Georgian Mathematical Journal Volume 8 (2001), Number 4, 697–712

SUSPENSION AND LOOP OBJECTS IN THEORIES AND COHOMOLOGY

H.-J. BAUES AND M. JIBLADZE

Abstract. We introduce theories in which suspension, resp. loop objects, are defined and we describe the cohomology of such theories. Cohomology classes in degree 3 classify track theories.

2000 Mathematics Subject Classification: 18D05, 55P35, 55P40.

Key words and phrases: Theory with suspension objects, theory with loop objects, Σ-cohomology, Ω-cohomology, track theory.

Since the work of Lawvere [16] it is well known that many algebraic categories (like the categories of monoids, groups, algebras, Lie algebras, etc.) are actually categories of models of a theory. In topology there are fundamental examples of theories as in (1) and (2) below in which also suspension objects ΣX or loop objects ΩX are defined. Such theories with suspension objects are also endowed with a partial suspension operator E and dually theories with loop objects are endowed with a partial loop operator L, compare for example Baues [5, 4, 2].

For example the homotopy category S of one point unions of spheres

S^{n_{1}} ∨ ... ∨ S^{n_{k}}  (1)

with n_{1}, ..., n_{k} ≥ 1, k ≥ 0, is a theory with suspension objects and partial suspension operator E. Moreover the homotopy category K(ℤ/2) of products of Eilenberg–Mac Lane spaces

K^{n_{1}} × ... × K^{n_{k}}  (2)

with n_{1}, ..., n_{k} ≥ 0, k ≥ 0 and K^{n} = K(ℤ/2, n) is a theory with loop objects and partial loop operator L.

In this paper we introduce a complete system of axioms defining a theory with suspension objects and partial suspension operator termed a Σ-theory. The categorical dual of a Σ-theory is a theory with loop objects and partial loop operator termed Ω-theory.

The theory S given by spheres in (1) is a Σ-theory and the theory K(ℤ/2) given by Eilenberg–Mac Lane spaces in (2) is an Ω-theory. These examples show the crucial significance of such theories. In [10] we show that the Ω-theory K(ℤ/2) is completely determined by the Steenrod algebra.

We introduce the Σ-cohomology H_{Σ}^{∗}(T) of a Σ-theory T and dually the Ωcohomology H_{Ω}^{∗}(T) of an Ω-theory T. The universal Toda brackets of Baues and Dreckmann [6] yield for the theories S and K(ℤ/2) above the cohomology

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classes

\begin{matrix}⟨S⟩ ∈ H_{Σ}^{3}(S), \\ ⟨K(ℤ/2)⟩ ∈ H_{Ω}^{3}(K(ℤ/2)). \end{matrix}  (3)

The cohomology class ⟨S⟩ determines all secondary homotopy operations in S like triple Toda brackets. The cohomology class ⟨K(ℤ/2)⟩ determines all secondary cohomology operations like triple Massey products or operations of Adams [1].

In [9] we study the concept of suspenders and loopers in groupoid enriched categories termed track categories. As a main result in this paper we show that a Σ-theory T together with an element α ∈ H_{Σ}^{3}(T) determines up to equivalence a track category which has the property of a “track theory with suspenders” and vice versa such track theories are classified by the Σ-cohomology H_{Σ}^{3}.

Dually an Ω-theory T together with an element α ∈ H_{Ω}^{3}(T) determines up to equivalence a track category which has the properties of a “track theory with loopers” and vice versa such track theories are classified by the Ω-cohomology H_{Ω}^{3}.

For example the topological track category given by spaces in (1) and maps between such spaces (1-cells) and by homotopy classes of homotopies between such maps (2-cells) is a track theory with suspenders corresponding to the pair (S, ⟨S⟩). Similarly the topological track category given by spaces in (2) is a track theory with loopers corresponding to the pair (K(ℤ/2), ⟨K(ℤ/2)⟩).

1. Theories

A theory for us is a category S in which finite sums exist. We assume that a collection of generating objects X_{i} in S is given such that all objects of S are finite sums of objects from the collection. A model M of the theory S in a category C is a contravariant functor M : S^{op} → C which carries sums to products. Hence a model can be thought of as a collection of objects M_{i} of C together with certain morphisms (“operations”) M_{i_{1}} × ... × M_{i_{n}} → M_{i} which satisfy certain “identities”, i. e. fit in certain commutative diagrams, according to the structure of S. In most cases we consider models of S in the category Set of sets, so that a model is a collection of sets with algebraic structure.

1.1. Example. Let S be the full subcategory of the homotopy category (Top^{∗})_{≃} consisting of finite one point unions of spheres S^{n} with n ≥ 1. Then S is a theory. Homotopy groups π_{n}(X) = [S^{n}, X] of a space X together with homotopy operations given by maps in S yield a model of S. Such models are also termed π-algebras in the work of Blanc [12], Stower [18], Dwyer–Kan–Stower [13], Baues–Goerss [7].

The categorical dual of a theory termed a theory with products is a category P with finite products denoted A × B. A model of P is a covariant functor P → C which preserves products. Originally Lawvere used such theories with products. But the example of homotopy groups above shows that also theories with coproducts arise naturally.

SUSPENSION AND LOOP OBJECTS IN THEORIES AND COHOMOLOGY 699

1.2. Example. Let K(ℤ/2) be the full subcategory of the homotopy category (Top^{∗})_{≃} consisting of finite products of Eilenberg–MacLane spaces K(ℤ/2, n) with n ≥ 1. Then K is a theory with products. The cohomology groups H^{n}(X; A) = [X, K(A, n)] of a space X together with cohomology operations given by maps in K(ℤ/2) yield a model of K(ℤ/2). For example, the cohomology of X as a module over the Steenrod algebra or the cohomology ring of X can be deduced from the model; see Gray [14].

Each object in S is a cogroup object and objects in K are group objects. Therefore S is a “theory of cogroups” as defined below and dually K is a “theory of groups”.

1.3. Definition. A theory of cogroups is a theory T all of whose objects S are equipped with a cogroup structure (0_{S} : S → ∗, m_{S} : S → S ∨ S, n_{S} : S → S) in a way which is compatible with sums, i. e.

0_{S∨T} = \binom{0_{S}}{0_{T}} : S ∨ T → ∗,

\xymatrix{ & S ∨ S ∨ T ∨ T \ar[dr]^{id_{S}∨\binom{i_{S}}{i_{T}}∨id_{T}} \\ S ∨ T \ar[ur]^{m_{S}∨m_{T}} \ar[rr]^{m_{S∨T}} &  & S ∨ T ∨ S ∨ T,}

n_{S∨T} = n_{S} ∨ n_{T} : S ∨ T → S ∨ T.

We shall deal more generally with the following theories of coactions which are fundamental in development of homotopy theory in [3]. Principal coactions are defined as in [9, Section 4].

1.4. Definition. A theory of coactions is a theory T all of whose objects X are assigned a cogroup S_{X} and a principal coaction with structure maps a_{X} : X → X ∨ S_{X}, d_{X} : S_{X} → X ∨ X. These structures are required to be compatible with sums, i. e. S_{X∨Y} = S_{X} ∨ S_{Y}, etc.

Clearly a theory of cogroups is a special case of a theory of coactions, with S_{X} = X, a_{X} = m_{X}, d_{X} = (n_{X} ∨ id_{X})m_{X}, where m_{X} and n_{X} are the comultiplication and the coinverse respectively of the cogroup X.

1.5. Definition. Let S and T be cogroups in a category C. We say that f : S → T is linear if f is compatible with the cogroup structure, i. e. m_{T} f = (f ∨ f)m_{S}. More generally a map f : S → X ∨ T is called linear in T if the diagram

\xymatrix{S \ar[rr]^{f} \ar[d]^{m_{S}} &  & X ∨ T \ar[d]^{id_{X}∨m_{T}} \\ S ∨ S \ar[r]_{f∨f} & X ∨ T ∨ X ∨ T \ar[r]^{( \begin{matrix}i_{X} \\ i_{1,T} \\ i_{X} \\ i_{2,T} \end{matrix} )} & X ∨ T ∨ T}

commutes.

We point out that in a theory of cogroups T, morphisms in T are not required to be linear.

2. Theories with suspension

In this section we introduce and study “theories with suspension objects and partial suspension operator” termed Σ-theories. There are three different ways to characterize a Σ-theory. The categorical dual of a Σ-theory is termed a Ω-theory.

Let T be a theory of cogroups or more generally a theory of coactions.

For an object X of T we define the category T(X) as follows. Objects of T(X) are the objects of T having a cogroup structure. A morphism s from S to T in T(X) is given by a commutative diagram

(2.1)  \xymatrix{ & X \ar[dl]_{i_{X}} \ar[dr]^{i_{X}} \\ X ∨ S \ar[rr]^{s} \ar[dr]_{id_{X}∨0_{S}} &  & X ∨ T \ar[dl]^{id_{X}∨0_{T}} \\  & X ∨ ∗ & .}

Note that each map f : X → Y induces a functor T(f) : T(X) → T(Y) which is identity on objects and carries a : X ∨ S → X ∨ T to

\binom{i_{Y}}{(f ∨ T)ai_{S}} : Y ∨ S → Y ∨ T.

In particular T(∗) is the category whose objects are cogroups in T and whose morphisms are maps respecting counits. It is thus a theory of cogroups, and it is easy to see that the functor T(!_{X}) : T(∗) → T(X) induced as above preserves finite coproducts. Hence also T(X) is a theory of cogroups with the same structure on objects as T(∗).

There is an alternative description of morphisms in T(X) as follows: assigning to s : X ∨ S → X ∨ T the map si_{S} : S → X ∨ T gives an exact sequence of groups

\xymatrix{0 → hom_{T(X)}(S, T) \ar[r] & hom_{T}(S, X ∨ T) \ar[r]^{hom_{T}(S,id_{X}∨0_{T})} & hom_{T}(S, X ∨ ∗),}

where the group structures are induced by the cogroup structure of S. In other words, there are isomorphisms

(2.2)  hom_{T(X)}(S, T) ≅ [S, X ∨ T]_{X}

where by definition [S, X ∨ T]_{X} = hom_{Pair(T)}(0_{S}, id_{X} ∨ 0_{T}). This is the set of those maps S → X ∨T which are “trivial on X”, i. e. those maps g : S → X ∨T which render the diagram

\xymatrix{S \ar[r]^{g} \ar[d]_{0_{S}} & X ∨ T \ar[d]^{id_{X}∨0_{T}} \\ ∗ \ar[r]^{!} & X ∨ ∗}

commutative. The inverse isomorphism [S, X ∨ T]_{X} ≅ hom_{T(X)}(S, T) carries such a map g to

\binom{i_{X}}{g} : X ∨ S → X ∨ T.

It is easy to see that s : X ∨S → X ∨T is linear as a morphism of T(X) iff si_{S} : S → X ∨ T is linear in T as a morphism in T; see 1.5. The following definition of a Σ-theory is canonically derived from the notion of a Σ-representable track category as we shall see in our main result 4.2 below.

2.1. Definition. A theory with suspensions (or a Σ-theory for short) is a theory of coactions T together with a system of functors

E_{X} : T(X) → T(X)

which preserve finite sums and carry each map to a linear map. These functors must be natural in the sense that for all maps f : X → Y the diagram of functors

\xymatrix{T(X) \ar[r]^{E_{X}} \ar[d]^{T(f)} & T(X) \ar[d]^{T(f)} \\ T(Y) \ar[r]^{E_{Y}} & T(Y)}

commutes. In particular for X = ∗ one has the suspension functor

Σ = E_{∗} : T(∗) → T(∗).

Hence for each cogroup S in T the suspension ΣS is defined and is a cogroup in T. Considering the above diagram for f =!_{X} : ∗ → X one sees that all functors E_{X} coincide with Σ on objects. Moreover using the identification (2.2) above one may describe the effect of E_{X} on morphisms by maps which we will call partial suspension maps

(2.3)  E : [S, X ∨ T]_{X} → [ΣS, X ∨ ΣT]_{X}

so that

E_{X} \binom{i_{X}}{g} = \binom{i_{X}}{Eg}.

By definition, for any map f : S → T in T(∗), the map Σf is linear. More generally for any g : S → X ∨ T with (id_{X} ∨ 0_{T})g = i_{∗}0_{S} : S → X ∨ ∗, the map Eg : ΣS → X ∨ ΣT is linear in ΣT. Considering the comultiplication m_{S} : S → S ∨ S as a morphism in T(∗) one sees that

Σm_{S} : ΣS → Σ(S ∨ S) = ΣS ∨ ΣS

is linear. This implies that the cogroup structure on ΣS is coabelian.

2.2. Remark. The categorical dual of a theory with suspensions (Σ-theory) is a theory with loop objects termed an Ω-theory for short. Hence an Ω-theory T is a category with finite products in which each object X is endowed with an action of a group object G, X × G → X, so that T is a theory of actions with properties dual to 1.4. Moreover T(X) is the category of diagrams

\xymatrix{ & X \\ X × G \ar@{-->}[ur]^{p_{X}} &  & X × H \ar[ul]_{p_{X}} \ar[ll] \\  & X × ∗ \ar[ul] \ar[ur]}  (1)

dual to (2.1) and a system of functors

L_{X} : T(X) → T(X)  (2)

is given with properties dual to 2.1. The loop functor is given by

Ω = L_{∗} : T(∗) → T(∗)  (3)

which carries a group object G to the group objectΩG in T. Moreover partial loop operations

L : [X × H, G]_{X} → [X × ΩH, ΩG]_{X}  (4)

are defined dually to (2.3) by L_{X} in (2) above.

There are alternative ways to characterize a Σ-theory:

2.3. Lemma. Equipping a theory of coactions T with the structure of a Σtheory is equivalent to assigning a cogroup ΣS to each cogroup S in T and functions

E : [S, X ∨ T]_{X} → [ΣS, X ∨ ΣT]_{X}

to each object X and cogroups S, T in such a way that the following conditions are satisfied.

E (\binom{f}{b}a) = \binom{f}{Eb}Ea

for any f : X → Y, a ∈ [S, X ∨ T]_{X}, b ∈ [T, Y ∨ U]_{Y}, and furthermore Ea is linear in ΣT. The functor Σ obtained from the particular case of this condition when X = Y = ∗, must preserve sums. And, for i_{S} : S → X ∨ S one must have Ei_{S} = i_{ΣS} : ΣS → X ∨ ΣS.

Proof. We already saw how to derive partial suspension maps from the Σ-struc-ture and the maps E_{X} from E. Conditions that they must obey are equivalent, to properties of E_{X}, namely, to preservation of composition, sums and identities, respectively.

Conversely, given the maps E, one defines E_{X} as above and then the required properties will correspond to properties of E as indicated. □

For any Σ-theory T one has the functor

(2.4)  Σ_{∗} : T → T,

termed the suspender of T, as follows. For an object X of T with the coaction structure (S_{X}, a_{X}, d_{X}) let Σ_{∗}(X) = X ∨ ΣS_{X}. Moreover define the difference map ∇(f) ∈ [S_{X}, Y ∨ S_{Y}]_{Y} for any map f : X → Y by

∇(f) = ([figure: S_{X}; d_{X}; X ∨ X; f∨f; Y ∨ Y; ^{\binom{i_{Y}}{a_{Y}}}; Y ∨ S_{Y}] ).

Then the partial suspension in T yields E∇(f) ∈ [ΣS_{X}, Y ∨ ΣS_{Y}] and one defines

Σ_{∗}(f) = \binom{i_{Y} f}{E∇(f)} : X ∨ ΣS_{X} → Y ∨ ΣS_{Y} .

This indeed defines a functor Σ_{∗} which also can be used to characterize the Σ-theory completely.

2.4. Lemma. Let T be a theory of coactions. Then structures of a Σ-theory on T are in a one-to-one correspondence with sum-preserving endofunctors

Σ_{∗} : T → T

equipped, for each coaction (X, S_{X}, a_{X}, d_{X}) in T, with isomorphisms

ϑ_{X} : Σ_{∗}(X) ≅ X ∨ ΣS_{X}

for some cogroups ΣS_{X} depending only on the cogroups S_{X}. These isomorphisms must be natural in the sense that the inclusions

j_{X} = ϑ_{X}^{−1}i_{X} : X → Σ_{∗}(X)

and the projections

p_{X} = \binom{id_{X}}{0_{ΣS_{X}}}ϑ_{X} : Σ_{∗}(X) → X

are natural in X. Moreover the composite maps

\xymatrix{ΣS_{X} \ar[r]^{i_{ΣS_{X}}} & X ∨ ΣS_{X} \ar[r]^{ϑ_{X}^{−1}} & Σ_{∗}(X) \ar[r]^{Σ_{∗}(f)} & Σ_{∗}(Y) \ar[r]^{ϑ_{Y}} & Y ∨ ΣS_{Y}}

must be linear in ΣS_{Y}.

Proof. Given the functor Σ_{∗} as above, for any cogroup S consider S as a coaction via S_{S} = S coacting on itself via its own comultiplication; let ΣS = ΣS_{S}. These come equipped with maps q_{S} : Σ_{∗}(S) → ΣS given by

q_{S} = ( \xymatrix{Σ_{∗}(S) \ar[r]^{ϑ_{S}} & S ∨ ΣS \ar[r]^{0_{S}∨id_{Σ_{S}}} & ∗ ∨ ΣS ≅ ΣS} ).

We call Σ the suspension and define the partial suspension Eg for g ∈ [S, X∨T]_{X} by

Eg = ( \xymatrix{ΣS \ar[r]^{i_{ΣS}} & S ∨ ΣS \ar[r]^{ϑ_{S}^{−1}} & Σ_{∗}(S) \ar[r]^{Σ_{∗}(g)} & Σ_{∗}(X ∨ T)}

\xymatrix{≅ Σ_{∗}(X) ∨ Σ_{∗}(T) \ar[r]^{p_{X}∨q_{T}} & X ∨ ΣT} ) .

It is clear that Eg is linear in ΣT and also other properties required of E and Σ are all satisfied. The converse procedure has been already described above. □

3. Cohomology of Σ-theories and Ω-theories

Recall that a natural system D on a category T is a functor D : F T → Ab where F T is the category of factorizations of T; see [8, 2.1]. Here D carries the object f : A → B in F T to an abelian group D_{f} and carries maps [figure: g; f; h] to induced action maps g_{∗} : D_{f} → D_{gf}, ξ ↦ gξ, and

h^{∗} : D_{f} → D_{fh}, ξ ↦ ξh.

For objects A, B in T let

[A, B] = hom(A, B)

be the set of morphisms A → B in T. If T is a Σ-theory we have for the coaction on each object X in T the structure maps

\begin{cases}a_{X} : X → X ∨ S_{X} \\ d_{X} : S_{X} → X ∨ X \end{cases}

as in 1.4.

3.1. Proposition. Each Σ-theory T comes equipped with a canonical natural system D^{T} defined as follows: For a map f : X → Y let

D_{f}^{T} = [ΣS_{X}, Y],

with the abelian group structure induced by the abelian cogroup ΣS_{X}. Left action of a map y : Y → Y′ on a : ΣS_{X} → Y is given by

y_{∗}(a) = ya.

The right action of x : X′ → X on a is given by the composite

[figure: ΣS_{X′}; E∇(f); X ∨ ΣS_{X}; ^{\binom{f}{a}}; Y,]

where ∇(f) is the composite

∇(f) = ( [figure: S_{X}; d_{X}; X ∨ X; f∨f; Y ∨ Y; ^{\binom{i_{Y}}{a_{Y}}}; Y ∨ S_{Y}] ).

Proof. It turns out that naturality of actions is evident if one uses the functor Σ_{∗} in (2.4) above, whereas the fact that this is indeed the natural system of abelian groups and not just sets is clear if one uses the suspensions Σ. So let us first define the actions using the former and then transform them using the latter.

For any morphism f : X → Y, put

D_{f}^{T} = {a : Σ_{∗}(X) → Y | aj_{X} = f},

as in the diagram

\xymatrix{ & X \ar[d]^{j_{X}} \ar[dl]_{f} \\ Y & Σ_{∗}(X). \ar@{-->}[l]_{a}}

Actions of this natural system are defined as follows: for y : Y → Y′ let the induced action D_{f}^{T} → D_{yf}^{T} be given by a ↦ ya, as in the diagram

\xymatrix{ &  & X \ar[d]^{j_{X}} \ar[dl]_{f} \\ Y′ & Y \ar[l]_{y} & Σ_{∗}(X); \ar[l]_{a} \ar[ll]^{ya}}

whereas for x : X′ → X let the action D_{f}^{T} → D_{fx}^{T} be given by a ↦ aΣ_{∗}(x), as in the diagram

\xymatrix{ & X \ar[d]^{j_{X}} \ar[dl]_{f} & X′ \ar[l]_{x} \ar[d]^{j_{X′}} \\ Y & Σ_{∗}(X) \ar[l]_{a} & Σ_{∗}(X′), \ar[l]_{Σ_{∗}(x)} \ar[ll]^{x^{∗}(a)}}

where the square commutes by 2.4.

Now let us use the isomorphisms

ϑ_{X} : Σ_{∗}(X) ≅ X ∨ ΣS_{X}

inducing bijections

D_{f}^{T} ≈ [ΣS_{X}, Y],

where the latter sets have the group structure induced by the canonical cogroup structure on ΣS_{X}. In fact as we have seen ΣS_{X} is an abelian cogroup (see 2.1), so the groups D_{f}^{T} are indeed abelian. Moreover the linearity conditions imply that the actions are indeed homomorphisms.

For convenience let us give explicit expressions for the actions in these new terms. Left actions are again given by composition: for y : Y → Y′ the induced homomorphism is

[ΣS_{X}, Y] → [ΣS_{X}, Y′], given by a ↦ ya

for any a : ΣS_{X} → Y.

Right actions are more involved. For x : X′ → X and a : ΣS_{X} → Y the map x^{∗}(a) : ΣS_{X′} → Y is given by the lower horizontal composition in the diagram

[figure: Σ_{∗}(X); ≅; ϑ_{X}; Σ_{∗}(X′); Σ_{∗}(f); ≅; ϑ_{X′}; Y; X ∨ ΣS_{X}; _{\binom{f}{a}}; X′ ∨ ΣS_{X′}; ΣS_{X′}.; i_{ΣS_{X′}}; x^{∗}(a)]

This yields the result by use of the definition of Σ_{∗} in (2.4). □

For each category C with natural system D the cohomology H^{n}(C; D) is defined as in [11]. This leads to the following notion of Σ-cohomology.

3.2. Definition. Let T be a Σ-theory and let D^{T} be the natural system determined by T as in 3.1. Then the Σ-cohomology of T is given by the abelian group (n ≥ 0)

H_{Σ}^{n}(T) = H^{n}(T; D^{T}).

The categorical dual of a Σ-theory is an Ω-theory T for which we have accordingly the Ω-cohomology

H_{Ω}^{n}(T) = H^{n}(T; D^{T}),

where D^{T} is the natural system associated to T with D_{f}^{T} = [A, ΩB] for f : A → B in T.

4. Track theories with suspensions.

A track theory 𝒯 is a track category with sums A ∨ B in the weak sense so that for hom-groupoids in 𝒯 the map

\xymatrix{[[A ∨ B, X]] \ar[r]^{∼} & [[A, X]] × [[B, X]]}

is an equivalence of groupoids; see [8, §4]. Moreover 𝒯 is Σ-representable if suspenders (Σ_{f}, i_{f}, υ_{f}) as defined in [9] exist for all maps f : A → B in 𝒯 .

4.1. Definition. A Σ-track theory is a track theory 𝒯 which is Σ-representable and for which the homotopy category 𝒯_{≃} is a theory of coactions.

Recall that linear track extensions are defined as in [8, 2.2]. It is clear that a Σ-track theory 𝒯 has abelian hom-groupoids so that 𝒯 is a linear track extension of its homotopy category 𝒯_{≃} as in [8].

4.2. Theorem. For any Σ-track theory 𝒯 its homotopy category carries a canonical structure of a Σ-theory and 𝒯 is a linear track extension of 𝒯_{≃} by the canonical natural system D^{𝒯_{≃}} associated to the Σ-theory 𝒯_{≃}. Conversely, any linear track extension of a Σ-theory by its canonical natural system is a Σ-track theory.

Proof. Consider a track category 𝒯 satisfying the hypothesis. To equip 𝒯_{≃} with a structure of a Σ-theory it suffices, according to 2.1, to define the functor Σ_{∗}. We do this by declaring Σ_{∗}(X) = Σ_{id_{X}} for all objects X and Σ_{∗}(f) = Σ_{∗}(f, f) for all maps f : X → Y, where (f, f) is considered as a morphism in Pair(𝒯 ) from id_{X} to id_{Y}. By [9, 2.1] this indeed defines an endofunctor, equipped with natural maps j_{X} = i_{id_{X}} : X → Σ_{∗}(X). Σ_{∗} preserves sums by [9, 2.4]. Moreover the isomorphisms ϑ are obtained, in view of [9, 2.2], by virtue of [9, 4.3]. The latter also implies that under these isomorphisms the maps j_{X} correspond to the coproduct inclusions i_{X} : X → X ∨ΣS_{X}. To prove naturality of the projections

p_{X} = \binom{id_{X}}{0_{ΣS_{X}}}ϑ_{X} : Σ_{∗}(X) → X

it suffices to construct, for any map f : X → Y, a track fp_{X} ≃ p_{Y} Σ_{∗}(f). By the universal property of the suspender Σ_{id_{X}}, such track exists iff there is a track in G_{id_{X}} between fp_{X}υ_{X} ∈ Aut(fp_{X}j_{X}) and p_{Y} Σ_{∗}(f)υ_{X} ∈ Aut(p_{Y} Σ_{∗}(f)j_{X}). By [9, 2.1], one may take ζ_{(f,f)} for this purpose. Finally, linearity condition from 2.1 means commutativity in 𝒯_{≃} of the diagram

\xymatrix{ΣS_{X} \ar[rr]^{E} \ar[d]^{Σm_{SX}} &  & Y ∨ ΣS_{Y} \ar[d]^{id_{Y} ∨Σm_{SY}} \\ ΣS_{X} ∨ ΣS_{X} \ar[r]_{E∨E} & Y ∨ ΣS_{Y} ∨ Y ∨ ΣS_{Y} \ar[r]^{( \begin{matrix}i_{Y} \\ i_{1,ΣSY} \\ i_{Y} \\ i_{2,ΣSY} \end{matrix} )} & Y ∨ ΣS_{Y} ∨ ΣS_{Y} ,}

for f : X → Y, where E is the composite

\xymatrix{ΣS_{X} \ar[r]^{i_{ΣS_{X}}} & X ∨ ΣS_{X} \ar[r]^{ϑ_{X}^{−1}} & Σ_{∗}(X) \ar[r]^{Σ_{∗}(f)} & Σ_{∗}(Y) \ar[r]^{ϑ_{Y}} & Y ∨ ΣS_{Y} ,}

as in 2.1. In other words, for any object Z the induced diagram

\xymatrix{[ΣS_{X}, Z] &  & [Y, Z] × [ΣS_{Y} , Z] \ar[ll]_{E^{∗}} \\ [ΣS_{X}, Z]^{2} \ar[u]_{+} & ([Y, Z] × [ΣS_{Y} , Z])^{2} \ar[l]_{_{E^{∗}_{2}}} & [Y, Z] × [ΣS_{Y} , Z]^{2} \ar[l]_{(id_{[Y,Z]}×p_{1},id_{[Y,Z]}×p_{2})} \ar[u]_{id_{[Y,Z]}×+}}

must commute. Now using universal property of suspenders, the entries [Σ?, Z] may be replaced by the isomorphism classes of groupoids G_{?}(Z), thus obtaining

\xymatrix{π_{0}G_{0_{S_{X}}} (Z) &  & [Y, Z] × π_{0}G_{0_{S_{Y}}} (Z) \ar[ll] \\ π_{0}G_{0_{S_{X}}} (Z)^{2} \ar[u] & ([Y, Z] × π_{0}G_{0_{S_{Y}}} (Z))^{2} \ar[l] & [Y, Z] × π_{0}G_{0_{S_{Y}}} (Z)^{2} \ar[l] \ar[u]}

Calculating actions of maps in this diagram in the same way as in [9, 4.3], we see that commutativity of the diagram amounts to conjugatedness of the tracks

id_{zf}\(id_{z} · ω_{1})f + id_{zf}\(id_{z} · ω_{2})f and id_{zf}\(id_{z} · (ω_{1} + ω_{2}))f

in Aut(!_{Z}0_{S_{X}}) for an element (z, ω_{1}, ω_{2}) ∈ [Y, Z] × G_{0_{S_{Y}}} (Z)^{2} given by a map z : Y → Z and two tracks in Aut(!_{Z}0_{S_{Y}}), where · and \ denote coaction and division maps, as in [9, 4.3]. But in fact these tracks are easily seen to be equal.

We now turn to the converse statement. Given a linear track category 𝒯 with a structure of a Σ-theory on 𝒯_{≃}, we know from [8, 4.6] that 𝒯 has weak sums compatible with the sums in 𝒯_{≃}. Moreover as 𝒯_{≃} is a theory of coactions, it is clear that each object of 𝒯 has a structure of a weak principal coaction. Thus to have all suspenders it suffices by [9, 4.3] to construct suspenders of the maps 0_{S} : S → ∗, for weak cogroups S in 𝒯 . For i_{0} : ∗ → Σ_{0}, take the suspension ΣS = E_{∗}(0_{S} : S → ∗). Define the universal track

υ_{0} ∈ Aut( \xymatrix{S \ar[r]^{0_{S}} & ∗ \ar[r]^{!_{ΣS}} & ΣS} )

by υ_{0} = σ_{!_{ΣS}0_{S}}(id), utilizing the linear track extension action

D_{!_{ΣS}}^{𝒯_{≃}} = \xymatrix{[ΣS, ΣS] \ar[r]^{σ_{0_{S}}} & Aut(!_{ΣS}0_{S})} ,

where the equality follows from 3.1. To show that this indeed has the universal property of a suspender, we have to show the following:

• for any object Z and any track η ∈ Aut(!_{Z}0_{S}) there exists a map Σ_{η} : ΣS → Z with η = Σ_{η}υ_{0};

• for any h, h′ : ΣS → Z with hυ_{0} = h′υ_{0} there is a track δ : h → h′. Using 3.1, the first condition becomes a tautology, whereas in the second one

obtains h = h′, so trivially one can take for δ the identity track. □

5. Classification of Σ-track categories

An equivalence between theories of coactions F : T → T′ is an equivalence of the underlying categories which preserves sums and which carries the coaction structure of an object X in T to the coaction structure of the object F (X) in T′.

5.1. Definition. A weak equivalence between Σ-track theories is a weak equivalence of the underlying track categories as in [8, 1.5] which preserves weak sums and suspenders and for which the induced functor between homotopy categories is an equivalence between theories of coactions.

5.2. Definition. An equivalence (F, Φ) between Σ-theories is an equivalence F : T → T′ between the underlying theories of coactions which is compatible with the suspension structure, that is for the diagram of functors

\xymatrix{T(X) \ar[r]^{F_{X}} \ar[d]^{E_{X}} & T′(F X) \ar[d]^{E_{F X}} \\ T(X) \ar[r]^{F_{X}} & T′(F X)}

with F_{X} induced by F one has a natural isomorphism Φ_{X} : E_{F X}F_{X} ≅ F_{X}E_{X}.

5.3. Definition. A classifying pair is a pair (T, τ) where T is a Σ-theory and τ is an element in the Σ-cohomology H_{Σ}^{3}(T). Two such pairs (T, τ) and (T′, τ′) are equivalent if there exists an equivalence (F, Φ) with F : T′ → T such that F^{∗}τ = I_{∗}τ′ in H^{3}(T′; F^{∗}D^{T}). Here the natural isomorphism I : D^{T′} ≅ F^{∗}D^{T} is given by Φ.

Using [8, 4.8] and 4.2 we obtain the following classification result.

5.4. Classification. There is a 1-1 correspondence between weak equivalence classes of Σ-track theories and equivalence classes of classifying pairs.

6. Examples

Since the track category associated to a Quillen model category is both Σrepresentable and Ω-representable (see [9, 1.6]) we get the following result.

6.1. Proposition. Let Q_{cf} be the track category of cofibrant and fibrant objects in a Quillen model category and let 𝒯 ⊂ Q_{cf} be a full track subcategory. If 𝒯_{≃} is a theory of coactions then 𝒯 is a Σ-track theory and if dually 𝒯_{≃} is a theory of actions then 𝒯 is an Ω-track theory. In particular 𝒯_{≃} is a Σ-theory and a Ω-theory respectively.

A similar result is available for a cofibration category and a fibration category in the sense of Baues [4]. This, in particular, yields the following examples.

6.2. Example. Let 𝒮 (k) be the track subcategory of Top^{∗} consisting of onepoint unions of spheres S^{n} with n ≥ k. Then 𝒮 (k) is a Σ-track theory and hence the homotopy category 𝒮 (k)_{≃} is a Σ-theory, see 1.1. The partial suspension in 𝒮 (k)_{≃}, in fact, coincides with the partial suspension used in [5, 4, 2]. The partial suspension can be described totally by the ordinary suspension for homotopy groups of spheres and by Whitehead products. The Σ-track theory 𝒮 (k) is classified by an element τ ∈ H_{Σ}^{3}(𝒮 (k)_{≃}) in the Σ-cohomology. This is the universal Toda bracket ⟨𝒮 (k)⟩ = τ also considered in [6] and Baues [2].

6.3. Example. Let 𝒦 (k) be the track subcategory of Top^{∗} consisting of products of Eilenberg-MacLane spaces K(A, n), n ≤ k, and A a finitely generated abelian group. Then 𝒦 (k) is an Ω-track theory and hence the homotopy category 𝒦 (k)_{≃} is an Ω-theory. The partial loop operation in 𝒦 (k)_{≃}, in fact, coincides with the loop operation L used in [5, 4, 2]. Here L can be described totally by the ordinary loop operation (suspension) in cohomology and by cup products, see [4]. The Ω-track theory 𝒦 (k) is classified by an element τ ∈ H_{Ω}^{3}(𝒦 (k)_{≃}) in the Ω-cohomology. This is the universal Toda bracket ⟨𝒦 (k)⟩ = τ also considered by Baues and Dreckmann [6] and Baues [2].

Recall that an additive category A is a category enriched in abelian groups which has sums. These sums are then also products (termed direct sums) denoted A ⊕ B; see [17]. Each object is canonically an abelian group object by (1, 1) : A ⊕ A → A and all maps in A are linear. A functor between additive categories is additive if it preserves biproducts.

6.4. Lemma. An additive category A together with an additive functor σ : A → A has canonically the structure of both a Σ-theory and a Ω-theory.

Proof. We simply define the suspender in 2.4 by Σ_{∗}(X) = X ⊕ σ(X). Dually we define a looper by Ω_{∗}(X) = X ⊕ σ(X). One readily checks the properties in 2.4 and the dual properties respectively. □

The Σ-cohomology of the Σ-theory (A, σ) in 6.4 is the cohomology

H_{Σ}^{n}(A, σ) = H^{n}(A; Hom(σ, −)),

where Hom(σ, −) is the bifunctor A^{op} × A → Ab which carries (X, Y) to Hom(σX, Y). Moreover the Ω-cohomology of the Ω-theory (A, σ) is the cohomology

H_{Ω}^{n}(A, σ) = H^{n}(A; Hom(−, σ)),

where the bifunctor Hom(−, σ) carries (X, Y) to Hom(X, σY). Here Hom denotes the set of morphisms in A which is an abelian groups since A is an additive category. The cohomology groups above are special cases of those by Baues and Wirsching [2] which are in particular studied by Jibladze and Pirashvili [15].

Let R be a commutative ring with unit and let Mod_{∗}(R) be the category of ℤ-graded R-modules X = (X_{i})_{i∈ℤ} with the graded tensor product X ⊗ Y given by

(X ⊗ Y)_{n} = ⨁_{i+j=n} X_{i} ⊗_{R} Y_{j}.

Here X is non-negatively graded if X_{i} = 0 for i < 0. A non-negatively graded R-module A which has the structure of a monoid in (Mod_{∗}(R), ⊗) is termed a graded algebra. Let R be the graded R-module concentrated in degree 0 and given by the ring R. Then of course R is also a graded algebra; in fact, the initial object in the category of graded algebras. A map ε : A → R between graded algebras is termed an augmentation of A.

6.5. Example. Let A be a graded algebra and let T_{A} be the category of ℤgraded finitely generated free left A-modules. Objects are direct sums of objects A[n] with n ≥ 0, where A[n] is the free left A-module generated by a single element [n] in degree n. The elements of A[n] are of the form a · [n] with a ∈ A and degree |a·[n]| = |a|+n. Morphisms f : A[n] → A[m] are determined by the element α ∈ A with f[n] = α · [m] and we write f = α^{∗} if f(a · [n]) = a · α · [m]. We define the shift functor

σ : T_{A} → T_{A}

to be the unique additive functor which carries A[n] to A[n + 1] and carries f = α^{∗} : A[n] → A[m] to α^{∗} : A[n + 1] → A[m + 1]. Hence by 6.4 the category T_{A} together with σ is canonically both a Σ-theory and a Ω-theory.

Acknowledgement

The second author gratefully acknowledges hospitality of the Max Planck Institut für Mathematik, Bonn and of the Université Catholique de Louvain, Louvain-la-Neuve.

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(Received 28.11.2000)

Authors’ addresses:

H.-J. Baues Max-Planck-Institut fuer Mathematik Vivatsgasse 7, D-53111 Bonn Germany E-mail: baues@mpim-bonn.mpg.de

M. Jibladze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: jib@rmi.acnet.ge