_{B×B} = ℰ x(B, B).

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For f: B ⟶ A, have base change spectra

S_{f}: B ⟶ A and tS_{f}: A ⟶ B Theorem 27. (tS_{f}, S_{f}) is a dual pair.

Ignoring source and target, for r: B ⟶ ∗, S_{r} ≃ S_{B} ≃ tS_{r}.

(f × id)_{!}B ≃ S_{f} ≃ (id ×f)^{∗}A
(id ×f)_{!}B ≃ tS_{f} ≃ (f × id)^{∗}A.

\underline{Change of source}: For f: A ⟶ A′ Y ⊙ S_{f} ≃ (id ×f)^{∗}Y ≃ tS_{f} ▷ Y : A ⟶ B,

X ⊙ tS_{f} ≃ (id ×f)_{!}X: A′ ⟶ B,

S_{f} ▷ X ≃ (id ×f)_{∗}X: A′ ⟶ B

for 1-cells X: A ⟶ B and Y : A′ ⟶ B. \underline{Change of target}: For g: B ⟶ B′ tS_{g} ⊙ Z ≃ (g × id)^{∗}Z ≃ Z ◁ S_{g}: A ⟶ B,

S_{g} ⊙ X ≃ (g × id)_{!}X: A ⟶ B′,

X ◁ tS_{g} ≃ (g × id)_{∗}X: A ⟶ B′

for 1-cells X: A ⟶ B and Z: A ⟶ B′.

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6. Dictionaries

Let X, Y be spectra over B. Let Z be a spectrum. X ⊙ tY is a spectrum. tY ⊙ X is a spectrum over B × B.

Translate ⊙, ◁, ▷ into base change, ∧_{B}, and the internal hom F_{B}(X, Y) in Ho𝒮_{B}: As spectra,

Y ⊙ tX ≃ r_{!}(Y ∧_{B} X)

tY ◁ tX ≃ r_{∗}F_{B}(X, Y) ≃ X ▷ Y.

As spectra over B,

tX ⊙ Z ≃ X ⊼ Z
Z ⊙ X ≃ Z ⊼ X

Z ◁ X ≃ F_{B}(X, r^{∗}Z) ≃ tX ▷ Z.

As spectra over B × B,

tY ⊙ X ≃ Y ⊼ X.

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Translate ∧_{B}, F_{B} into ⊙, ◁, ▷.

Δ_{!}X ⊙ tY ≃ X ∧_{B} Y ≃ X ⊙ Δ_{!}Y, Δ_{!}X ▷ Y ≃ F_{B}(X, Y) ≃ Δ^{∗}(X ▷ Y), X ◁ Δ_{!}Y ≃ F_{B}(Y, X) ≃ Δ^{∗}(X ◁ Y).

Translate base change into ⊙, ◁, ▷. Let f: B ⟶ A be a map, X be a spectrum over A, and Y be a spectrum over B. 1-cells

A ⟶ ∗ and B ⟶ ∗. f_{!}Y ≃ Y ⊙ tS_{f} tf_{!}Y ≃ S_{f} ⊙ tY

X ⊙ S_{f} ≃ f^{∗}X ≃ tS_{f} ▷ X tS_{f} ⊙ tX ≃ tf^{∗}X ≃ tX ◁ S_{f}

f_{∗}Y ≃ S_{f} ▷ Y tf_{∗}Y ≃ tY ◁ tS_{f}.

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 29

7. Parametrized duality theory

\underline{Fiberwise duality}: symmetric monoidal duality in Ho𝒮_{B}. S = unit in ℰ (∗, ∗) Δ_{!}S_{B}=unit in ℰ (B, B). Duals wrt S:

D_{ℓ}X = S ◁ X: ∗ ⟶ B

D_{r}tX = tX ▷ S: B ⟶ ∗.

Duals wrt Δ_{!}S_{B}: D_{ℓ}tX = Δ_{!}S_{B} ◁ tX: B ⟶ ∗

D_{r}X = X ▷ Δ_{!}S_{B}: ∗ ⟶ B.

\underline{Duality wrt S}: tX is right dualizable iff X is left dualizable, and tD_{ℓ}X ≃ D_{r}tX. Viewed in Ho𝒮_{B}, these are

D_{B}X = F_{B}(X, S_{B}) = F̄(X, Δ_{∗}S_{B}). (F̄ is external hom that goes with ⊼). (tX, Y) is a dual pair iff (tY, X) is a dual pair, and then Y ≃ D_{B}X. η: Δ_{!}S_{B} ⟶ tX ⊙ Y ε: Y ⊙ tX ⟶ S

Proposition 28. (tX, Y) is a dual pair iff (X, Y) is a fiberwise dual pair.

30 J.P. MAY

\underline{Duality wrt Δ_{!}S_{B}}: X is right dualizable iff tX is left dualizable, and D_{ℓ}tX ≃ tD_{r}X. Viewed in Ho𝒮_{B}, these are

D_{B}^{CW}X = F̄(X, Δ_{!}S_{B}).

(X, tY) is a dual pair iff (Y, tX) is a dual pair, and then Y ≃ D_{B}^{CW}X.

η: S ⟶ X ⊙ tY ε: tY ⊙ X ⟶ Δ_{!}S_{B}

Right dualizable ≡ Costenoble-Waner dualizable, abbreviated CW-dualizable.

Proposition 29. X is CW-dualizable with dual Y iff Y is CW-dualizable with dual X, and then X ≃ D_{B}^{CW}D_{B}^{CW}X.

Proposition 30. If X is CW dualizable and J is any spectrum over B, then

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

is an equivalence of spectra.

Definition 31. An ex-space K over B is CW-dualizable if Σ_{B}^{∞}K is CW dualizable.

Theorem 32. If M is a smooth compact n-manifold, then S_{M}^{0} = M ⨿ M over M is CW-dualizable. Its dual is Σ_{M}^{−q}S^{ν}.

S^{ν} is the fiberwise one-point compactification of the normal bundle ν of M ⊂ ℝ^{q}.

Proposition 33. If S_{K}^{0} is CW-dualizable and (K, p) is a space over B, then

(K, p) ⨿ B ≅ p_{!}S_{K}^{0}

is CW-dualizable.

Theorem 34. If X is a wedge summand of a “finite cell spectrum”, then X is CW-dualizable.

Proof. A cell is (D^{n}, p) ⨿ B; induction. □

Duality in ℰ x when A = B = ∗ is SW-duality (Spanier-Whitehead).

Proposition 35. If S_{B} is CW-dualizable, then r_{!}S_{B} = Σ^{∞}B_{+} is SW-dualizable, so B is equivalent to a finite CW-complex.

If B infinite, then S_{B} is not CW-dualizable. It is invertible, hence fiberwise dualizable. Let M be a smooth closed manifold. Then (S_{M}, tΣ_{M}^{−q}S^{ν}) is a dual pair. Also, Σ_{M}^{−q}S^{ν} is invertible in Ho𝒮_{M} with inverse Σ_{M}^{∞}S^{τ}:

Σ_{M}^{∞}S^{τ} ∧_{M} Σ_{M}^{−q}S^{ν} ≃ Σ^{−q}(S^{τ} ∧_{M} S^{ν}) ≃ S_{M}

since τ ⊕ ν is trivial and S^{τ} ∧_{M} S^{ν} ≅ S_{M}^{V}. Combining these dualities leads to homotopical Poincaré duality.

Theorem 36. For a spectrum k,

k ∧ M_{+} ≃ S_{M} ▷ (k ∧ S^{τ}).

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 33

8. Parametrized homology and cohomology

For spectra J and X over B, define

J_{n}(X) = π_{n}(r_{!}(J ∧_{B} X))
J^{n}(X) = π_{−n}(r_{∗}F_{B}(X, J))
≅ [S_{B}^{−n}, F_{B}(X, J)]_{B}.

For a spectrum k,

(r^{∗}k)_{∗}(X) ≅ k_{∗}(r_{!}X) ≡ k_{∗}^{B}(X)
(r^{∗}k)^{∗}(X) ≅ k^{∗}(r_{!}X) ≡ k_{B}^{∗}(X).
r_{!}(J ∧_{B} X) ≃ J ⊙ tX
r_{∗}F_{B}(X, J) ≃ X ▷ J.
J_{∗}(X) = π_{∗}(J ⊙ tX)
J^{∗}(X) = π_{−∗}(X ▷ J).

If (X, Y) is a dual pair,

J ⊙ tY ≃ X ▷ J

Theorem 37 (Costenoble-Waner duality). J_{∗}(Y) ≅ J^{−∗}(X).

Let M be a smooth closed n-manifold. Have

k ∧ M_{+} ≃ S_{M} ▷ (k ∧ S^{τ}).

Take X = S_{M} and J = k ∧ S^{τ}.

Theorem 38 (Poincaré duality).

k_{∗}(M_{+}) ≅ (k ∧ S^{τ})^{−∗}(S_{M}).

Definition 39. The Thom complex of an n-plane bundle ξ over B is

T ξ = S^{ξ}/s(B) = r_{!}S^{ξ}.

Let k be a commutative ring spectrum. A korientation of ξ is a class μ ∈ k^{n}(T ξ) whose fiber restriction

μ_{b} ∈ k^{n}(T ξ_{b}) ≅ k^{n}(S^{n}) ≅ k^{0}(S^{0}) = π_{0}(k)

is a unit in the ring π_{0}(k) for each b ∈ B.

DUALITY IN BICATEGORIES AND TOPOLOGICAL APPLICATIONS 35

Proposition 40. A k-orientation μ of ξ induces an equivalence of spectra over B k∧S^{ξ} ≃ r^{∗}k∧_{B}S^{ξ} ⟶ r^{∗}k∧_{B}S_{B}^{n} ≃ k∧S^{n}.

“ξ is trivial to the eyes of k” r_{!}S_{B}^{0} = B_{+} and thus

(k∧S_{B}^{n})^{−∗}(S_{B}) ≅ (r^{∗}Σ^{n}k)^{−∗}(S_{B}^{0}) ≅ k^{n−∗}(B_{+}).

Take B = M. A k-orientation of M is a k-orientation of its tangent bundle.

Theorem 41 (Poincaré duality). Let k be a commutative ring spectrum and M be a k-oriented smooth closed n-manifold. Then

k_{∗}(M_{+}) ≅ k^{n−∗}(M_{+}).

New result by the same methods.

Theorem 42. Let L^{d} be a smooth closed submanifold of a smooth closed manifold M^{n}, both k-oriented. Then

k_{n−d+p}(T ν_{M,L}) ≅ k^{d−p}(L_{+}) ≅ k_{p}(L_{+}).