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Homology and cohomology via enriched bifunctors

We show that the category of numerically generated pointed spaces is complete, cocomplete, and monoidally closed with respect to the smash product, and then utilize these features to establish a simple but flexible method for constructing generalized homology and cohomology theories by using the notion of enriched bifunctors.


Introduction
On the category of pointed topological spaces, generalized homology and cohomology theories are usually constructed by using spectra. However, the definition of a spectrum is not suitable for more general type of model categories, e.g. the category of equivariant spaces, because it is too tightly connected to the framework of classical stable homotopy theory. In this paper, we present an alternative and more category theoretical approach to homology and cohomology theories which is based on the notion of a linear enriched functor instead of a spectrum.
Our strategy is as follows: We first replace the category Top 0 of pointed topological spaces by a convenient full subcategory NG 0 which fulfills our requirements. More explicitly, we call a topological space X numerically generated if it has the final topology with respect to its singular simplexes, and define NG 0 to be the full subcategory of Top 0 consisting of numerically generated spaces. Then NG 0 is complete, cocomplete, and is monoidally closed in the sense that there is an internal hom Z Y satisfying a natural bijection hom NG 0 (X ∧ Y, Z) ∼ = hom NG 0 (X, Z Y ).
Moreover, there exist a reflector ν : Top 0 → NG 0 such that the natural map νX → X is a weak equivalence, and a sequence of weak equivalences where map 0 (X, Y ) is the set of pointed maps from X to Y equipped with the compact-open topology. Thus NG 0 is eligible, from the viewpoint of homotopy theory, as a convenient replacement for Top 0 .
A functor T : NG 0 → NG 0 is called enriched if the correspondence which maps f : X → Y to T f : T X → T Y induces a morphism Y X → T Y T X between internal homs, and is called linear if it converts a cofibration sequence into a homotopy fibration sequence. Theorem 6.4 states that every linear enriched functor T defines a generalized homology theory X → {h n (X; T )} such that h n (X; T ) is isomorphic to π n T X for n ≥ 0, and to π 0 T (Σ −n X) for n < 0. We see that every homology theory represented by a spectrum comes from a linear enriched functor, and vice versa. In fact, the functor which assigns to a linear enriched functor its derivative, in the sense of Goodwillie [2], at S 0 induces a Quillen equivalence between the model categories of linear enriched functors and spectra. Thus the homotopy category of linear enriched functors is equivalent to the stable category. A benefit of our approach is its flexibility: For example, the enrichedness of T implies the orthogonality, hence the symmetricity, of its derivatives {T (Σ n X) | n ≥ 0}.
Our method for constructing cohomology theories is based on the notion of a bifunctor NG op 0 × NG 0 → NG 0 , contravariant with respect to the first argument and covariant with respect to the second argument. There are bivariant version of the notions of enrichedness and linearity, and Theorem 7.3 states that every bifunctor F which is bivariantly enriched and linear determines a pair of a generalized homology theory X → {h n (X; F )} and a cohomology theory X → {h n (X; F )} such that hold whenever k and n + k are non-negative. Just as a spectrum represents both homology and cohomology theories, every linear enriched functor T : NG 0 → NG 0 produces homology and cohomology theories through the bilinear functor F given by the formula F (X, Y ) = (T Y ) X . However, there exist enriched bifunctors not of the form as above but induce interesting bivariant homology-cohomology theories. (Cf. Remark after Theorem 7.3.) Since our approach is based solely on the standard category theoretical notions such as enrichedness and linearity, it can be easily extended to other model categories consisting of simplicial sets, diffeological spaces, equivariant (topological or diffeological) spaces, etc. Some of such extensions will be discussed in a subsequent paper.

Diffeological spaces
To derive the properties of the category of numerically generated spaces we use an adjunction between the categories of topological and diffeological spaces, respectively. Recall from [3] that a diffeological space consists of a set X together with a family D of maps from open subsets of Euclidean spaces into X satisfying the following conditions: Covering. Any constant parametrization R n → X belongs to D. Locality. A parametrization σ : U → X belongs to D if every point u of U has a neighborhood W such that σ|W : W → X belongs to D.
Smooth compatibility. If σ : U → X belongs to D, then so does the composite σf : V → X for any smooth map f : V → U between open subsets of Euclidean spaces.
We call D a diffeology of X, and each member of D a plot of X. A map between diffeological spaces f : X → Y is called smooth if for any plot σ : U → X of X the composite f σ : U → Y is a plot of Y . In particular, if D and D ′ are diffeologies on a set X then the identity map (X, D) → (X, D ′ ) is smooth if and only if D is contained in D ′ . In that case, we say that D is finer than D ′ , or D ′ is coarser than D. Clearly, the class of Diffeological spaces and smooth maps form a category Diff . A category is complete if it has equalizers and small products, and is cocomplete if it has coequalizers and small coproducts. Therefore, the theorem follows from the basic constructions given below.
Products. Given diffeological spaces X j , j ∈ J, their product is given by a pair ( j∈J X j , D), where D is the set of parametrizations σ : U → j∈J X j such that every its component σ j : U → X j is a plot of X j . Coproducts. The coproduct of X j , j ∈ J, is given by ( j∈J X j , D), where D is the set of parametrizations σ : U → j∈J X j which can be written locally as the composite of the inclusion X j → j∈J X j with a plot of X j . Subspaces. Any subset A of a diffeological space X is itself a diffeological space with plots given by those parametrizations σ : U → A such that the post composition with the inclusion A → X is a plot of X. Quotients. Let p : X → Y be a surjection from a diffeological space X to a set Y . Then Y inherits from X a diffeology consisting of those parametrizations σ : U → Y which lifts locally, at every point u ∈ U , along p. Exponentials. Given diffeological spaces X and Y , the set hom Diff (X, Y ) has a diffeology D X,Y consisting of those σ : U → hom Diff (X, Y ) such that for every plot τ : V → X of X, the composite Then there is a natural map α : C ∞ (X × Y, Z) → C ∞ (X, C ∞ (Y, Z)) given by the formula: α(f )(x)(y) = f (x, y) for x ∈ X and y ∈ Y . The following exponential law implies the cartesian closedness of Diff .

Numerically generated spaces
Given a topological space X, let DX be the diffeological space with the same underlying set as X and with all continuous maps from open subsets of Euclidean spaces into X as plots. Clearly, a continuous map f : X → Y induces a smooth map DX → DY . Hence there is a functor D : Top → Diff which maps a topological space X to the diffeological space DX.
On the contrary, any diffeological space X determines a topological space T X having the same underlying set as X and is equipped with the final topology with respect to the plots of X. Any smooth map f : X → Y induces a continuous map T X → T Y , hence we have a functor T : Diff → Top. Proof. Let X be a diffeological space and Y a topological space. Then a map f : T X → Y is continuous if and only if the composite f • σ is continuous for every plot σ of X. But this is equivalent to say that f : X → DY is smooth. Thus the natural map is bijective for every X ∈ Diff and Y ∈ Top. Proposition 3.2. A topological space X is numerically generated if and only if the counit of the adjunction T DX → X is a homeomorphism.
Proof. The condition T DX = X holds if and only if X has the final topology with respect to all the continuous maps from an open subset of a Euclidean space into X. But this is equivalent to say that X has the final topology with respect to the singular simplexes of X.
Let us write ν = T D, so that X is numerically generated if and only if νX = X holds. Proof. Every plot σ : U → X of X lifts to a plot of νX, since νU = U holds for any open subset U of R n . Thus νX has the same plots as X, and hence ν(νX) has the same topology as νX.
It follows that NG is reflective in Top, and the correspondence X → νX induces a reflector ν : Top → NG.
Similar argument shows that T (colim J DF ) is a colimit of F . But in this case we have T (colim J DF ) ∼ = colim J νF = colim J IF , because the left adjoint functor T preserves colimits.

Exponentials in NG
A map f : X → Y between topological spaces is said to be numerically continuous if the composite f • σ : ∆ n → Y is continuous for every singular simplex σ : ∆ n → X. Clearly, we have the following.
an open subset U of a Euclidean space into X.
Let us denote by map(X, Y ) the set of continuous maps from X to Y equipped with the compact-open topology, and let smap(X, Y ) be the set of numerically continuous maps equipped with the initial topology with respect to the maps where σ : ∆ n → X runs through singular simplexes of X. More explicitly, the space map(X, Y ) has a subbase consisting of those subsets where K is a compact subset of X and U is an open subset of Y . On the other hand, smap(X, Y ) has a subbase consisting of those subsets where σ : ∆ n → X is a singular simplex, L a compact subset of ∆ n , and U an open subset of Y . As we have Proof. If X is numerically generated then a numerically continuous map f : X → Y is automatically continuous. Hence map(X, Y ) → smap(X, Y ) is surjective for all Y . Conversely, if map(X, νX) → smap(X, νX) is surjective, then the unit of the adjunction X → νX is continuous, implying that X is numerically generated.
Proof. Since X has the weak topology with respect to the family of closed cells, a map f from X to Y is continuous if and only if it is numerically continuous. Hence map(X, Y ) → smap(X, Y ) is a bijection. To prove the continuity of its inverse, we have to show that every subset of the form The proposition above implies that any CW-complex X is numerically generated. Thus we have the following. Remark. If X and Y are CW-complexes then so is the product X × Y in NG. In fact, if {e j | j ∈ J} and {e ′ k | k ∈ K} are the sets of closed cells of X and Y , respectively, then X × Y has the weak topology with respect to the set of closed cells Thus the subcategory of NG consisting of CW-complexes is closed under finite products.
For numerically generated spaces X and Y , let us denote Then there is a map α : This clearly implies the following.
Corollary 4.6. The category NG is a cartesian closed category.
To prove the theorem, we use the relationship between Y X and the exponentials in Diff .
is continuous for every singular simplex τ : ∆ m → X. But τ * σ corresponds to the the composite Thus σ is a plot of D smap(X, Y ) if and only if ev(σ, τ ) is continuous for every τ . which is equivalent to say that σ is a plot of C ∞ (DX, DY ).
We are now ready to prove Theorem 4.5.
Proof of Theorem 4.5. The map α is a homeomorphism because it coincides with the composite of homeomorphisms in which (1) follows from Theorem 2.2, and (2) is induced by the numerical isomorphism smap(Y, Z) → ν smap(Y, Z).

The space of basepoint preserving maps
Let Top 0 and Diff 0 be the categories of pointed objects in Top and Diff , respectively. Evidently, the adjunction (T, D) between Top and Diff induces an adjunction (T 0 , D 0 ) between Top 0 and Diff 0 . Thus the category NG 0 of pointed objects in NG can be identified with a full subcategory of Top 0 consisting of those pointed spaces (X, x 0 ) such that νX = X holds. Clearly, NG 0 is complete and cocomplete.
Given pointed spaces X and Y , let map 0 (X, Y ) and smap 0 (X, Y ) denote, respectively, the subspace of map(X, Y ) and smap(X, Y ) consisting of basepoint preserving maps. Then Proposition 4.3 implies the following.
Proposition 5.1. If X is a pointed CW-complex, then the inclusion map map 0 (X, Y ) → smap 0 (X, Y ) is a homeomorphism for any pointed space Y .
As before, let us denote Y X = ν smap 0 (X, Y ). By taking the constant map as basepoint, Y X is regarded as an object of NG 0 . Recall that the smash product X ∧ Y of pointed spaces X = (X, x 0 ) and Y = (Y, y 0 ) is defined to be the quotient of X × Y by its subspace X ∨ Y = X × {y 0 } ∪ {x 0 } × Y . We now define a pointed map where the middle term (Z, z 0 ) (X×Y,X∨Y ) denotes the subspace of Z X×Y consisting of those maps f : X × Y → Z such that f (X ∨ Y ) = {z 0 } holds, p * is induced by the natural map p : X × Y → X ∧ Y , and α ′ is the restriction of the homeomorphism ν smap(X × Y, Z) ∼ = ν smap(X, ν smap(Y, Z)).
Since p : X ×Y → X ∧Y is universal among continuous maps f : X ×Y → Z satisfying f (X ∨ Y ) = {z 0 }, the induced map p * is bijective, hence so is α 0 .
is a plot of C. But for any v ∈ V there exist a neighborhood W of v and a plot τ : W → A × B such that p τ = τ |W hold. Therefore, the composite ev( σ × τ ) coincides, on U × W , with the plot This implies that ev( σ × τ ) is locally, hence globally, a plot of C. Thus σ = (p * ) −1 · σ is a plot of C ∞ ((A ∧ B, * ), (C, c 0 )).

The proposition implies a natural bijection
Hence we have the following. (2) If X ∈ NG 0 , then the bijection ι : map 0 (X, Y ) → smap 0 (X, Y ) is a weak homotopy equivalence.
(2) We have a commutative diagram is bijective. Thus smap 0 (X, Y ) has the same n-loops as map 0 (X, Y ). Moreover, a similar diagram as above, but S n replaced by I + ∧ S n , shows that smap 0 (X, Y ) has the same homotopy classes of n-loops as map 0 (X, Y ). It follows that is an isomorphism for every f ∈ map 0 (X, Y ) and n ≥ 0.
Corollary 5.5. For all X, Y ∈ NG 0 , the space Y X is weakly equivalent to the space of maps map 0 (X, Y ) equipped with the compact-open topology.

Homology theories via enriched functors
Let C 0 be a full subcategory of NG 0 . Then C 0 is an enriched category over NG 0 with hom- Let T : C 0 → NG 0 be an enriched functor. Then for any pointed map f : X ∧ Y → Z such that Y and Z are objects of C 0 the composite induces, by adjunction, a pointed map X ∧T Y → T Z. Similarly, an enriched cofunctor T : C op 0 → NG 0 assigns X ∧ T Z → T Y as an adjunct to the composite X → F 0 (Y, Z) → F 0 (T Z, T Y ). Proposition 6.1. Enriched functors and cofunctors from C 0 to NG 0 preserve homotopies.
Proof. Let h : I + ∧ X → Y be a pointed homotopy between h 0 and h 1 . Then an enriched functor T : C 0 → NG 0 induces a homotopy I + ∧ T X → T Y between T h 0 and T h 1 . Similarly, an enriched cofunctor T induces a homotopy I + ∧ T Y → T X between T h 0 and T h 1 .
Corollary 6.2. If T : C 0 → NG 0 is an enriched functor then a homotopy equivalence f : X → Y induces isomorphisms T f * : π n T X ∼ = π n T Y for n ≥ 0. Similarly, if T is an enriched cofunctor then f induces isomorphisms T f * : π n T Y ∼ = π n T X for n ≥ 0.
From now on, we assume that C 0 satisfies the following conditions: (i) C 0 contains all finite CW-complexes. (ii) C 0 is closed under finite wedge sum. (iii) If A ⊂ X is an inclusion of objects in C 0 then its cofiber X ∪CA belongs to C 0 ; in particular, C 0 is closed under the suspension functor X → ΣX. The category FCW 0 of finite CW-complexes is a typical example of such a category.
Given a continuous map f : X → Y , let induced by the cofibration sequence A ⊂ X ⊂ X ∪ CA, is a homotopy fibration sequence with respect to the null homotopy of T A → T (X ∪ CA) coming from the contraction of A within the reduced cone CA. Likewise, an enriched cofunctor T : C op 0 → NG 0 is called linear if the induced sequence is a homotopy fibration sequence.
If T is a linear functor then every pair (X, A) gives rise to an exact sequence of pointed sets terminated at π 0 T (X ∪ CA). Here i and j denote the inclusions A ⊂ X and X ⊂ X ∪ CA, respectively, and ∆ is the composite Similarly, a linear cofunctor T induces an exact sequence terminated at π 0 T A. Theorem 6.4. For every linear functor T : C 0 → NG 0 , there exists a generalized homology theory X → {h n (X; T )} defined on C 0 such that h n (X; T ) is isomorphic to π n T X if n ≥ 0, and to π 0 T (Σ −n X) otherwise.
Proof. We first show that the map T (X ∨ Y ) → T X × T Y induced by the projections of X ∨ Y onto X and Y is a weak equivalence. This means that the functor Γ → NG 0 , which maps a pointed finite set k = {0, 1, . . . , k} to T (X ∧ k) = T (X ∨ · · · ∨ X), is special in the sense that the natural map T (X ∧ k) → T (X) k is a weak equivalence for all k ≥ 0. Hence π n T X is an abelian monoid with respect to the multiplication π n T X × π n T X ∼ = π n T (X ∨ X) T ∇ * − −− → π n T X induced by the folding map ∇ : X ∨ X → X. This multiplication coincides with the standard multiplication of π n T X because they are compatible with each other. In particular, π 1 T X is an abelian group for all X. Moreover, any pointed map f : X → Y induces a natural transformation T (X ∧ k) → T (Y ∧ k), hence a homomorphism of abelian monoids T f * : π n T X → π n T Y for all n ≥ 0.
To see that T (X ∨ Y ) → T X × T Y is a weak equivalence, it suffices to show that the sequence T X → T (X ∨ Y ) → T Y , induced by the inclusion X → X ∨Y and the projection X ∨Y → Y , is a homotopy fibration sequence.
Next consider the homotopy exact sequence associated with the sequence T X → T (CX) → T (ΣX). As T (CX) is weakly contractible, we obtain for every n ≥ 0 a short exact sequence of abelian monoids 0 → π n+1 T (ΣX) ∆ − → π n T X → 0.
Since π n+1 T (ΣX) is an abelian group, the homomorphism ∆ is injective, hence an isomorphism. But this in turn means that π n T X is an abelian group for n ≥ 0. Therefore, h n (X; T ) is an abelian group for all n ∈ Z.
For a pointed map f : X → Y , we define to be the homomorphism induced by T f : T X → T Y for n ≥ 0, and T (Σ −n f ) for n < 0. It is easy to see that the functor X → {h n (X; T )} together with a natural isomorphism h n (X; T ) ∼ = h n+1 (ΣX; T ), given by ∆ −1 for n ≥ 0 and the identity for n < 0, satisfies the homotopy and exactness axioms.
(1) Let AG(X) denote the topological free abelian group generated by X with basepoint identified with 0. Then the correspondence X → AG(X) defines a linear enriched functor NG 0 → NG 0 . By the Dold-Thom theorem, h n (X, AG) is isomorphic to the singular homology group for a CW-complex X. But this is not the case for general X. In fact, the singular homology theory cannot be represented by a linear enriched functor T : NG 0 → NG 0 , because the singular homology does not satisfy the wedge axiom (cf. [1, 6.11]), contradicting to the fact that the homology theory given by a linear enriched functor must satisfy the axiom.
(2) Labeled configuration spaces also give rise to linear enriched functors. Let C M (R ∞ ) be the configuration space of finite points in R ∞ with labels in a partial abelian monoid M . Then ΩC ΣX∧M (R ∞ ) is a linear enriched functor of X, and hence defines a connective homology theory. We have shown in [5] that the classical homology, the stable homotopy, and the connective K-homology theories can be obtained in this way.
If T : C 0 → NG 0 is an enriched functor then for every X ∈ C 0 the spaces T (Σ n X) together with the maps S 1 ∧ T (Σ n X) → T (Σ n+1 X) induced by the homeomorphism S 1 ∧ Σ n X ∼ = Σ n+1 X form a prespectrum ∂T X. Following Goodwillie [2], we call ∂T X the derivative of T at X. Let L(∂T X) be the spectrification of ∂T X (cf. [4]). Then its zeroth space L(∂T X) 0 is an infinite loop space and the correspondence X → L(∂T X) 0 defines an enriched functor LT : C 0 → NG 0 . If, moreover, T is linear then Theorem 6.4 implies that the natural map T X → LT X is a weak equivalence for every X; hence LT defines the same homology theory as T .
An enriched functor T is called stable if the natural map T X → LT X is a homeomorphism for every X. In particular, LT is stable for any T . Let SLEF be the category of stable linear enriched functors C 0 → NG 0 with enriched natural transformations as morphisms. Then there is a functor D from SLEF to the category Spec of spectra which maps a stable functor T to its derivative ∂T S 0 at S 0 . We have shown that the homology theory h • (−; T ) induced by a linear enriched functor T is represented by the spectrum D(LT ) = L(∂T S 0 ).
Conversely, any homology theory represented by a spectrum comes from a linear enriched functor. In fact, D has a left adjoint I : Spec → SLEF defined as follows: For a spectrum E = {E n }, IE is the enriched functor which maps X to the zeroth space L(E ∧ X) 0 of the spectrification of the prespectrum E ∧ X = {E n ∧ X}. The unit E → DIE and the counit IDT → T of the adjunction are weak equivalences given by the maps where µ : ∂T S 0 ∧ X → ∂T X is a map of prespectra consisting of the maps T (S n ) ∧ X → T (Σ n X) induced by the identity of S n ∧ X.
Let us regard Spec and SLEF as a model category with respect to the classes of weak equivalences, fibrations, and cofibrations consist, respectively, of level weak equivalences, level fibrations, and morphisms that have the left lifting property with respect to the class of trivial fibrations. Then we have the following. Proposition 6.6. The functor D : SLEF → Spec is a right Quillen equivalence, and hence induces an equivalence between the homotopy categories.
Corollary 6.7. The homotopy category of SLEF is equivalent to the stable category.
Remark. Let T : C 0 → NG 0 be an enriched functor. Then for every real inner product space V the standard orthogonal group action on V induces an orthogonal group action on the one-point compactification S V of V , and hence on T (Σ V X) where Σ V X = S V ∧ X. With respect to the orthogonal group action on each T (Σ V X), the prespectrum {T (Σ V X)} indexed by real inner product spaces V form an orthogonal prespectrum. Thus the derivative ∂T X = {T (Σ n X)} at X extends to an orthogonal spectrum, and consequently is a symmetric spectrum with respect to the symmetric group action on T (Σ n X) induced by the embedding of the symmetric group into the orthogonal group as the subgroup of coordinate permutations.

Bivariant homology-cohomology theories
We now introduce the notion of a bilinear functor, and describe a passage from bilinear functors to generalized cohomology theories. In fact, we shall show that a bilinear functor gives rise to a pair of generalized homology and cohomology theories, or in other words, a bivariant homology-cohomology theory.
Let F : C 0 op × C 0 → NG 0 be a bivariant functor which is contravariant with respect to the first argument, and is covariant with respect to the second argument. We say that F is enriched (over NG 0 ) if for all pointed spaces X, X ′ , Y , and Y ′ , the map is continuous and is pointed in the sense that if either f or g is constant then so is F (f, g).
are homotopy fibration sequences.
Example 7.2. Given a linear functor T : C 0 → NG 0 , there is a bilinear functor F T given by the formula F T (X, Y ) = F 0 (X, T Y ). Such a bilinear functor is said to be of the normal form. Conversely, a bilinear functor F determines a linear functor T : C 0 → NG 0 by putting T X = F (S 0 , X), called the covariant part of F . There is a natural transformation F → F T such that F (X, Y ) → F T (X, Y ) = F 0 (X, F (S 0 , Y )) maps every ξ ∈ F (X, Y ) to the map X → F (S 0 , Y ), x → F (x, 1 Y )(ξ). Here we identify x ∈ X with the evident map S 0 → X.
Theorem 7.3. For every bilinear functor F : C 0 op ×C 0 → NG 0 , there exist a generalized homology theory X → {h n (X; F )} and a generalized cohomology theory X → {h n (X; F )} such that (7.1) h n (X; F ) ∼ = π 0 F (S n+k , Σ k X), h n (X; F ) ∼ = π 0 F (Σ k X, S n+k ) hold whenever k, n + k ≥ 0. Moreover, h n (X; F ) is naturally isomorphic to the n-th homology group h n (X; T ) given by the covariant part T of F .
Proof. Since F (X, Y ) is linear with respect to Y , π n F (X, Y ) is an abelian group for all X, Y and n ≥ 0. Clearly, this abelian group structure is natural with respect to both X and Y . Moreover, the bilinearity of F implies natural isomorphisms π n F (X, Y ) ∼ = π n+1 F (X, ΣY ), π n F (ΣX, Y ) ∼ = π n+1 F (X, Y ) Consequently, there is a natural isomorphism π 0 F (X, Y ) ∼ = π 0 F (ΣX, ΣY ), called the suspension isomorphism.
For every pointed space X and every integer n, let us define h n (X; F ) = colim k→∞ π 0 F (S n+k , Σ k X), h n (X; F ) = colim k→∞ π 0 F (Σ k X, S n+k ) where the colimits are taken with respect to the suspension isomorphisms. Clearly (7.1) holds, and we have h n (X; F ) ∼ = h n (X; T ) where T is the covariant part of F . Thus the functor X → {h n (X; F )} together with the evident natural isomorphism h n (X; F ) ∼ = h n+1 (ΣX; F ) defines a generalized homology theory. Similarly, the covariant functor X → {h n (X; F )} together with the natural isomorphism h n+1 (ΣX; F ) ∼ = h n (X; F ) defines a generalized cohomology theory, because it satisfies the homotopy and exactness axioms.
Example 7.4. (1) Let F : NG 0 op × NG 0 → NG 0 be the bilinear functor given by the formula F (X, Y ) = AG(Y ) X . Then h n (X, F ) is the singular cohomology group for a CW-complex X. But there exists no bilinear functor representing the singular cohomology groups of all X. (Cf. Example 6.5 (2).) (2) The second author constructs in [6] a bilinear functor F such that h n (X; F ) is isomorphic to theČech cohomology group for all X, and h n (X; F ) is isomorphic to the Steenrod homology group for any compact metrizable space X.