Co-operational bivariant theory

For a covariant functor W. Fulton and R. MacPherson defined \emph{an operational bivariant theory} associated to this covariant functor. In this paper we will show that given a contravariant functor one can similarly construct a ``dual"version of an operational bivariant theory, which we call a \emph{co-operational} bivariant theory. If a given contravariant functor is the usual cohomology theory, then our co-operational bivariant group for the identity map consists of what are usually called ``cohomology operations". In this sense, our co-operational bivariant theory consists of \emph{``generalized"} cohomology operations.


Introduction
In [7] W. Fulton  ) is called a bivariant theory. 1 In §2 below we make a quick recall of Fulton-MacPherson's bivariant theory.Here we just remark that in [7, §2.2 Axioms for a bivariant theory] the value B * (X f − → Y ) is in the category of graded abelian groups, but as remarked in [7,Remark,p. 22] B * (X f − → Y ) can be valued in an arbitrary category, e.g., the category of abelian groups (see [7, §6.1.2Definition of F, p. 60]), the category of sets (see [7, §4.3 Differentiable Riemann-Roch]), the category of derived categories (see [7,§7 Grothendieck Duality and Derived Functors]), etc.
As a typical example of such a bivariant theory, Fulton and MacPherson defined a bivariant homology theory H * (X f − → Y ) constructed from the usual cohomology theory H * , which is a multiplicative cohomology theory.Here the multiplicativity is crucial.Given a homology theory, i.e., a covariant functor with values in the category of (graded) abelian groups, in [7, §8.1] they defined what is called an operational bivariant theory or simply an operational theory.Here we note that in fact there are at least three more kinds of operational bivariant theories requiring some other extra conditions, depending on covariant functors to be treated: • [7, §8.3] requiring one more extra condition, • [6, §17.1] and [2,1] requiring two more extra conditions, and • [8] requiring three more extra conditions.In this paper we do not consider these refined ones for the sake of simplicity. 1 As recalled in §2, a bivariant theory B * assigns to each map f : X → Y a graded abelian group (sometimes, a set, etc.) B * (X For the sake of simplicity, sometimes we write a bivariant theory B * (X → Y ) or B(X → Y ) , instead of a bivariant theory B * or B, unless some confusion is possible.Shoji

YOKURA
In this paper, motivated by the definition of an operational bivariant theory given in [7, §8.1], we define a co-operational bivariant theory 2 from a contravariant functor which is not necessarily a multiplicative cohomology theory.Let F * be a contravariant functor such as the usual cohomology theory H * , the K-theory K (X ) of complex vector bundles.Then the co-operational bivariant theory B coop F * (X f − → Y ) is defined in a way analogous to the definition of Fulton-MacPherson's operational bivariant theory associated to a covariant functor F * .In this paper Fulton-MacPherson's operational bivariant theory shall be denoted by B op F * (X f − → Y ).For example, let F * be the usual cohomology groups H * .It turns out that for the identity id X : X → X the co-operational bivariant theory B coop H * (X id X − − → X ) consists of elements c's such that each c is a collection of homomorphisms c g : H * (X ′ ) → H * (X ′ ) for all g : X ′ → X such that for maps (1.1) the following diagram commutes: (1.2) In other words, c g : H * (X ′ ) → H * (X ′ ) is nothing but what is usually called a cohomology operation 3 (see, e.g., [24,25,26] and [18]) for the given cohomology group H * .Here we note that we consider such operations for all objects X ′ over X , i.e., X ′ → X (in other words, we consider such operations on what is usually called the category over X ), as we define above.In this paper, we can consider a general contravariant functor F * with values in a certain category.As Fulton and MacPherson consider a homology theory, i.e., a covariant functor with values in the category of graded abelian groups in [7, §8.1], in §4 below we consider a contravariant functor F * with values in the category of graded abelian groups as a model case.So, in this case, c g : F * (X ′ ) → F * (X ′ ) is a homomorphism of graded abelian groups.If F * is a contravariant functor with values in an arbitrary category, such as the category of sets, abelian groups, derived categories, then c g : F * (X ′ ) → F * (X ′ ) is a morphism in the category.Even if we consider such an arbitrary category, a morphism c g : F * (X ′ ) → F * (X ′ ) shall be still called "cohomology" operation, although it should be called "a contravariant functor operation" if we "follow" the way of naming "cohomology operation".A natural transformation τ : F * → G * between two contravariant functors can be extended to a Grothendieck transformation and depends on τ.In particular, for the identity map id X : X → X , 2 A finer co-operational bivariant theory, motivated by the other refined or sophisticated operational bivariant theories listed above, will be given and discussed in a different paper. 3Speaking of "operation", in [29] A. Vishik considers an operation between two oriented cohomology theories [14] (cf.[15]): An operation from A * to B * is a natural transformation from A * to B * considered as contravariant functors on the category of smooth schemes, i.e., operations commute with pull-backs (but not necessarily with push-forwards).On the other hand, in [22] P. Sechin considers a multiplicative operation between two oriented cohomology theories.
Co-operational bivariant theory 23 consists of elements c's such that each c is a collections of homomorphisms c g : F * (X ′ ) → F * (X ′ ), for all g : X ′ → X such that the above diagram (1.2) commutes with H * being replaced by F * and the following diagram also commutes for some And the homomorphism (for the identity id X : X → X ) (1.4) A natural transformation τ : F * → G * of contravariant functors F * and G * , of course, means that it assigns a homomorphism τ : F * (X ) → F * (X ) to each object X and the following diagram commutes for h : X ′′ → X ′ : (1.5) Obviously the homomorphism τ : F * (X ) → G * (X ) for each X can be interpreted as the following trivial commutative diagram If we adopt this interpretation, then the above commutative diagram (1.5) meaning "naturality" of τ of two contravariant functors can be expressed as the following commutative cubic diagram: (1.7) Here we emphasize that the identity map id F * (X ) : F * (X ) → F * (X ) for any contravariant functor F * is obviously a cohomology operation, which shall be called the "identity cohomology operation" in this paper.Let us consider the following class ) g if we use the above notation c g , and the above diagram (1.7) becomes (1.8) Therefore, by (1.4) we get (1.9) Namely, the natural transformation τ : F * → G * , in other words (1.7), can be interpreted as (1.9), using the Grothendieck transformation τ coop : B coop τ F * → B coop G * of the associated co-operational bivariant theories.Here we give two more non-trivial examples: Firstly, consider the Chern character ch : K (−) → H ev (−; Q) (see, e.g., [17,§3 Properties]).Then for each positive integer k we have the Adams operation and the Adams-like operation on the even-degree rational cohomology which is defined by Ψ k H (a) := k r • a for each a ∈ H 2r (−; Q) and we have the following commutative diagram (1.10) which is natural with respect to the base change, namely, we do have the following commutative cube for h : X ′′ → X ′ : (1.11) Co-operational bivariant theory

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The above commutative diagram (1.10) is nothing but a part of the homomorphism (see Remark 1.12 below) Remark 1.12.Here we remark that the Adams operation Ψ k : K (−) → K (−) is a ring-homomorphism, hence the addition of two Adams operations is not necessarily an Adams operation, since the addition of two ringhomomorphisms is not necessarily a ring-homomorphism because in general although the composition of two Adams operations is an Adams operation.Thus B coop K * (X id X − − → X ) is a set and cannot be an Abelian group, provided that the contravariant functor K (−) is considered to have values in the category of commutative rings.However, if we consider only the abelian group structure, ignoring the structure of product making K (−) a ring, then Here we emphasize that "a part" means that it is possible that there are many other cohomology operations on K -theory and rational even-dimensional cohomology and natural transformations of them like (1.10).For example, in (1.10) Ψ k and Ψ k H can be respectively replaced by the identity cohomology operations id K (X ) and id H ev (X ,Q) .Therefore is an extension of the above natural transformation (1.10) of cohomology operations of K -theory and rational even-dimensional cohomology to a Grothendieck transformation.
Secondly, we consider the projection pr : Ω * (−) → C H * (−)/2 from Levine-Morel's algebraic cobordism Ω * (X ) to the Chow group C H * (X )/2, which is the Chow cohomology group C H * (X )/2 modded out by 2-torsions (see [28]).Then, as shown in [28] (see Brosnan [4], M. Levine [13] and A. Merkurjev [16]) there exists (unique) operations S i : C H * (X )/2 → C H * +i (X )/2, called "Steenrod operation", such that the following diagram commutes: (1.13) Here S i LN is a Landwever-Novikov operation (see [14]) and both Steenrod operation and Landwever-Novikov operations commute with pullbacks, namely, we have a commutative cube as the above (1.11),which is, as in the case of the above Chern character, nothing but a part of the homomorphism Hence we have the Grothendieck transformation is an extension of the above natural transformation (1.13) of Levine-Morel's algebraic cobordism and the Chow group C H * (X )/2 to a Grothendieck transformation.
The up-shot is that (1) a co-operational bivariant theory "captures" or "interprets" cohomology operations "bivariant theoretically", or a co-operational bivariant theory is a reasonable general setting to study cohomology operations and (2) a Grothendieck transformation is an extension or a generalization of a natural transformation of two kind cohomology operations, as shown by the above two examples.Finally we remark that Fulton-MacPherson's operational bivariant theory is actually described by using the notion of "homology operation"4 instead of "cohomology operation".In this sense both theories are really "operational" bivariant theories.Furthermore works will be done in a different paper.

Fulton-MacPherson's bivariant theories
We make a quick review of Fulton-MacPherson's bivariant theory [7], since we refer to some axioms required on the theory in later sections.
Let C be a category which has a final object pt and on which the fiber product or fiber square is welldefined.Also we consider the following classes: (1) a class C of maps, called "confined maps" (e.g., proper maps, in algebraic geometry), which are closed under composition and base change, and contain all the identity maps, and (2) a class I nd of commutative diagrams, called "independent squares" (e.g., fiber square, "Torindependent" square, in algebraic geometry), satisfying that (i) if the two inside squares in are independent, then the outside square is also independent, (ii) any square of the following forms are independent: where f : X → Y is any morphism.
Remark 2.1.Given an independent square, its transpose is not necessarily independent.For example, let us consider the category of topological spaces and continuous maps.Let any map be confined, and we allow a fiber square to be independent only if g is proper (hence g ′ is also proper).Then its transpose is not independent unless f is proper.(Note that the pullback of a proper map by any continuous map is proper, because "proper" is equivalent to "universally closed", i.e., the pullback by any map is closed.)Definition 2.2.A bivariant theory B on a category C with values in the category of graded abelian groups 5is an assignment to each morphism which is equipped with the following three basic operations.The (1) Product: For morphisms f : X → Y and g : Y → Z , the product (2) Pushforward: For morphisms f : X → Y and g : Y → Z with f confined, (3) Pullback : For an independent square An element α ∈ B(X f − → Y ) is sometimes expressed as follows: These three operations are required to satisfy the following seven compatibility axioms ([7, Part I, §2.2]): (A 12 ) Product and pushforward commute: Pushforward and pullback commute: We also require the theory B to have multiplicative units: (Units) For all X ∈ C , there is an element for all morphisms W → X and all α ∈ B(W → X ), and such that 1 X •β = β for all morphisms X → Y and all β ∈ B(X → Y ), and such that g * 1 X = 1 X ′ for all g : X ′ → X .

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A bivariant theory unifies both a covariant theory and a contravariant theory in the following sense: For a bivariant theory B, its associated covariant functors and contravariant functors are defined as follows: (1) B * (X ) := B(X a X − − → pt ) is covariant for confined morphisms and the grading is given by B i (X ) := is contravariant for all morphisms and the grading is given by B j (X ) := A typical example of a bivariant theory is the bivariant homology theory H(X f − → Y ) constructed from the singular cohomology theory H * (−), which unifies the Borel-Moore homology and the singular cohomology Here the underlying category C is the category of spaces embeddable as closed subspaces of some Euclidean spaces R n and continuous maps between them (see [7,§3 Topological Theories]).More generally, Fulton-MacPherson's (general) bivariant homology theory (here, using their notation) is constructed from a multiplicative cohomology theory h * (−) [7, §3.1]Here the cohomology theory h * is either ordinary or generalized.A cohomology theory h * is called multiplicative if for pairs (X , A), (Y , B ) there is a graded pairing (exterior product) such that it is associative and graded commutative, i.e., α × β = (−1) i + j β × α.A typical example of a multiplicative ordinary cohomology theory is the singular cohomology theory.The topological complex K -theory K (−) and complex cobordism theory Ω * (−) are multiplicative generalized cohomology theories.We consider the category of spaces embeddable as closed subspaces in some Euclidian spaces R N and continuous maps.For example, Whitney's embedding theorem says that any manifold of real dimension m can be embedded as a closed subspace of R 2m .We also note that a complex algebraic variety is embeddable 7  as a closed subspace of some Euclidean space R N .We let a confined map be a proper map and an independent square be a fiber square.
(1) By the definition (2.4) we have h 7 This is because the variety X is covered by finitely many affine varieties, which are embedded (as closed subsets) into R n for some n, thus it follows from [5, §8.8 Proposition] that the variety X is itself embedded (as a closed subset) into R N for some N .
8 Such a map always exist, since our space is embeddable as a closed subspace of some R N , thus this embedding is considered as is the Borel-More homology group (see, e.g., [6] and [20, B.1]).
the category C , which preserves the above three basic operations: (1) γ(α be a natural transformation of two multiplicative cohomology theories.Then we get the associated Grothendieck transformation (2.9) For later use, we introduce the following "image" of a Grothendieck transformation γ : B → B ′ : Then it is easy to see that Im γ is a bivariant subtheory of B ′ .This in fact follows from the above three properties: (1) γ(α . Indeed, we have: Here we consider the fiber square Unless some confusion is possible, we may use the symbol Im γ B ′ for Im γ in order to emphasize that Im γ is a subtheory of B ′ , thus recording B ′ .

Operational bivariant theory
Given a covariant functor or a homology theory, Fulton and MacPherson have defined what is called an operational bivariant theory or an operational theory [7, §8 Operational Theories] (also, see [6, §17. 1 and §17.2]).As we remarked in Introduction, in this paper we consider the operational bivariant theory defined in [7, §8.1].The case of the other refined operational bivariant theories will be considered in a different paper in which we define a co-operational bivariant theory "corresponding" to these refined operational bivariant theories.
Let h * be a homology theory, i.e., a covariant functor with values in graded abelian groups such that the functorial (pushforward) homomorphism f * : h * (X ) → h * (Y ) is defined for a confined map f : X → Y .Then its operational bivariant theory, denoted by for all m ∈ Z, all g : Y ′ → Y and the fiber square These homomorphisms c g are required to be compatible with pushforward (for confined maps), i.e., for a fiber diagram (3.1) where k is confined and thus k ′ is confined as well, the following diagram commutes Here we recall the definitions of the three operational-bivariant-theoretic operations: (1) Product: The product Here we consider the following fiber squares: (3.4) (2) Pushforward: Here we use the above commutative diagram (3.4).

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(3) Pullback: For a fiber square Here we use the above commutative diagrams (3.1).
Let B be a bivariant theory.Then its associated operational bivariant theory B op is defined to be the operational bivariant theory constructed from the covariant functor B * (X ) = B(X → pt ).If we use the above Then we have the following canonical Grothendieck transformation Here we note that (op(α)) g = (g * α) * : B * (Y ′ ) → B * (X ′ ), which is defined by (g * α) * (b) := (g * α) • b, for which see [7, §2.5 Gysin homomorphisms].
For the sake of convenience for the reader and later presentation, we prove the following: Proposition 3.5.The above map op : B → B op is a Grothendieck transformation.Proof.
(1) op(α , where we consider the above diagram (3.4) and α ∈ B(X Consider the above commutative diagram (3.1) and let In a sense, the operational bivariant product, pushforward and pullback are defined like that in Definition 3.3 above so that the above map op : B → B op becomes a Grothendieck transformation.
It is not clear whether one could construct a Grothendieck transformation of the associated operational bivariant theories from a natural transformation of two covariant functors.To be more precise, if t : h * (−) → h * (−) is a natural transformation of two covariant functors, then it is not clear whether one could construct a Grothendieck transformation t : which is an "operational bivariant theoretic analogue" of (2.9).A kind of similar problem is discussed in [7, §8.2].Suppose that B is a bivariant theory, h * is a covariant functor and there are homomorphisms φ(X ) : B * (X ) = B(X → pt ) → h * (X ), covariant for confined maps, and taking 1 ∈ B * (pt ) = B * (pt ) to 1 ∈ h * (pt ).The homomorphism ev X : B op h * (X → pt ) → h * (X ) defined by ev X (c) := (c id pt )(1) where 1 ∈ h * (pt ) is called the evaluation homomorphism.Then a question is whether there exists a unique Grothendieck transformation The answer to this question is negative, however a positive answer to a modified question is affirmative, details for which see [31] (cf.[3]).
Let γ : B → B ′ be a Grothendieck transformation between two bivariant theories B and B ′ .Then we have a natural transformation γ * : B * → B ′ * between two covariant functors B * and B ′ * .As observed above, it is not clear whether one could construct a Grothendieck transformation t : However, if we take smaller subtheories of both B op and B ′op , then we can show the following results.Proposition 3.7.Let B be a bivariant theory and B op be its associated operational bivariant theory.
(1) Define B op := Image(op : B → B op ), that is Then B op is an operational bivariant theory, which is a subtheory of B op , and op : B → B op defined by op(α) = op(α) and the inclusion ι op B : B op → B op are Grothendieck transformations.
(2) For a map to a point, X → pt , we have the isomorphism: Proof.Here we show only (2).Let α, β ∈ B(X → pt ).It suffices to show that op(α) = op(β) implies α = β, i.e., (g * α) * = (g * β) * for all g : Y → pt implies that α = β.Let us consider the case id pt : pt → pt .Then we are supposed to have (id In particular, let x be the unit Here the left and right equalities are due to the property of the unit (see (Units) in §2, or [7, §2.2 Axioms for a bivariant theory, p. 22].)Therefore we have that op(α) = op(β) implies α = β.□ Using this subtheory B op , given a Grothendieck transformation γ : B → B ′ , one might be tempted to (quite naturally) think that there would be a canonical Grothendieck transformation γ op : B op → B ′ op such that the following diagram commutes: where γ op is defined by γ op (op(α)) := op(γ(α)).Then it follows from the above Proposition 3.5 that γ op : B op → B ′ op is a Grothendieck transformation.However, the definition of γ op (op(α)) := op(γ(α)) turns out to be not well-defined due to the following reason.Let α, β ∈ B(X → Y ) such that op(α) = op(β).In order for the above definition γ op (op(α)) := op(γ(α)) to be well-defined, we should have that op(γ(α)) = op(γ(β)), but this cannot be automatically guaranteed, although it is the case for α, β ∈ B(X → pt ) as shown in the proof of Proposition 3.7 (2).Indeed, we get the following commutative diagram: ) is surjective, then we do have (g * (γ(α))) * = (g * (γ(β))) * , namely, we have Therefore we get the following: Proposition 3.8.Let γ : B → B ′ be a Grothendieck transformation.Then there is a canonical Grothendieck transformation which is defined by γ op (op(α)) := op(γ(α)).Hence, the following is a commutative diagram of Grothendieck transformations: In the above proposition we need the fact that γ : ) is surjective.So, we introduce the following notions., one might be tempted to consider the following slightly modified one: otheriwse by which we would obtain γ op : B op → * B ′ op .Unfortunately * B ′ may not be a bivariant theory, to be more precise, the bivariant product may not be well-defined, although the pushforward and the pullback are well-defined.

Co-operational bivariant theory
In this section, motivated by the above Fulton-MacPherson's operational bivariant theory, we consider another kind of operational bivariant theory associated to a contravariant functor which may not be a multiplicative cohomology theory.This new bivariant theory shall be called a co-operational bivariant theory.
If a contravariant functor h * is a multiplicative cohomology theory and t : h * → h * is a natural transformation between two multiplicative cohomology theories, then, as remarked in Remark 2.8, one can extend t : h * → h * to a Grothendieck transformation t : h * (X → Y ) → h * (X → Y ) between their associated bivariant theories in such a way that for an identity map id If it is not multiplicative, one cannot make such a construction.For example, the Chern class c : K 0 (−) → H * (−; Z) is not a natural transformation between two multiplicative cohomology theories, as proved by B. Totaro [27] (cf.[9] and [23]).Our co-operational bivariant theory is defined as follows: Theorem 4.1.("Co-operational" bivariant theory) Let F * (−) be a contravariant functor with values in the category of graded abelian groups, defined on a category C with all fiber squares independent.For each map f : is defined to be a collection of homomorphisms for all m ∈ Z, all g : Y ′ → Y and the fiber square And these homomorphisms c g are required to be compatible with pullback, i.e., for a fiber diagram (4.2) the following diagram commutes We define the following bivariant-theoretic operations: (1) Product: The product Here we consider the following fiber squares: Here we use the above commutative diagram (4.4).(3) Pullback: For a fiber square Here we use the above commutative diagrams (4.2).
Then B coop F * becomes a bivariant theory, i.e., satisfies the seven axioms of bivariant theory, listed in §2.
Proof.The proofs could be omitted.However, since the definition of our co-operational bivariant theory is different from that of Fulton-MacPherson's operational bivariant theory, for the sake of convenience or time-saving for the reader, we write down proofs, giving just comments for trivial ones.
(1) First we remark that these three operations are compatible with pullbacks.Here we show only the compatibility of the pushforward.For which we consider the following fiber squares: (4.5) and consider the following diagrams: The outer square, the upper trapezoid, the left and right triangles are commutative, therefore the lower trapezoid is commutative, which means that the pushforward is compatible with pullback.(2) (A 1 ) Product is associative: It is because the product is defined as composition of homomorphisms and composition is associative. ( Consider the following fiber squares: (4.7) Then by the definition we have (4) (A 3 ) Pullback is functorial: It is clear by the definition of pullback.
(5) (A 12 ) Product and pushforward commute: Let us consider the above diagrams (4.7): ) (A 13 ) Product and pullback commute: Let us consider the above diagrams (4.5): Co-operational bivariant theory 41 (8) (A 123 ) Projection formula: Let us consider the following diagrams: (4.8) □ Remark 4.9.In Fulton-MacPherson's bivariant theory B * the bivariant pushforward f * : B * (X ) is defined for a confined map f : X → Y .As one can see, in our co-operational bivariant theory, for the bivariant pushforward f * : B coop F i (X ) we do not need the confined-ness of the map f : X → Y .

Gysin homomorphisms]). For fiber squares
the following diagram commutes: Then considering the collection g * c g : Y ′ → Y gives us an operational bivariant class from the given bivariant class.So, conversely Fulton-MacPherson's operational bivariant theory is one obtained from a covariant functor.Similarly, our co-operational bivariant theory is motivated by considering the following: [7, §2.5 Gysin homomorphisms]).Here we remark that, as pointed out in the first paragraph of [7, §2.5 Gysin homomorphisms, p. 26], when we consider the Gysin homomorphism θ * induced by a bivariant element θ ∈ B(X f − → Y ) the map f is assumed to be confined.For fiber squares the following diagram commutes: The following is a co-operational version of Proposition 3.5, i.e., of the Grothendieck transformation op : B → B op .Since the proof of it is similar to that of the proof of Proposition 3.5, it is omitted, left for the reader.  1As remarked above, the reason why we need to consider confined maps is that (g * α) * is defined by namely, we use the pushforward f ′ * , which is defined for a confined map.

Co-operational bivariant theory 43
Here (g * α) * is defined by ) and we consider the following fiber square Then we have We have the following co-operational versions of Proposition 3.7, Proposition 3.8 and Corollary 3.10, respectively.Proposition 4.12.Let B be a bivariant theory and B coop be its associated co-operational bivariant theory.
(1) Then there exits a Grothendieck transformation of co-operational bivariant theories such that for the identity map id X : X → X the following composition of homomorphisms Here e Bh ′ : Co-operational bivariant theory

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(2) For each α ∈ h * (X ) and each g : X ′ → X , the homomorphism (g * α) * , an element of coop(α), is the homomorphism defined by taking the cup-product with g * α: and the following diagram commutes: Proof.
(1) It follows from the above Proposition 4.14, applying it to the Grothendieck transformation t : , that there exists a Grothendieck transformation of co-operational bivariant theories Now by the above construction we have (2) First we note that for α ∈ Bh(X Here we emphasize that the homomorphism which is equal to the cup-product x ′ ∪ g * α, by the definition of the associated bivariant theory Bh * (X → Y ) constructed from the multiplicative cohomology theory h * (see [7, §3.1.4 - §3.1.9],in particular [7, §3.1.7]for x ′ • g * α being equal to the cup-product x ′ ∪ g * α.) Hence the homomorphism (g * α) * is one defined by taking the cup-product with g * α and the commutativity of the above diagram follows from the property of the cup-product.□ Remark 4.18.The diagram (4.17) means that the above homomorphisms (−) , cohomology operations defined by taking the cup-product with g * α and g * (t (α)), are compatible with the natural transformation t : h * → h ′ * .Speaking of the cup-product, the following k-times cup-product is also a cohomology operation: and it is also compatible with the natural transformation t : h * → h ′ * .To be more precise, if we write the degree, we have ⊔ k : h m (X ) → h km (X ).And the following diagram commutes: because we define a cooperational bivariant class to be a collection of homomorphisms.So, if we consider a collection of maps instead of homomorphisms, then we get another larger co-operational bivariant theory B coop h(X → Y ), then we can have The above theorem means the following: (1) In the case of a natural transformation t : h * → h ′ * between multiplicative cohomology theories h * and h ′ * , using the associated bivariant theories Bh and Bh ′ constructed by Fulton and MacPherson [7, §3.1 Construction of a bivariant theory from a cohomology theory], we can extend the natural transformation t : h * → h ′ * to a Grothendieck transformation to co-operational bivariant theories by restricting to the subtheories Bh coop ⊂ Bh coop and Im t Bh ′ coop ⊂ Bh ′coop , although we may not be able to extend it to the whole co-operational bivariant theories Bh coop and Bh ′coop .
(2) Each element c g of a co-operational bivariant class c = c g : ) is a cohomology operation of h * , which is a simple homomorphism defined by taking the cup-product with g * α for α ∈ h * (X ), i.e., (4.20) Since we have the isomorphism Bh coop (X id X − − → X ) ∼ = h * (X ), coop(α) ←→ α, we can denote the cooperational bivariant class c by α, thus the above equality (4.20) could be expressed by, abusing notation or symbol, (3) However, by this construction, another co-operational bivariant class So, after observing these, we want to pose the following question: Co-operational bivariant theory 47 Question 4.21.Can one construct another associated bivariant theory Bh (which is different from Fulton-MacPherson's bivariant theory Bh) for a multiplicative cohomology theory h * such that Suppose that we have natural transformation of two contravariant functors which are not necessarily multiplicative cohomology theories: From now on a co-operational bivariant class is a collection of maps, not necessarily homomorphisms, and also the co-operational bivariant theory B coop F * (X → Y ) obtained from a contravariant functor is considered as a set, even if it can have finer algebraic structures.As we see above that even in the case of a Grothendieck transformation γ : B → B ′ it is not clear whether there is a Grothendieck transformation between the associated co-operational bivariant theories B coop and B ′coop , it is not clear at all whether there exists a Grothendieck transformation For that we need to consider a subtheory B coop T F * of B coop F * , which is defined as follows.Motivated by (4.13), (4.17), and (4.19), we define the following: is defined to satisfy that there exists an element such that the following diagram commutes:for m ∈ Z and for g : Remark 4.24.We emphasize that we therefore automatically have the following commutative cube: (4.25) F * is a bivariant theory.Namely, the following product, pushforward, and pullback are all well-defined.
(1) Product: (3) Pullback: For a fiber square Proof.As stated in the above statement, it suffices to show the well-defined-ness of the above operations, since the properties of the seven axioms of a bivariant theory already hold in B coop F * . ( such that the following diagram commutes: Here we consider the following fiber squares: (4.28) In the diagram (4.29) below we have that Co-operational bivariant theory 49 (4.29) Thus, the outer square of the above diagram (4.29) is commutative.So, if we set then we obtain the above commutative square (4.27).Hence c ( Here we consider the above squares (4. Thus, the outer square of the above diagram (4.31) is commutative.So, if we set then we obtain the above commutative square (4.30).Hence where we consider the following fiber squares: So, if we set (g * c) T := g * (c T ), we get the above commutative square (4.32).Thus we get g □ Remark 4.34.In the proof of the above Theorem 4.24 we set (1) (g * c) T := g * (c T ).However, we see that if T : F * → G * is surjective, i.e., T : F * (W ) → G * (W ) is surjective for any object W , then these equalities automatically hold.For example, as to (1), we can show it as follows.We have the following commutative diagrams: Hence we have The same argument can be applied to (2) and (3).(

Co-operational bivariant theory
Then we have a Grothendieck transformation 5. B coop F * (X id X − − → X ) and cohomology operations Once again we emphasize that B coop F * (X id X − − → X ) consists of a collection of homomorphisms c g : F m (X ′ ) → F m+i (X ′ ) for all g : X ′ → X and for X ′′ h − → X ′ g − → X , the following diagram commutes: In other words, c g : F m (X ′ ) → F m+i (X ′ ) is what is called a cohomology operation, which is a natural selftransformation of a contravariant functor.Therefore B coop F * (X id X − − → X ) consists of natural self-transformations of the contravariant functor F * .The above commutative cube (4.25) means that T is a natural transformation between two contravariant functors equipped with natural self -transformations.This fact is sometimes expressed as "cohomology operations of F * and G * are compatible with the natural transformation T : F * → G * ", as seen above (see, e.g., [17]).
Remark 5.1.Corresponding to the identity id X : X → X , c g : F m (X ′ ) → F m+i (X ′ ) is a cohomology operation which is associated to the identity id X : X → X .So, for a general map f : X → Y , c g : F m (X ′ ) → F m+i (Y ′ ) could be called a "generalized" cohomology operation 11 associated to the map f : X → Y , where we consider the fiber square Remark 5.2.As we remarked above, if F * (−) has values in an arbitrary category, then should be a collection of morphisms, hence, for example, if we consider the category of rings, then it is a collection of ring homomorphisms; if we consider the category of sets, then it is a collection of maps.Even if we consider the category of rings or (graded) abelian groups, if we ignore some structures, e.g., ignore all the structures and consider only just sets, then it has to be a collection of maps.In other words, it depends on what kind of situations we consider.
Remark 5.3.As pointed out by T. Annala (in private communication), if we consider a more refined cooperational bivariant theory B coop F * requiring certain extra conditions, then we will obtain an isomorphism , which is similar to the case of the refined operational bivariant theory of Chow homology theory treated in [6, §17.3].Details of this and some other refined co-operational bivariant theories will be treated in a different paper [30].
Here are some examples.for the topological K-theory is a ring homomorphism and it turns out that the Adams operation is the only operation satisfying a ring homomorphism. 12The Adams-like operation Ψ k H : H ev (X ; Q) → H ev (X ; Q) for the even cohomology with rational coefficients is defined by Ψ k H (x) := k r x for x ∈ H 2r (X , Q).The Chern character ch : K (X ) → H ev (X ; Q) is compatible with these Adams operations, i.e., the following diagram commutes: 11 The referee pointed out that it is natural to consider cohomology operations that map between cohomologies of different spaces: power operation is such an example that occurs naturally (see, e.g., [21]). 12If we consider only the additive structure, i.e., a group structure, ignoring the product structure, then there may be other operations.

Co-operational bivariant theory
53 which is compatible with pullback, i.e., for any continuous map h : X → Y we have the commutative cube: Then for g : X ′ → X , we have the following commutative diagram: That is, if we use the previous notation, for any g : X ′ → X , we have (Ψ k ) g = Ψ k and (Ψ k ) ch = Ψ k H . From the Chern character ch : K (X ) → H ev (X ; Q) we get the following Grothendieck transformation Here we note that in this case − − → X ), which consists of only Adams operations, provided that we require the homomorphism c g : K (X ′ ) → K (X ′ ) to be a ring homomorphism.Similarly, for g : X ′ → X , we have the following commutative diagram: That is, if we use the previous notation, for any g : X ′ → X , we have (LN I ) g = LN I and (LN I ) ch = LN C H I .
and R. MacPherson have introduced bivariant theory B * (X f − → Y ) with an aim to deal with Riemann-Roch type theorems for singular spaces and to unify them.B * (X ) := B − * (X − → pt ) becomes a covariant functor and B * (X ) := B * (X id X − − → X ) a contravariant functor.In this sense B * (X f − → Y
y y which is due to the following computation, e.g., for the first diagram: for ∀y ∈ B

Remark 4 . 10 .
Let B * (X → Y ) be a bivariant theory and let c ∈ B i (X f − → Y ).Then for any morphism g : Y ′ → Y and consider the fiber square

Proposition 4 . 11 .
Let B be a bivariant theory.Let B * (X ) := B * (X id X − − → X ) be the associated contravariant functor and let B coop (X → Y ) be the co-operational bivariant theory defined by B coop (X → Y ) := B coop B * (X → Y ) as above.Then, restricted to confined 10 morphisms f : X → Y , there exists a canonical Grothendieck transformation coop : B → B coop which is defined by is a co-operational bivariant theory constructed from a multiplicative cohomology theory h * , thus a co-operational bivariant class c ∈ Bh coop (X id X − − → X ) has to satisfy the above formula (4.20), but clearly ⊔ k does not satisfy (4.20).

Theorem 4 . 26 .
The above B coop T (a) the upper and lower trapezoids are both commutative, since c ∈ B the left and right triangles are both commutative, since by the definition of the co-operational bivariant product • we have 28) again.In the diagram (4.31) below we have that (a) the upper trapezoid is commutative, since T is a contravariant functor, (b) the lower trapezoid is commutative, since c ∈ B coop T F i (X g • f −−→ Z ), (c) the left and right triangles are both commutative, since by the definition of the pushforward we have

51A
Grothendieck transformation which we are looking for is the following: Corollary 4.35.Let T : F * → G * be a natural transformation between two contravariant functors.For B coop T F * we assume that bivariant elements c T in Definition 4.22 satisfy the following three properties (if T is surjective, they are automatically satisfied, as pointed out in Remark 4.34): Furthermore γ T (c) = c T means that for m ∈ Z and for g : Y ′ → Y we have the commutative diagram (4.23), i.e.,