Stabilization and costabilization with respect to an action of a monoidal category
aa r X i v : . [ m a t h . C T ] F e b STABILIZATION AND COSTABILIZATION WITH RESPECT TOAN ACTION OF A MONOIDAL CATEGORY
MEHMET AK˙IF ERDAL AND ¨OZG ¨UN ¨UNL ¨U
Abstract.
Given a monoidal category V that acts on a 0-cell A in a 2-category M , we give constructions of stabilization and costabilization of A with respectto the V -action. This provide a general unified treatment for the stabilizationof homotopy theories. The constructions of stabilization and costabilizationare defined via universal properties and they define two endofunctors on the2-category of 0-cells with V -actions in M . We show that several examplesthat exist in the literature fit into our setting after fixing the 0-cell A and the V -action on it. In particular, our constructions establish a duality betweenstable homotopy categories and Spanier-Whitehead categories. Introduction
Given a relative category A equipped with a family of suspension functors Σ α : A → A its stabilization is obtained by formally inverting all these suspensions. Thestandard construction is given as a category generated by spectrum objects in A .If there is only one such suspension functor to invert, spectrum objects consistsof sequence of objects X n in A together with structure maps Σ X n → X n +1 . Thestabilization construction that assigns a homotopy theory to a stable homotopytheory, however, is not functorial, see e.g., [14, Sec. 7]. In order to get over thenon-functoriality of the stabilization one uses Goodwillie calculus to get functorialapproximations via Goodwillie-Taylor tower, [20, 13].The non-functoriality of the stabilization is due to non-equivariance of the func-tors between homotopy theories. This can be understood best in set level construc-tions. Let A be a finite set on which N acts. Let µ denote the map µ : A → A given by µ ( a ) = 1 · a , where · is the N -action. If we universally lift such a N actionon A to a Z action (i.e., a set on which N acts by bijections) then we obtain thelargest subset of A on which the restriction of µ is a bijection, see [6]. Now, if f : A → B is a map between finite N -sets, then f does not have to induce a mapon such maximal subsets unless f is N -equivariant. In other words, the associationthat sends a finite N -set to its largest subset on which N acts by bijections doesnot define an endofunctor from the category of N -sets and functions; however, itdefines an endofunctor on the category of N -sets and N -equivariant functions.In this paper, by introducing a notion of equivariant functors between categorieswith monoidal category actions, we give another way of tackle the non-functorialityof stabilization. In fact, one can consider a collection suspension functors (or loopspace functors) on a relative category A as an action of a monoidal category on Mathematics Subject Classification.
Primary 57Q20; Secondary 55Q45.
Key words and phrases. action, monoidal category, stabilization, (co)homology and(co)homotopy theories.Partially supported by T ¨UB˙ITAK-TBAG / A and construct stabilization as a functor from a category of relative categoriesequipped with actions of a fixed monoidal category and certain equivariant func-tors between them. The essential ingredient of our viewpoint is this notion of V -equivariance. We state our main result in an arbitrary 2-category M . Given amonoidal category V , we consider a 2-category V M of V -0-cells, V -equivariant 1-cells and 2-cells, and take the full sub-2-category st V M of stable 0-cells. Then thestabilization is the universal way of turning a V -0-cell into a stable V -0-cell, thathas the 2-categorical terminal property. Thus, the stabilization defines a 2-functorfrom V M to st V M . We also obtain a dual version of the stabilization, which wecall call costabilization. Statements of the main results.
Let V be a monoidal category and let M beany (possibly large) strict 2-category. In section 2, we discuss V -actions and weintroduce the notions of centralizer and cocentralizer of a V -action and using thiswe introduce the notion of V -equivariant 1-cells between 0-cells. We show that thedata defined by 0-cells of M with V -biactions, V -equivariant 1-cells between 0-cellsand 2-cells between equivariant 1-cells form a 2-category, which we denote by V M .A 0-cell in this 2-category will be called V -object. There is a distinguished sub-2-category of V M that consists of stable V -objects, where a V -object is stable if the V -action on A has an inverse up to 2-cells, see Definition 2.6. This sub-2-categoryis denoted by st V M . One of the main results of the present paper is the following: Theorem 1.1. If M is a complete (resp. cocomplete) -category, then st V M iscoreflective (resp. reflective) in V M . Here by (co)completeness we mean 2-categorical completeness; i.e.,
Cat -(co)complete.Thus, M is powered and copowered over Cat . The coreflector in the statement ofthe theorem is called stabilization and the reflector is called costabilization . Thetheorem is still valid in the lax sense, and in this case the coreflector is called laxstabilization and the reflector is called colax costabilization . The above theorem hasa particular interest in homotopy theory, when we choose M to be the 2-category RelCat of relative categories, relative functors and relative natural transforma-tions, see [1]. Under certain conditions, an action on a relative category induces anaction on the category of functors that satisfy generalized versions of (co)homologyaxioms of Eilenberg-Steenrod. In particular, we show that (co)homology theoriesare objects in the stabilization of (co)homology functors with respect to actionsinduced by suspensions.We later establish that stable homotopy categories indexed by a symmetricmonoidal category V are homotopy categories of lax stabilizations of pointed rela-tive categories with respect to V -actions and Spanier-Whitehead categories indexedby V are homotopy categories of colax costabilizations with respect to V actions. Organization of the paper.
In Section 2 we give the definition of V -actions andthe central notion of equivariance. We introduce the 2-category V M of V -0-cells, V -equivariant 1-cells and 2-cells between them. In section 3 we give definitionsand constructions of stabilizations and cotabilizations with respect to V -actions.We also give the proof of our main result. In section 4 we show that homologyand cohomology theories are stabilizations of V -actions on categories of homologyand cohomology functors. In section 5 we consider the special case of actions onrelative categories and obtain the various categories of spectra as stabilizations, and CTIONS OF MONOIDAL CATEGORIES 3
Spanier-Whitehead categories as homotopy categories of costabilizations, of relativecategories with respect to actions of symmetric monoidal categories.2.
Actions of monoidal categories and equivariance
In this section, we present a categorification of the definitions of monoid actionson sets and equivariant functions given in [6]. In particular, we define the action ofa monoidal category on an object in a strict 2-category, generalizing the definitionin [10], and explore the properties of this notion.2.1. V -actions. Let ( V , ⊗ , ) be a strict monoidal category. There is an associatedstrict 2-category B V with a single 0-cell ∗ , called the delooping bicategory of V .The 1-cells (i.e., 1-morphisms) from ∗ to ∗ are objects of V and the compositionof two 1-cells in B V is given by the monoidal product on V . We write B V op for1-cell dual of the 2-category B V ; that is, the 2-category obtained by reversing its1-cells but not the 2-cells. For the rest of the paper we use B W for the 2-category B V × B V op . Let M be a strict 2-category and A be a 0-cell in M . An action of V on A (or a V -action on A ) is a stirct 2-functor α : B W → M where α ( ∗ , ∗ ) = A .This definition is, in fact, the same as the definition of ordinary biaction. How-ever, we construct a 2-category endowed with an exotic notion of equivariance sothat the category equivalent to one sided actions, see Section 2.2.Clearly, ordinary actions of monoids, considered as 2-categories in the trivialway, are trivial examples. Remark 2.1.
Note that we can define actions of a nonstrict monoidal category afterstrictification. In other words, if V is a monoidal category (not necessarily strict)we can define a V -action as a str V -action where str V denotes the strictification ofthe monoidal category V . Equivalently, for a arbitrary monoidal category V we canconstruct B W as above and any pseudofunctor from B W to a strict 2-category canbe considered as a V -action by the strictification adjunction given in [3].Some examples where higher morphisms are more interesting are given in [10]which can be considered as examples to our actions by composing with the projec-tion from B V × B V op onto its left component B V and considering Remark 2.1. Example 2.2. If V is a strict monoidal category, then V (or any of its monoidalsubcategory) acts on V by its tensor product. One can define an action by ten-soring from the left, or from the right or from both sides. More generally, if A is(co)powered over V , then the (co)power defines a V -action on A . Example 2.3.
Any 1-endomorphism on a 0-cell A in a strict 2-category M gen-erates a N action by considering N as a monoidal category with only identity mor-phisms. More generally, one can choose a collection of 1-endomorphisms of A and take the monoidal category generated by these endomorphisms. The resultingmonoidal category acts on A .Some examples that have interest in homotopy theory are discussed in Section4.2. Notation 2.4.
We use the notation A α for a 0-cell A with V -action α . MEHMET AK˙IF ERDAL AND ¨OZG¨UN ¨UNL¨U V -equivariant -cells. Since M is a strict 2-category, we have a functor from M op × M to Cat which sends ( A , B ) to the category M ( A , B ) which we simplydenote by [ A , B ]. Given a 2-functor F : C op × C → M between strict 2-categories,its lax end › x : C F ( x, x ) is defined by the categorical equivalence[ Z, “ x : C F ( x, x )] ≃ Lax( C op × C , Cat ) (hom C ( − , − ) , [ Z, F ( − , − )])natural in Z . Here Lax( C op × C , Cat ) denotes the 2-category of 2-functors, lax-natural transformations and modifications. Similarly, the end ¸ x : C F ( x, x ) is definedby replacing Lax with Psd, the category of 2-functors, pseudo-natural transforma-tions and modifications; that is,[ Z, ˛ x : C F ( x, x )] ≃ Psd( C op × C , Cat ) (hom C ( − , − ) , [ Z, F ( − , − )])In other words, the (lax) end of F is a hom C weighted (lax) bilimit of F .Let A be a 0-cell in the strict 2-category M , and α : B W → M be a V -actionon A . Since B V has a single 0-cell, The categoryPsd( B W , Cat ) (hom B V op ( − , − ) , [ Z, α ( − , − )])on the right hand side of the categorical equivalece that defines the end ˛ x : B V op α ( x, x )is equivalent to the category whose objects are 1-cells ω : Z → A and invertible 2-cells (i.e., a 2-isomorphisms) σ ( u ) : α ( , u op ) ◦ ω ⇒ α ( u, op ) ◦ ω assigned to all objects u in V so that we have σ ( v ) • ( α (id , f ) ⊲ ω ) = ( α ( f, id op ) ⊲ ω ) • σ ( u ) (2.1)for every morphism f : u → v in V and the following diagram of 2-cells α ( , v op ) ◦ α ( , u op ) ◦ ωα ( , v op ) ◦ α ( u, op ) ◦ ω = α ( u, op ) ◦ α ( , v op ) ◦ ωα ( u, op ) ◦ α ( v, op ) ◦ ωα ( , ( u ⊗ v ) op ) ◦ ω α ( u ⊗ v, op ) ◦ ωσ ( u ⊗ v ) α ( , v op )( σ ( u )) α ( u, op )( σ ( v )) (2.2)commutes for every u , v in V and1 α ( , op ) ◦ ω = σ ( ) . (2.3)Morphisms from ( ω, σ ) to ( ω, σ ) are 2-cells θ from the 1-cell ω to the 1-cell ω sothat σ ( u ) • ( α ( , u op ) ⊳ θ ) = ( α ( u, op ) ⊳ θ ) • σ ( u )for all objects u in V . CTIONS OF MONOIDAL CATEGORIES 5
Remark 2.5.
In the case when M has an object ∗ such that [ ∗ , A ] ∼ = A for every0-cell A , than one obtains that ˛ x : B V op α ( x, x )has objects as pairs ( a, σ ) where a is a 1-cell ∗ → A and σ is the natural transfor-mation as above that satisfies 2.1, 2.2 and 2.3. A morphism from ( a, σ ) to ( a, σ ) isa morphism θ : a → a in A that satisfies σ ( u ) • ( α ( , u op ) ⊳ θ ) = ( α ( u, op ) ⊳ θ ) • σ ( u ) . M = Cat , that is A is a category, by choosing Z as the terminal category Nowlet ψ : B W −→ B W op × B W (2.4)be the 2-functor given by ψ ( u, v op ) = (( v, u op ) op , ( u, v op )). Given two V -actions α and β on 0-cells A α and B β , we define a V -action on [ A α , B β ] by the composition B W ψ → B W op × B W ( α op ,β ) −→ M op × M [ − , − ] −→ Cat . (2.5)We denote this V -action by [ α, β ]. We define the category Fun V ( A α , B β ) of V -equivariant -cells from A α to B β as the endFun V ( A α , B β ) = ˛ x : B V op [ α, β ]( x, x )provided that it exists. Any object in Fun V ( A α , B β ) is called V -equivariant 1-cell from A α to B β . The category Fun V ( A α , B β ) will in fact be considered asthe hom-category of a 2-category. Therefore, we give an explicit description ofFun V ( A α , B β ) that is functorial on ( A α , B β ). Objects of Fun V ( A α , B β ) are pairs( f, τ ) such that f is an object in [ A α , B β ] and τ is a natural isomorphism from[ α, β ]( , − op )( f ) : V → [ A α , B β ] to [ α, β ]( − , op )( f ) : V → [ A α , B β ]. We define ω ( f, τ ) = f and for every object u in V , we define the ( f, τ ) component of the2-cell σ ( u ) as τ ( u ). So that the above compatibility conditions are satisfied. Inparticular, given a V -equivariant 1-cell ( f, τ ) from A α to B β we have an morphism σ ( u ) ( f,τ ) = τ ( u ) : β ( , u op ) ◦ f ◦ α ( u, op ) → β ( u, op ) ◦ f ◦ α ( , u op )in [ A α , B β ] for every u in V .2.3. The -category V M . We define two functors π l , π r : B W → B W as π l :( u, v op ) ( u, op ) and π r : ( u, v op ) ( , v op ) on 1-cells and corresponding projec-tions on 2-cells. We say a V -action α is a right (respectively left) action if α ◦ π r = α (respectively α ◦ π l = α ). An action will be called an action mute on one side if itis a right action or a left action. We will say a left action is mute on right and aright action is mute on left. Now we start defining the 2-category V M . The 0-cellsof V M are the 0-cells of M equipped with a V -action mute on at least one side.The morphisms of V M are V -equivariant 1-cells as defined in the previous section.Now we define the composition of two equivariant 1-cells.Let A α , B β , and C γ be 0-cells in V M . Assume ( f, τ f ) is a V -equivariant 1-cellfrom A α to B β and ( g, τ g ) is a V -equivariant 1-cell from B β to C γ . We define thecomposition of ( f, τ f ) and ( g, τ g ) as the pair ( g ◦ f, τ g ◦ f ) where τ g ◦ f ( u ) for u in V is defined as follows: In case β is mute on left and hence a right action, for u in V , MEHMET AK˙IF ERDAL AND ¨OZG¨UN ¨UNL¨U we have β ( u, op ) = 1 B , so we can define τ g ◦ f ( u ) as the isomorphism given by thevertical composition of the isomorphisms: A A B B C CA A B B C CA A B B C C α ( u, op ) f β ( u, op ) g γ ( , u op ) α ( u, op ) f β ( , u op ) g γ ( u, op ) α ( , u op ) f β ( u, op ) g γ ( u, op ) g ◦ fg ◦ f τ g ( u ) τ f ( u ) In case β is mute on the right and hence a left action, for u in V we have β ( , u op ) =1 B and so we can define τ g ◦ f ( u ) as the isomorphism given by the vertical composi-tion of the following isomorphisms: A A B B C CA A B B C CA A B B C C α ( u, op ) f β ( , u op ) g γ ( , u op ) α ( u, op ) f β ( u, op ) g γ ( u, op ) α ( , u op ) f β ( , u op ) g γ ( u, op ) g ◦ fg ◦ f τ g ( u ) τ f ( u ) Notice in case β is mute on both sides, these definitions coincide by the inter-change law.Let A α , B β , C γ and D δ be 0-cells in V M and ( f, τ f ) : A α → B β , ( g, τ g ) : B β → C γ and ( h, τ h ) : C γ → D δ be V -equivariant 1-cells. In the case when β and γ are both CTIONS OF MONOIDAL CATEGORIES 7 mute on the left, the associativity can be obtained from the following diagram:
A A B B C C D DA A B B C C D DA A B B C C D DA A B B C C D D α ( u, op ) f β ( u, op ) g γ ( u, op ) h δ ( , u op ) α ( u, op ) f β ( u, op ) g γ ( , u op ) h δ ( u, op ) α ( u, op ) f β ( , u op ) g γ ( u, op ) h δ ( u, op ) α ( , u op ) f β ( u, op ) g γ ( u, op ) h δ ( u, op ) h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ fh ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f τ h ( u ) τ g ( u ) τ f ( u ) If β is mute on the left and γ is mute on the right then τ h ◦ ( g ◦ f ) ( u ) is obtainedfrom the following diagram A A B B C C D DA A B B C C D DA A B B C C D DA A B B C C D D α ( u, op ) f β ( u, op ) g γ ( , u op ) h δ ( , u op ) α ( u, op ) f β ( , u op ) g γ ( u, op ) h δ ( , u op ) α ( , u op ) f β ( u, op ) g γ ( u, op ) h δ ( , u op ) α ( , u op ) f β ( u, op ) g γ ( , u op ) h δ ( u, op ) h ◦ ( g ◦ f ) h ◦ ( g ◦ f ) τ h ( u ) τ g ( u ) τ f ( u ) MEHMET AK˙IF ERDAL AND ¨OZG¨UN ¨UNL¨U and τ ( h ◦ g ) ◦ f ( u ) is obtained from the following diagram A A B B C C D DA A B B C C D DA A B B C C D DA A B B C C D D α ( u, op ) f β ( u, op ) g γ ( , u op ) h δ ( , u op ) α ( u, op ) f β ( , u op ) g γ ( u, op ) h δ ( , u op ) α ( u, op ) f β ( , u op ) g γ ( , u op ) h δ ( u, op ) α ( , u op ) f β ( u, op ) g γ ( , u op ) h δ ( u, op )( h ◦ g ) ◦ f ( h ◦ g ) ◦ f τ h ( u ) τ g ( u ) τ f ( u ) Using the exchange law, since every unlabeled square is identity, we obtain that τ ( h ◦ g ) ◦ f ( u ) = τ h ◦ ( g ◦ f ) ( u ) . The remaining cases are similar to one of the cases above. It is also straightforwardto check that the diagram in 2.2 commutes. Therefore, we obtain that the compo-sition in V M is strictly associative. It is now easy to see that the unitors in V M are also identity, which makes V M a strict 2-category.2.4. Stable -cells with respect to a V -action. We define the notion of stabilityfor V -objects with respect to a V -action. Definition 2.6.
We say a 0-cell A α in V M is stable if α is mute on the right (resp.left) and there exits a V -action β on a 0-cell B that is mute on the left (resp. right)such that the 0-cells of A α and B β are 1-equivalent in V M . The full sub-2-categoryof stable V -objects is denoted by st V M .3. Stabilizations and costabilizations with respect to V -actions Suppose that M is a strict 2-category that is powered and copowered over Cat .We denote by ⋔ : Cat op × M → M : ( C , A ) ⋔ ( C , A )the powering of a 0-cell A in M by a category C , and we denote by ⊠ : Cat × M → M : ( C , A )
7→ C ⊠ A the copowering of A by C .We define two functors µ r , µ l : B W × B W op → Cat op as follows: For s in { l, r } define µ s (( ∗ , ∗ ) , ( ∗ , ∗ )) = V considered as a category after forgetting the monoidalstructure and µ s (( x, y op ) , ( z, w op ) op ) : V → V is the functor given by µ s (( x, y op ) , ( z, w op ) op )( u ) = (cid:26) ( y ⊗ u ) ⊗ w , if s = rz ⊗ ( u ⊗ x ) , if s = l (3.6) CTIONS OF MONOIDAL CATEGORIES 9
Let α : B W → M be a V -action on a 0-cell A in M . Define Θ s,α as thecomposition B W × B W id × ψ → B W × B W op × B W µ s × α −→ Cat op × M ⋔ −→ M . (3.7)Define a V -action Inv s ( α ) : B W → M asInv s ( α )( − ) = ˛ x : B V op Θ s,α ( − , ( x, x )) (3.8)We use the notation Inv s ( A α ) for Inv s ( α )( ∗ , ∗ ).Let ¨ µ l = µ l (( , op ) , − ) and ¨ µ r = µ l ( − , ( , op ) op ). Let s, s ′ ∈ { l, r } with s = s ′ .Note that Θ s,α ( ∗ , ∗ , − ) is equal to the composition given by B W ψ → B W op × B W ¨ µ s ′ × α −→ Cat op × M ⋔ −→ M . which we denote by ⋔ (¨ µ s ′ , α ). Thus, Inv s ( A α ) is isomorphic to ˛ x : B V op ⋔ (¨ µ s ′ , α )( x, x ) (3.9)The inclusion { } → V induces a V -equivariant 1-cell ǫ s : Inv s ( A α ) → A α called the evaluation at , given by the composition of ω s : Inv s ( A α ) → ⋔ ( V , A ),the universal wedge of the end, and ε : ⋔ ( V , A ) → ⋔ ( , A ) ∼ = A , the usualevaluation. We can define σ : [Inv s ( A α ) , α ]( , u op )( ǫ s ) → [Inv s ( A α ) , α ]( u, op )( ǫ s )as follows: Let ˜ µ l = µ l ( − , ( , op )) and ˜ µ r = µ l (( , op ) op , − ). For every u in V ,we have a map ε ◦ ⋔ (¨ µ s ′ , α )( , u op ) ◦ ω s → ε ◦ ⋔ (¨ µ s ′ , α )( u, op ) ◦ ω s By naturality of ε , we obtain α ( , u op ) ◦ ε ◦ ⋔ (¨ µ s ′ , id A )( , u op ) ◦ ω s → α ( u, op ) ◦ ε ◦ ⋔ (¨ µ s ′ , id A )( u, op ) ◦ ω s Notice that ε ◦ ⋔ (˜ µ s ′ , id A )( u, op ) = ε ◦ ⋔ (¨ µ s ′ , id A )( , u op ) and ε ◦ ⋔ (˜ µ s ′ , id A )( , u op ) = ε ◦ ⋔ (¨ µ s ′ , id A )( u, op ). Then we obtain α ( , u op ) ◦ ε ◦ ⋔ (˜ µ s ′ , id A )( u, op ) ◦ ω s → α ( u, op ) ◦ ε ◦ ⋔ (˜ µ s ′ , id A )( , u op ) ◦ ω s By Fubini theorem, we get α ( , u op ) ◦ ε ◦ ω s ◦ Inv s ( α )( u, op ) → α ( u, op ) ◦ ε ◦ ω s ◦ Inv s ( α )( , u op )We define σ as σ ( u ) : α ( , u op ) ◦ ǫ s ◦ Inv s ( α )( u, op ) → α ( u, op ) ◦ ǫ s ◦ Inv s ( α )( , u op ) . Theorem 3.1.
Let α be an left (respectively right) V -action on a -cell A in astrict -catgeory M that is powered and copowered over Cat . Then A α is stable ifand only if ǫ r (respectively ǫ l ) is a -equivalence in V M .Proof. The “if” part follows from the definition of being stable. Now for the ”onlyif” part. Assume that A α is stable. Without loss of generality assume α is aright action. Then A α is 1-equivalent to B β for some left V -action β . Now it isstaright forward to check that ǫ l from Inv l ( B β ) to B β is a 1-equivalence. Also noticethat the 1-equivalence from A α to B β induces a 1-equivalence from Inv l ( A α ) and Inv l ( B β ). Hence by naturality of the evaluation map ǫ l from Inv l ( A α ) to A α is a1-equivalence. (cid:3) Let A α be a 0-cell in V M where V and M are as above. Definition 3.2.
The stabilization of A α is a stable 0-cell Stab V ( A ) α , togetherwith a V -equivariant 1-cell ǫ : Stab V ( A ) α → A α such that for every stable 0-cell B β , the induced functor ǫ ∗ : Fun V ( B β , Stab V ( A ) α ) → Fun V ( B β , A α ) is a categoricalequivalence. Remark 3.3.
Notice that the definition above implies for every V -equivariant 1-cell f : B β → A α there exists a V -equivariant 1-cell e f : Stab V ( A ) α → B β togetherwith a 2-isomorphism Stab V ( A ) α B β A αǫ e f f (3.10)where e f is unique up to unique 1-isomorphism.Given any 0-cell A α in V M letInv V ( A α ) = Inv l (Inv r ( A α ))and ǫ = ǫ r ◦ ǫ l . Let Inv V ( A α ) = Inv l ( A α ) and Inv n V ( A α ) = Inv V (Inv n − V ( A α )) forevery n >
1. Define Inv ∞V ( A α ) as the limit of the diagram · · · −→ Inv V ( A α ) ǫ −→ Inv V ( A α ) ǫ −→ Inv V ( A α ) , in the 2-category [ B W , M ]. When limits in M exists, so does this limit. Notethat every 0-cell in the above diagram is mute on the right, so that the object isisomorphic to an object in [ B V , M ] (which is a subcategory of V M ) under theinclusion induced by the functor B W → B V : ( u, v op ) u .Let ǫ ∞ : Inv ∞V ( A α ) → A α be the ǫ l composed with the transfinite compositionsof the maps in the diagram. Theorem 3.4.
Assume that limits in M exist. Then Inv ∞V ( A α ) together with ǫ ∞ : Inv ∞V ( A α ) → A α is a stabilization of A α .Proof. Since Inv r is given by an end of a powering, it commutes with limits. There-fore, Inv r (Inv ∞V ( A α )) is isomorphic to the limit lim n Inv r (Inv n V ( A α )). For each n ≥
1, we have 1-cells ǫ r : Inv r (Inv n V ( A α )) → Inv n V ( A α ), so that there is a universal1-cell F : Inv r (Inv ∞V ( A α )) → Inv ∞V ( A α )and we have 1-cells ǫ l : Inv n +1 V ( A α ) → Inv r (Inv n V ( A α )), so that there is a universal1-cell G : Inv ∞V ( A α ) → Inv r (Inv ∞V ( A α )) . Hence, Inv ∞V ( A α ) is stable.Given any stable B β in V M we have a V -equivariant 1-isomorphism B β → Inv V ( B α ), so that Inv V ( B α ) is a also stable. Inductively, we obtain V -equivariant1-isomorphisms B β → Inv n V ( B α ) for all n ≥
1. Thus, there is a 1-isomorphisms
CTIONS OF MONOIDAL CATEGORIES 11 B β → Inv ∞V ( B α ). Then, for any given V -equivariant 1-cell f : B → A , there exist aunique ˜ f : B → Inv ∞V ( A α ) given by the composition˜ f : B f −→ Inv ∞V ( B α ) Inv ∞V ( f ) −→ Inv ∞V ( A α )such that ǫ ∞ ◦ ˜ f is naturally isomorphic to f . This implies that ǫ ∞∗ : Fun V ( B β , Inv ∞V ( A ) α ) → Fun V ( B β , A α )is an equivalence of categories. (cid:3) In the case when V is a symmetric monoidal category, we can consider the bi-reverse as a stabilization functor due to the following theorem. Theorem 3.5.
Assume that V is a symmetric monoidal category. Then Inv V ( A α ) is equivalent to Inv ∞V ( A α ) .Proof. Let E : Inv V ( A α ) → Inv l (Inv V ( A α )) be the functor induced by π × µ ◦ ( π × π ) : V × V × V → V × V : ( u, v, w ) ( v, u ⊗ w ) . It is straightforward to check that E with ǫ r : Inv r (Inv V ( A α )) → Inv V ( A α ) is a1-equivalence. (cid:3) If M = Cat , then the objects in this limit can be viewed as follows: Theobjects of Inv ∞V ( A α ) are sequences of pairs { ( f n , φ n ) } n ∈ N where f n ∈ Inv n V ( A α )and φ n : f n +1 ( )( ) → f n is an isomorphism in Inv n V ( A α ). A morphism ζ betweentwo objects { ( f n , φ n ) } and { ( g n , ϕ n ) } is a set of arrows ζ n : f n → g n satisfying theevident compatibility conditions. The action on Inv ∞V ( A α ) is defined pointwise;that is, α ∞ ( u, v op ) { ( f n , φ n ) } = { ( e α nr ( u, v op )( f n ) , e α nr ( u, v op )( φ n )) } where e α nr is theaction on Inv n V ( A α ).We can also see that Inv ∞V ( A α ) is stable under the action by using the object-wisedescription given above. Define ψ : Inv ∞V ( A α ) → Inv r (Inv ∞V ( A α ))by ψ ( { ( f n , φ n ) } )( u ) = { ( f n +1 ( )( u ) , φ n +1 ( )( u )) } for any { ( f n , φ n ) } in Inv ∞V ( A α ). Then( ǫ l ◦ ψ )( { ( f n , φ n ) } ) = ψ ( { ( f n , φ n ) } )( ) = { ( f n +1 ( )( ) , φ n +1 ( )( )) } ∼ = { ( f n , φ n ) } and for any g : V →
Inv ∞V ( A α ) in Inv r (Inv ∞V ( A α )) such that g ( u ) = { ( f g ( u ) n , φ g ( u ) n ) } .Note that g ( u ) ∼ = e α nr ( u, v op )( g ( )) = e α nr ( u, v op )( { ( f g ( ) n , φ g ( ) n ) } ) ∼ = e α nr ( u, v op )( { ( f g ( ) n +1 ( )( ) , φ g ( ) n +1 ( )( )) } ) ∼ = { ( e α nr ( u, v op )( f g ( ) n +1 ( )( u )) , φ g ( ) n +1 ( )( u )) } Therefore ( ψ ◦ ǫ l )( g )( u ) = ( ψ ( g ( ))( u ) = ψ ( { ( f g ( ) n , φ g ( ) n ) } )( u )= ( f g ( ) n +1 ( )( u ) , φ g ( ) n +1 ( )( u )) ∼ = ( f g ( u ) n +1 ( u )( u ) , φ g ( u ) n +1 ( u )( u )) ∼ = ( f g ( u ) n +1 ( )( ) , φ g ( u ) n +1 ( )( ))= ( f g ( u ) n , φ g ( u ) n ) = g ( u ) Hence by induction φ ( ǫ l ( e f )) = e f .In the case when V is a symmetric monoidal category, the inverse equivalence E : Inv V ( A α ) → Inv l (Inv V ( A α )) of ǫ r : Inv r (Inv V ( A α )) → Inv V ( A α ) in the proofof Theorem 3.5 is given by E ( f )( u )( v )( w ) = f ( v )( u ⊗ w ). Observe that for every g in Inv r (Inv V ( A α )), E ( ǫ ( g ))( u )( v )( w ) = ǫ ( g )( v )( u ⊗ w )= g ( )( v )( u ⊗ w ) ∼ = g ( u )( u ⊗ v )( u ⊗ w ) ∼ = g ( u )( v ⊗ u )( u ⊗ w ) ∼ = g ( u )( v )( w ) . Dualizing the definition of the stabilization and obtain the costabilization.
Definition 3.6.
The costabilization of A α is a stable 0-cell coStab V ( A ) α , togetherwith a V -equivariant 1-cell η : A α → coStab V ( A ) α in V M such that for every stable0-cell B β , the induced functor η ∗ : Fun V ( A α , B β ) → Fun V (coStab V ( A ) α , B β ) is anequivalence of categories. Remark 3.7.
Similar to the definition of stabilization, the definition above impliesfor every V -equivariant 1-cell f : B β → A α in V M such that β is stable, there existsa 1-cell e f : B β → Stab V ( A ) α together with a 2-isomorphismcoStab V ( A ) α B β A α f e fη (3.11)where e f is unique up to unique 1-isomorphism.Let ∆ : B W → B W × B W be the diagonal functor. Let ν : B W × B W →
Cat be the functor given by ν s (( x, y op ) , ( z, w op ))( u ) = (cid:26) ( x ⊗ u ) ⊗ w , if s = rz ⊗ ( u ⊗ y ) , if s = l (3.12)Given a V -action α : B W → M on a 0-cell A , define Υ s,α as the composition B W × B W id × ∆ → B W × B W × B W ( ν s ,α ) −→ Cat × M ⊠ −→ M . (3.13)Define a V -action coInv s ( α ) : B W → M ascoInv s ( α )( − ) = ˛ x : B V op Υ s,α ( − , ( x, x )) (3.14)We write coInv r ( A α ) for coInv r ( α )( ∗ , ∗ ).Let ¨ ν l = ν l (( , op ) , − ) and ¨ ν r = ν l ( − , ( , op )). Then, for s = s ′ in { l, r } ,Υ s,α ( ∗ , ∗ , − ) is naturally isomorphic to the composition B W ∆ → B W × B W (¨ ν s ′ × α ) −→ Cat × M ⊠ −→ M . which we denote by ¨ ν s ′ ⊠ α . Thus, coInv s ( A α ) is isomorphic to ˛ x : B V op (¨ ν s ′ ⊠ α )( x, x ) CTIONS OF MONOIDAL CATEGORIES 13
In this case, the inclusion { } → V induces a V -equivariant 1-cell η s : A α → coInv s ( A α )which we call the coevaluation at . The V -equivarience of η s is similar to the caseof ǫ s above. Theorem 3.8.
Let α be an left (respectively right) V -action on a -cell A in astrict -category M that is powered and copowered over Cat . Then A α is stable ifand only if η r (respectively η l ) is a -equivalence in V M .Proof. This is dual to Theorem 3.1. (cid:3)
Given any 0-cell A α in V M let η = η l ◦ η r . Let coInv V ( A α ) = coInv l ( A α ) andcoInv n V ( A α ) = coInv V (coInv n − V ( A α )) for every n >
1. Define coInv ∞V ( A α ) as thecolimit of sequencecoInv V ( A α ) η −→ coInv V ( A α ) η −→ coInv V ( A α ) −→ · · · , provided that it exists. Let η ∞ : A α → coInv ∞V ( A α ) be η l composed with thetransfinite compositions of the maps in the diagram. Theorem 3.9.
Assume that colimits in M exist. Then coInv ∞V ( A α ) together with η ∞ : A α → coInv ∞V ( A α ) is a costabilazation of A α .Proof. The proof is dual to the proof of Theorem 3.4. (cid:3)
In the case when V is a symmetric monoidal category, we can consider the bi-reverse as a stabilization functor due to the following theorem. Theorem 3.10. If V is a symmetric monoidal category, then coInv V ( A α ) is equiv-alent to coInv ∞V ( A α ) .Proof. The proof is dual to the proof of Theorem 3.5. (cid:3)
Let V be a symmetric monoidal category and M = Cat . Note that if A α ismute on the right, then coInv ∞V ( A α ) is equivalent to coInv r ( A α ). The categorycoInv r ( A α ) can be constructed (up to equivalence of categories) as follows. LetcoInv r ( A α ) be the category whose objects are the collections of all pairs ( u, a ) andtriples ( u, v, a ) where u, v are objects in V and a an object in A α . A morphismbetween pairs is a morphism in V × A and a morphism triples is a morphismin
V × V × A . A morphism from the triple ( u, v, a ) to a pair ( w, a ) is eithera morphism from ( u, a ) to ( w, a ) or a morphism from ( u ⊗ v, α ( v, op )( a )) to( w, a ). There is no morphism from a pair to a triple in coInv r ( A α ). Let S bethe collection of morphisms from triples to pairs for which are identity in bothcoordinates; that is, a morphism from ( u, v, a ) to ( w, a ) is in S if either a = a and u = w and the corresponding map ( u, a ) to ( w, a ) is identity or u ⊗ v = w and α ( v, op )( a ) = a and the corresponding map from ( u ⊗ v, α ( v, op )( a )) to( w, a ) is identity. Then the category coInv r ( A α ) is given by the localization ofcoInv r ( A α ) at S . Observe that every triple in coInv r ( A α ) become isomorphic toa pair after passing to localization, coInv r ( A α ). Besides, every pair of the form( u ⊗ v, α ( v, op )( a )) becomes isomorphic to the pair ( u, a ). The V action oncoInv r ( A α ) is given by tensoring of first coordinate from the left; that is, if x is in V , thenIf A is stable, then η r : A α → coInv r ( A α ) is an equivalence of categories. Thiscan be realized via the isomorphism given by ( u, a ) α ( u, op )( a ). Combining the results of Theorems 3.4 and 3.9, we obtain our main result.
Corollary 3.11 (Theorem 1.1) . If M is a complete (resp. cocomplete) -category,then st V M is coreflective (resp. reflective) in V M . Lax stabilization and spectrification.
Let V be symmetric monoidal and α : B W → M be a V -action. Assume { s, s ′ } = { l, r } . Define a V -action ℓ Inv s ( α ) : B W → M as ℓ Inv s ( α )( − ) = “ x : B V op Θ s,α ( − , ( x, x )) (3.15)so that ℓ Inv s ( α )( ∗ , ∗ ) = ℓ Inv s ( A α ) is isomorphic to “ x : B V op ⋔ (¨ µ s ′ , α )( x, x ) (3.16)Assume that µ is mute on the right (left) and s = r ( s = l ). Then we can call ℓ Inv s ( A α ) the lax stabilization of A α . There is a canonical 1-cell ι α : Inv s ( A α ) → ℓ Inv s ( A α ) . We call the left adjoint of ι α in V M the spectrification whenever it exists.Assume that α factors through the category of adjunctions Adj( M ) whose 1-cellsare adjunctions in M . Then, there is an adjoint action ¯ α on A so that for each u, v in V , α ( u, v op ) is right adjoint to ¯ α ( v, u op ). Then there exist a equivariant 1-cellˆ¯ α : A ¯ α → ℓ Inv s ( A α )induced by the isomorphism [ A , ⋔ ( C , A )] ∼ = Cat ( V , [ A , A ]) . Therefore, if there is aspectrification, then there exist a V -equivariant 1-cell A ¯ α → Inv s ( A α ). Composingwith ǫ s we obtain a V -equivariant 1-cell A ¯ α → A α . In the case M = Cat and s = r , then ˆ α is a pair (ˆ α, σ ¯ α ) where ˆ α ( a )( u ) =¯ α (1 , u op )( a ) and σ ¯ α : a → α ( u, op )(¯ α (1 , u op )( a )) = α ( u, op )(ˆ α ( a ))is the adjunct of the identity map id ¯ α (1 ,u op )( a ) : ¯ α (1 , u op )( a ) → ¯ α (1 , u op )( a ). If L denotes the spectrification, then the V -equivariant functor A ¯ α → A α is given bythe composition A ¯ α ˆ α −→ ℓ Inv r ( A α ) L −→ Inv r ( A α ) ǫ r −→ A α . Examples of stable actions and stabilizations
Our primary examples of stable actions and stabilizations come from reduced(co)homology theories. Traditionally, (co)homology theories are measurement toolsthat compare homotopy properties of objects via comparing corresponding algebraicdata. However, one does not have to use algebraic data to make such comparisons.It can be done by means of any functor between any two category. If A and B aretwo categories, and F : A → B is a functor between them than if
F x ≇ F y in B then x ≇ y in A . More generally, if F is a functor that preserves certain structure (suchas homotopy) than F x and
F y not sharing this structure implies neither do x and y .In this paper we propose a more general and unorthodox definitions of a homologyand cohomology functors, whose source and target are homotopy categories, wherethe target has the notion of “exact sequence”. For the sake of computational tools,as in the classical (co)homology theories, we require these functors to satisfy certainaxioms similar to the classical Eilenberg-Steenrod axioms. Notions of suspension CTIONS OF MONOIDAL CATEGORIES 15 and loop functors, which are in fact actions of certain monoidal categories, are alsogeneralized accordingly.
Some conventions about categories.
We assume the axiom of Gr¨othendieckuniverses, cite [4]. For a universe U , when a monoidal category V acts on a category A , we assume V is U -small if A is a U -category, and the category A will belong toa U + -category of U -categories.4.1. V -graded (co)homology theories. For this section we let V be symmet-ric monoidal and consider V -actions on categories that are π of pointed ( ∞ , V acts, homotopy fiber and cofiber sequencesexists. Let f : A → S be a functor between such categories where S admits a notionof exact sequences (e.g., an Abelian category). We say f is left exact if it sendshomotopy fiber sequences to exact sequences, and right exact if it sends homotopycofiber sequences to exact sequences.We first consider a general definition of cohomology and homology functors inview of Eilenberg-Steenrod axioms. Let A and S be two pointed categories suchthat S has exact sequences. Cohomology Functors:
A product preserving right exact functor h : A op →S is called a cohomology functor. The full subcategory of cohomologyfunctors in [ A op , S ] is denoted by cohml( A , S ). Homology Functors:
A coproduct preserving left exact functor h : A → S is called a homology functor. The full subcategory of homology functors in[ A , S ] is denoted by hml( A , S ).Let α be a V -action on A op that is mute on the left. Consider S with thetrivial action. Then the functor category [ A op , S ] admits a V -action as describedin 2.5, which is mute on the right. We denote this action by [ α, α,
1] restricts to the subcategory cohml( A , S ) of cohomology functors.For example, it is enough to assume α is cocontinuous (preserves ( ∞ , A ).We define category of cohomology theories graded over V as follows: Definition 4.1.
The category of cohomology theories graded over V is defined asCOHML V ( A , S ) = Stab V (cohml( A , S ) [ α, );i.e., the stabilization of cohomology functors with respect to [ α, V is an object in the category of cohomologytheories.We can similarly define homology theories. Suppose now that α is a V -action on A that is mute on the left, so that the induced action [ α,
1] on [ A , S ] restricts tohml( A , S ) (e.g., α is cocontinuous). Definition 4.2.
The category of homology theories graded over V is defined asHML V ( A , S ) = Stab V (hml( A , S ) [ α, );i.e., the stabilization of homology functors with respect to action induced by [ α, V is an object in the category ofhomology theories. Axiomatic interpretation of the definitions above.
Unfolding the definitionsabove we can see that a cohomology theory graded over V consists of a functor h : V × A op → S such that each h u = h ( )( u ) is a cohomology functor, togetherwith natural isomorphisms σ u,v : h u → h u ⊗ v ◦ α ( , v op )such that the following diagram commutes h u ( a ) h u ⊗ v ⊗ w ( α ( , ( v ⊗ w ) op )( a )) h u ⊗ v ( α ( , v op )( a )) σ u ⊗ v,w σ u,v ⊗ w σ u,v (4.17)for every u, v, w in V . This diagram is induced by the commuting triangle 2.2.Moreover, naturality of σ implies that the following diagram commutes h u ( a ) h u ⊗ v ( α ( , v op )( a )) h u ⊗ w ( α ( , w op )( a )) h u ⊗ v ( α ( , w op )( a )) h u ⊗ m (id) σ u,v σ u,w h u ⊗ v ( α ( , m op )( a )) (4.18)for any morphism m : v → w in V .Similarly, a homology theory graded over V consists of a functor h : V × A → S such that for every u in V , the functor h u given by h u ( a ) = h ( , u )( a ) is a homologyfunctor, together with natural isomorphisms σ u,v : α ( , v op ) ◦ h u ⊗ v → h u satisfying similar conditions dual to the ones above.4.2. Some known examples of (co)homology theories.
Definitions of severalexisting cohomology and homology theories fit into the setting that described above,after choosing actions appropriately. Some examples of actions below are non-strict;however, they are particular cases of the setting above in view of Remark 2.1. Inthe examples below, a space means a compactly generated and weakly Hausdorfftopological space.4.2.1.
Generalized ordinary (co)homology theories.
Let N be the natural numbersconsidered as a monoidal category with identity morphisms as the only morphisms.Let A b denote the category of abelian groups. Consider A b with the trivial N -action. Let ho T be the homotopy category of pointed spaces with respect toQuillen model structure. Define the N -action Σ byΣ( n, m op )( X ) = X ∧ S m = Σ m X for n, m in N and X in ho T . Then a generalized (co)homology theory h in oursetting gives a generalized (co)homology theory satisfying the Eilenberg-Steenrod(co)homology axioms. CTIONS OF MONOIDAL CATEGORIES 17
Equivariant cohomology theories graded over representations.
Let G be acompact lie group and U be a complete G -universe; that is, a countably infinitedimensional orthogonal G -representation having non-zero G -fixed points and con-tains the direct sum V ⊕ λ for every finite dimensional representation V and everycardinal λ ≤ ℵ , see [22, Defn. 1.1.12] or [15, Ch. II, Defn. 1.1]. Let V bethe monoidal category RO ( G ; U ) whose objects are orthogonal G -representationsembeddable in U and whose morphisms are G -linear isometric isomorphisms andmonoidal product is the direct sum. For V an object in V , denote by S V the onepoint compactification of V . Let h RO ( G ; U ) be the quotient of RO ( G ; U ) withthe relation given by f ∼ g : V → W if the induced maps f ∗ ≃ s g ∗ : S V → S W are stably homotopic [17, see pp.130]. Let G T denote the category of pointed G -spaces and pointed G -maps with the standard model structure and ho G T beits homotopy category. Let A b be the category of abelian groups and 1 denote thetrivial V -action on A b . We define an action Σ on ho G T op as follows:Σ( V, W op )( X ) = Σ W X = X ∧ S W ;that is, the usual suspension action. For each V, W in V the suspension functorΣ W preserves homotopy colimits in G T , see [11]. Hence Σ induces an actionon cohml(ho G T , A b ). An RO ( G )-graded cohomology theory is an object in thestabilization of cohml(ho G T , A b ) with respect to the action [Σ , h, σ ) where h : h RO ( G ; U ) × ho G T op → A b is afunctor with h V = h ( V, − ) a cohomology functor for every V in h RO ( G ; U ), and σ V,W : h V → h W ⊕ V ◦ Σ W are natural isomorphisms. For each pair of representations W, Z , as above, we havethe following diagram commutes h V ⊕ W ⊕ Z (Σ ( W ⊕ Z ) X ) h V ( X ) h V ⊕ W (Σ W X ) σ V,W σ V ⊕ W,Z σ V,W ⊕ Z . If m : W → W ′ is a morphism in h RO ( G ; U ), then by naturality of σ we have thefollowing diagram commutes h V ⊕ W ′ (Σ W ′ X ) h V ⊕ W ′ (Σ W X ) h V ( X ) h V ⊕ W (Σ W X ) σ V,W h ( id ⊕ m ) σ V,W ′ h V ⊕ W ′ ( id ∧ S m ) This definition of cohomology theory is the same as the one given in [17, VII, Defn.1.1]. Dualizing the definition, one obtains RO ( G )-graded equivariant homologytheories.4.2.3. Equivariant cohomology theories graded over actions on spheres.
Let V bethe category whose objects are pointed spheres (with S − = ∗ ) with continuousbase-point preserving G -actions, morphisms are pointed G -isomorphisms betweenthem. It is a monoidal category with G -smash product with diagonal action and S − = ∗ as the unit. We denote an object in V by S mµ , where n is the dimension of the sphere and µ is the G -action. Note that RO ( G ; U ) faithfully embeds in V .Define Σ on ho G T op similar to the above; i.e., Σ ( S nξ , ( S mµ ) op )( X ) = Σ S mµ X = X ∧ S m ;with the diagonal action. Clearly, Σ induces an action on cohml(ho G T , A b ). An V -graded cohomology theory is an object in the stabilization of cohml(ho G T , A b )with respect to the action [ Σ, V -graded cohomology theory is a pair( h, σ ) where h : V × ho G T op → A b is a functor with h S nµ = h ( S nµ , − ) a cohomologyfunctor for every S nµ in V , and σ S mµ ,S nξ : h S nξ → h S mµ ∧ S nξ ◦ Σ S mµ are natural isomorphisms. The compatibility diagrams commute as above.4.2.4. Parameterized cohomology theories graded over vector bundles.
For the orig-inal definitions of parameterized reduced cohomology theories, see [18]. These the-ories are cohomology theories for the categories of ex-spaces. Let B be a space and T /B be the category of spaces over B . The terminal object in this category isthe identity of B . The category of based spaces over B , T B , is the under category T B = id B / ( T /B ). Objects of this category are often called ex-spaces of B , seee.g. [9]. Let ho T B be the homotopy category of T B with respect to the modelstructure of [18, Ch. 6]. Ordinary parameterized cohomology theories takes valuesfrom ho T B and graded over integers in [18]. On the other hand, this categoryadmits other obvious suspensions than the ordinary one.Let V be the monoidal category whose objects are real vector bundles over B with fiberwise inner products (i.e., taking values in B × R the trivial R -bundle over B ), fiberwise linear isometric embeddings as morphisms, Whitney sum ⊕ B as themonoidal product and 0-bundle as the monoidal unit. For ξ in V , denote by S ξ theassociated fiber-wise one point compactification, which is a sphere bundle over B with point as the section induced by the zero section of ξ . Given bundles ξ, η in V ,define Σ B by Σ B ( ξ, η op )( t : X → B ) = t ∧ B S η , see [18, Definition 1.3.3] for ∧ B . The construction Σ B defines an action on ho T B .Moreover, for every ξ, η in V Σ B ( ξ, η op ) preserves homotopy colimits in T B . Con-sider A b with the trivial V -action 1. Then the V -action [Σ B ,
1] on [ho T op B , A b ] in-duces a V -action on cohml(ho T B , A b ). A parameterized cohomology theory gradedover V is an object Stab V (cohml(ho T B , A b )). Since V is symmetric monoidal, thecategory Stab V (cohml(ho T B , A b )) is equivalent to Inv l (cohml(ho T B , S )). Then aparameterized cohomology theory graded over V is a pair ( h, σ ) where h : V → cohml(ho T B , A b ) is a functor and σ is the associated desuspension. More pre-cisely, a parameterized cohomology theory graded over V consists of a functor h : V × ho T op B → A b such that for every ξ in V h ξ = h ( )( ξ ) is a cohomologyfunctor in ho T B , together with natural isomorphisms σ ξ,η : h ξ → h ξ ⊕ B η ◦ Σ ξB CTIONS OF MONOIDAL CATEGORIES 19 for every ξ, η in V . Moreover, for every object τ : X → B in ho T B the followingdiagram commutes h ξ ⊕ B η ⊕ B ζ (Σ ( η ⊕ B ζ ) B τ ) h ξ ( τ ) h ξ ⊕ B η (Σ ηB τ ) σ ξ,η σ ξ ⊕ Bη,ζ σ ξ,η ⊕ Bζ . If f : η → η ′ is a morphism in V , then we have a commutative diagram as follows: h ξ ⊕ B η ′ (Σ η ′ B τ ) h ξ ⊕ B η ′ (Σ ηB τ ) h ξ ( τ ) h ξ ⊕ B η (Σ ηB τ ) σ ξ,η h ( id ⊕ B f ) σ ξ,η ′ h ξ ⊕ Bη ′ ( id ∧ B S f ) since σ is natural transformation. Passing to isomorphism classes in V , one ob-tains KO ( B )-graded parameterized cohomology theories (for a slightly differentdefinition see [21, Sec.1]).One can also pass to bundles fibrewise embeddable in a universe bundle forconvenience. Let u be a vector bundle in V , such that for every finite dimensionalvector bundle ξ in V and for every λ ≤ ℵ , there is a monomorphism ξ ⊗ B λ → u ofvector bundles. Let V u be the subcategory of V whose objects are such ξ ’s and whosemorphisms are isomorphisms in V between them. Denote by Σ B u the restrictionof the action Σ B . Then, the associated stabilization of cohomology functors withrespect to the action V u -action [Σ B u ,
1] gives the parametrized cohomology theoriesindexed by a universe.One can go further and grade parameterized cohomology theories over the sym-metric monoidal category V of sphere bundles over B admitting sections, withfiberwise isomorphisms of pointed bundle maps as morphisms. More precisely, ob-jects of V are pairs ( ξ, s ) where ξ : E → B is a sphere bundle and s : B → E is asection. The monoidal product is given by the fiber-wise smash product on the firstcoordinate and the unique section of the pushout in [18, Definition 1.3.4] on thesecond coordinate. The action is defined similarly; that is, given objects ( ξ, s ) , ( η, t )in V , Σ B is given by Σ B (( ξ, s ) , ( η, t ) op )( τ : X → B ) = τ ∧ B η. Then Σ B induces an action on cohml(ho T B , A b ) and an object in the stabilizationof cohml(ho T B , A b ) with respect to Σ B defines a parameterized cohomology theorygraded over sphere bundles admitting sections. The properties enjoyed by such atheory can be obtained similar to the one described above.5. (Co)stabilizations of relative categories with respect to V -actions The category
RelCat of relative categories and relative functors between themadmits a model structure due to Barwick-Kan [1, 6.1]. Besides,
RelCat is a carte-sian closed symmetric monoidal category [1, 7.1]. It is in particular, a 2-category(
Cat -enriched) and powered and copowered over any small category. If I is a smallcategory and A = ( A, W A ) is a relative category, then the powering ⋔ ( I, A ) is given by ( A I , W IA ), where W IA denotes the pointwise weak equivalences. Similarly,the copowering I ⊠ A is given by ( I × A, I × W A ).5.1. Stabilization of relative categories with respect to V -actions. Let α be a V -action on A = ( A, W A ) that is mute on the right (i.e., a left action). Then,by Theorem 3.5, Stab V ( A α ) is equivalent to Inv( A α ). If α is mute on the right,then we have also an equivalence Inv( A α ) ∼ = Inv l ( A α ). The objects of Inv l ( A α ) arepairs ( E, σ ) where E : V → A : u E u is a relative functor (with trivial relativestructure on V ) and for every v in V σ ( v ) : E ⇒ ⋔ (¨ µ r , α )( v, op )( E )is a natural isomorphism. Here, we note that ⋔ (¨ µ r , α )( , u op )( E ) = E as α ismute on the right. Write α v = α ( v, op ) and σ ( v ) u = σ u,v (while noting that α ( v, op ) = α ( v, w op ) for every w in V ). Then, for every u, v in V , σ defines anisomorphism σ u,v : E u → α v E u ⊗ v . The diagram 2.2 implies that for every w in V the following triangle commutes E u α v E u ⊗ v α v ⊗ w E u ⊗ v ⊗ wσ u,v σ u,v ⊗ w (5.19)where the bottom horizontal map is the composition α v E u ⊗ v α v ( σ u ⊗ v,w ) −→ α v α w E u ⊗ v ⊗ w ∼ = −→ α v ⊗ w E u ⊗ v ⊗ w Morphisms are natural transformations that are coherent with the V -action, andweak equivalences are defined levelwise. This category admits a right action ˜ α givenby ˜ α ( u, v op )( f, σ ) = ( f ◦ ¨ µ r ( u ) , ˜ σ ( u )) where ˜ σ ( u ) v,w = σ ( u ⊗ v ) w .On the other hand, the lax stabilization of A with respect to α is the same exceptthat σ in the above construction is not required to be an isomorphism but just anatural transformation. In the case when the underlying relative category A is amodel category and V acts by left Quillen functors, the lax stabilization coincideswith the lax homotopy limit in [2]; and therefore, admits a model structure whereweak equivalences and cofibrations defined levelwise.One equivalently uses Inv( A α ) for the stabilization, which has a natural leftaction (without using V is symmetric monoidal). In this case, objects can be viewedas pairs ( E, ς ) where E : V × V → A and is a functor and ς : E → ⋔ (¨ µ r × ¨ µ l , α )( E )is a natural isomorphism; so that, for every u, v, w, z in V , ς u,v ; w,z : E ( u, v ) → α ( w, z op ) E ( u ⊗ w, z ⊗ v ) is a natural isomorphism. Writing E u ⊖ v = E ( u, v ) and α w ⊖ z = α ( w, z op ), we can see that Inv( A α ) just extends the indexing in Inv l ( A α )to formal differences in V . Again, in the lax version, the assumption on ς beinginvertible is dropped.In some classical cases, the lax stabilization given in the present paper coincideswith the notion of prespectra in the literature while stabilization coincides withspectra.5.1.1. A special case: V is a monoidal groupoid acting on a model category. Con-sider the special case when the symmetric monoidal category V is a groupoid; thatis, the 2-category B V is a (2 , α is an action on a relative category A given by a 2-functor from B W to the 2-truncation of the ( ∞ , CTIONS OF MONOIDAL CATEGORIES 21 model category
RelCat with respect to the Barwick-Kan model structure. ThenInv l ( A α ), as given by a 2-end, coincides with the homotopy end (i.e., (2 , A is a model category and V acts on A by leftQuillen functors, then following [2] one obtains a construction of Inv l ( A α ) in thiscase same as above. Objects of Inv l ( A α ) are pairs ( E, σ ) where E : V → A : u → E u is a functor and σ ( v ) : E ⇒ ⋔ (¨ µ r , α )( v, op )( E )is a natural weak equivalence for every v in V ; i.e., σ u,v is a weak equivalence.The diagram 5.19 commutes, which corresponds to the compatibility conditionmentioned in [2, Defn. 3.1].5.1.2. A construction of spectrification.
For the case of the actions on relative cat-egories, we can construct spectrification as in [5] or [12]. Let V be symmetricmonoidal and α be a V -action of a relative category A . Assume that α is muteon the right. Recall that spectrification is left adjoint to the natural inclusion ι α : Inv s ( A α ) → ℓ Inv s ( A α ). Let T( V ) denote the transport category of V ; i.e., cat-egory whose objects are objects of V and a morphism u → v in T( V ) is a pair ( w, f )where w is an object and f : u ⊗ w → v is an isomorphism in V . The composition of( w, f ) and ( z, g ) is ( z ⊗ w, g ◦ f ⊗ id w ). Suppose that α is an action that commuteswith T( V ) shaped colimits. Then the inclusion ι α admits a left adjoint L given asfollows: For ( E, σ ) an object in ℓ Inv s ( A α ), let F α,E,z : T( V ) → A be the functorgiven by F α,E,z ( u ) = α ( u, E ( z ⊗ u ) and F α,E,z (( w, f )) is the composition α ( u, E ( z ⊗ u ) u · σ −→ α ( u ⊗ w, E ( z ⊗ u ⊗ w ) f ∗ ∼ = α ( v, E ( z ⊗ v )Then LE : V → A is the functor given by LE ( z ) = colim T( V ) F α,E,z We call the left adjoint of this inclusion spectrification . The x component of σ isgiven by the following composition α ( x, LE ( z ⊗ x )) = α ( x, u ∈ T( V ) α ( u, E ( z ⊗ x ⊗ u ))= colim u ∈ T( V ) α ( x ⊗ u, E ( z ⊗ x ⊗ u ) ∼ = colim u ∈ T( V ) α ( u, E ( z ⊗ u ) = LE ( z )so that LE is an 0-cell in V M . The .... follows from the naturality of colimit. The1-cell L is V -equivariant via LE ( x ⊗ − ) ∼ = L ( E ( x ⊗ − )).5.2. Some known examples of stabilizations of relative categories.
Sequential and Symmetric Spectra.
Let V = N with addition and identitymaps as morphisms, and let A = T be the category of pointed spaces with thestandard model structure. Let Ω : B W →
RelCat be the action given by usualloop space functors; that is, Ω( n, m )( X ) = Ω n X , the n -fold loop space of X .This action is in fact the action generated by the functor Ω : T → T . ThenStab V ( T ) has objects as pairs E : N → T : n E n functors together with(natural) isomorphisms σ m : E n → Ω m E n + m . The triangular diagram 5.19 abovealready commutes in this case. Such objects are known as spectra in the literature.If we consider the lax stabilization ℓ Stab V ( T ), then σ m is just a map. Objectsof ℓ Stab V ( T ) are then called prespectra. The model structure on ℓ Stab V ( T )coincides with the level model structure and the spectrification functor in 5.1.2 is the usual spectrification of prespectra. In this case, since V is a monoidal groupoid,the construction in 5.1.1 gives the category of Ω-spectra.If instead we choose V as the permutation category; whose objects are finite sets[ n ] = { , . . . , n } and whose morphisms are permutations (i.e., V is the skeleton ofthe groupoid of finite sets and permutations), and cardinal sum as the monoidalproduct and [0] = ∅ as the monoidal unit. Defining Ω([ n ] , [ m ])( X ) = Ω n X on T one obtains that ℓ Stab V ( T ) has objects of pairs E : N → T : [ n ] E n functorstogether with maps σ n,m : E n → Ω m E n + m . Since V ([ n ] , [ n ]) ∼ = S n ; the symmetricgroup, each E n admits an action of S n . By the naturality of σ , the map is σ n,m is S n × S m -equivariant where S n × S m action on E n + m the action is the restrictedaction of S n + m and on Ω m E n + m the S m -action is the conjugation action and on E n , the second coordinate of S n × S m acts trivially. This implies the S n × S m -equivariance of the dual maps σ ∗ n,m : Σ m E n → E n + m . The objects of ℓ Stab V ( T )are symmetric spectra [8].5.2.2. Coordinate Free and orthogonal Spectra.
Let I nn be the monoidal categoryreal inner product spaces with direct sum and linear isometric embeddings. Foran inner product space V , let S V denote its one point compactification. Define a I nn -action Ω on the relative category T by Ω( V, op )( X ) = Ω V X = T ( S V , X ).Then Stab I nn ( T ) is equivalent to the category with objects as pairs ( E, σ ) where E : I nn → T : V E V is a functor and σ V,W : E V → Ω W E V ⊕ W is a naturalisomorphism. Moreover, for any Z ∈ V the diagram E V Ω W ⊕ Z E V ⊕ W ⊕ Z Ω W E V ⊕ W Ω W Ω Z E V ⊕ W ⊕ Z Ω W ( σ V ⊕ W,Z ) σ V,W ⊕ Z σ V,W ∼ = commutes. Note that this diagram is the same as the diagram 5.19 after composingbottom arrow with inverse of the right vertical isomorphism.Such objects are closely related to the objects known as coordinate free spectra inthe literature. Let U = R ∞ be a countably infinite dimensional real inner productspace. Let V be the category whose objects are finite dimensional subspaces of U and morphisms are linear isometric isomorphisms. Again, define Ω on T byΩ(1 , V )( X ) = Ω V X = T ( S V , X ). Then the stabilization Stab V ( T ) is equivalentto the category of coordinate free spectra. In fact, if V ⊂ W by writing W − V for the orthogonal complement of V in W , we obtain structure maps of the form σ V,W − V : E V → Ω W − V E W . Similarly, the lax stabilization ℓ Stab V ( T ) gives thecoordinate free prespectra, which admits the level model structure in which weakequivalences and cofibrations defined levelwise. Since V is a monoidal groupoid,the construction in 5.1.1 gives the category of coordinate free Ω-spectra.Note that for each V in V , the hom-set V ( V, V ) ∼ = O ( V ), the orthogonal groupon V . Let O be the full-subcategory of V consisting of objects R ⊕ n for each n ∈ N .Then O is monoidal with unit R = 0 and the symmetric monoidal product givenas R ⊕ k ⊕ R ⊕ l = R ⊕ k ⊕ R ⊕ l = R ⊕ ( l + k ) , for every k, l ∈ N . On morphisms, theproduct is defined via the diagonal embedding O ( k ) × O ( l ) → O ( k + l ). Considerthe O -action Ω on T by Ω( R , op )( X ) = Ω X = T ( S , X ) and Ω( R ⊕ n , op )( X ) =Ω n X ∼ = T ( S n , X ) , where S n denotes the one point compactification of R ⊕ n . A CTIONS OF MONOIDAL CATEGORIES 23 morphism m : R ⊕ n → R ⊕ n is sent to the induced self map on loop spaces. In thiscase, an object Stab O ( T ) is a pair ( E, σ ) where E : O → T : R n E n is a functorand a natural isomorphism σ n,m : E n → Ω m E n + m . Since O ( R ⊕ n , R ⊕ n ) ∼ = O ( n ), E n admits an O ( n )-action. By naturality of σ n,m in n , for every element g : R ⊕ n → R ⊕ n in O ( n ), we have the following diagram commutes E n Ω m E n + m E n Ω m E n + mσ n,m g ∗ g ∗ σ n,m Therefore, σ n,m is O ( n )-equivariant with respect to the restricted action on codomainsvia the diagonal embedding. For any h : R ⊕ m → R ⊕ m , there is an induced map ofone point compactifications h S : S m → S m . Thus, there is an induced conjugationaction O ( m )-action on Ω m X = T ( S m , X ) for every X . The space E n + m admitsa O ( m + n ) action, and thus; an O ( m )-action after restriction, so that there is an O ( m )-action on Ω m E n + m by conjugation. By 2.1, the following diagram commutes E n Ω m E n + m E n Ω m E n + mσ n,m h ∗ S h ∗ = id σ n,m Thus, σ n,m is O ( n ) × O ( m )-equivariant. This gives the notion of the orthogonalspectra. In fact, the action Ω induces an adjoint action Σ on T , for which Σ(0 , n ) =Σ n is the left adjoint to Ω n ; namely, n -fold suspension functor. The adjunct of σ n,m gives a map σ ′ n,m : Σ m E n → E n + m . This map is O ( k ) × O ( l ) equivariant since σ ′ n,m is.Generalizing the idea we can find another model for the highly structured spectraby replacing O with G ; the groupoid whose objects are S n , n -sphere with a basepointfor all n ∈ N , whose morphisms are G n the group of self homeomorphisms of S n .This is topological version of the orthogonal spectra.5.2.3. Genuine G -Spectra. Now, let G be a group and G I nn be the monoidal cat-egory of G -representations with direct sum and G -equivariant linear maps. Con-sider the category G T of pointed G -spaces and pointed G -maps with the stan-dard model structure. Define a G I nn -action Ω : G I nn × G I nn op → RelCat on G T by Ω( V, op )( X ) = Ω V X = G T ∗ ( S V , X ), the set of pointed maps fromthe one point compactification of V to X considered with the conjugation action.Then Stab V ( G T ) is equivalent to the category with objects as pairs ( E, σ ) where E : G I nn → G T : V E V is a functor and σ V,W : E V → Ω W E V ⊕ W is a naturalisomorphism. The diagram analogues to 5.19 commutes.For convenience one considers the more classical notion of equivariant spectraindexed by a G -universe. Let V be as in 4.2.2. Define a V -action Ω on G T as above.Then Stab V ( G T ) is the category of genuine G -spectra and ℓ Stab V ( G T ) gives thecategory of genuine orthogonal G -prespectra, see [17, Ch. XII, 2]. Here, we notethat V ( V, V ) = O ( V ), so that for each object ( E, σ ) in ℓ Stab V ( G T ), E ( V ) admitsa O ( V )-action and σ V,W is O ( V ) × O ( W )-equivariant analogous to the coordinate free case above. The construction analogous to the one in 5.1.1 gives the categoryof G -Ω-spectra.5.2.4. Parameterized Spectra.
Let B be a space and T B be the model category givenin [18, Ch. 6]. Let V be the monoidal category as given in 4.2.4. Consider the modelstructure on T B [18, Ch. 6]. For ξ in V , denote by S ξ the associated fiber-wise onepoint compactification, which is a sphere bundle over B with point as the sectioninduced by the zero section of ξ . Define the V -action Ω B : B W →
RelCat on T B as follows: Given ξ, η in V , define Ω B byΩ B ( ξ, η op )( t : X → B ) = T B ( S ξ , t ) . Denote by Ω ξB the functor T B ( S ξ , − ). Then Stab V ( T B ) has objects as pairs ( ̥ , σ )where ̥ : V → T B : ξ ̥ ξ is a functor and σ ξ,η : ̥ ξ → Ω ηB ̥ ξ ⊕ B η such that forany ζ ∈ V the following diagram commutes ̥ ξ Ω ζ ⊕ B ηB ̥ ξ ⊕ B η ⊕ B ζ Ω ηB ̥ ξ ⊕ B η Ω ηB Ω ζB ̥ ξ ⊕ B η ⊕ B ζ Ω ηB ( σ ζ ) σ η ⊕ Bζ σ η ∼ = Such objects are called as genuine spectra parameterized by B . The lax stabilization ℓ Stab V ( T B ) then gives the category of genuine prespectra parameterized by B . Thecategory ℓ Stab V ( T B ) admits the level model structure as in the previous examples.5.2.5. Parameterized G -Spectra. Parameterized G -spectra and prespectra can beobtained just like the transition from coordinate free spectra to genuine G -spectra.5.2.6. Diagram Spectra.
Let T be a closed symmetric monoidal model category withinternal hom hom T and D be a T -enriched small symmetric monoidal category.Assume both T and D are pointed as categories. Let D T be the category of T -enriched functors from D to T . The category D T admits a symmetric monoidalproduct with Day convolution. Let R be any monoid in the symmetric monoidalcategory D T . Define a D action on Ω on T as Ω ( u, v op )( X ) = Ω u X = hom T ( R ( u ) , X ) . Since R is a monoid, the action is well-defined. The lax stabilization with respectto Ω , ℓ Stab D ( T ), has objects as pairs ( E, ς ) where E : D → T : u → E u and ς v : E v → Ω u E v ⊗ u is a natural transformation. If Ω factors through the categoryof adjunctions; i.e., there exist an adjoint action Σ on T , then an object in the laxstabilization coincides with the notion of D -spectra of [16].5.3. Costabilization of relative categories with respect to V -actions. Let α : B W →
RelCat be a V -action on A = ( A, W A ) that is mute on the left.The colax costabilization of A with respect to α , denoted by ℓ coStab V ( A ), canbe defined as the oplax coend of α , which can be seen as a oplax colimit over V .This oplax colimit can be given in terms of Gr¨othendieck construction. Since α ismute on the left, this 2-colimit can be given as ℓ coStab V ( A ) = ´ V (¨ µ r ⊠ α ) ◦ ι r .Then objects of ℓ coStab V ( A ) are pairs ( u, a ) with ( u ∈ V and a ∈ M , and amorphism between ( u, a ) → ( u ′ , a ′ ) is a triple ( v, ϕ, f ) where v ∈ V and ( ϕ, f ) :( u ⊗ v, α ( v, op )( a )) → ( u ′ , a ′ ). A triple ( v, ϕ, f ) is a weak equivalence if ( ϕ, f ) : CTIONS OF MONOIDAL CATEGORIES 25 ( u ⊗ v, α ( v, op )( a )) → ( u ′ , a ′ ) is a weak equivalence; that is, u ⊗ v ∼ = u ′ in V and f : α ( v, op )( a ) → a ′ is a weak equivalence in M . In particular, for every v in V ,( u, a ) is weakly equivalent to ( u ⊗ v, α ( v, op )( a )).In sufficiently nice cases, if A is a model category ℓ coStab V ( A ) admits a modelstructure; e.g., when V is bicomplete (so a model category with trivial model struc-ture), and α is relative and proper functor, see [7]. The construction coincideswith the homotopy colimit; and thus, homotopically correct with respect to theBarwick-Kan model structure (see [7] and also [19]).The costabilization is obtained by localizing ℓ coStab V ( A ) at the oplax cartesianmorphisms in the Gr¨othendieck construction.5.4. Some examples of (colax) costabilizations of relative categories.
Spanier Whitehead Category.
Let V = N with only identity maps as mor-phisms and A = T pointed topological spaces with with standard model struc-ture and Σ : B W →
RelCat be the action given by usual suspensions; thatis, Σ( n, m )( X ) = Σ n X the n -fold suspension of X . Then ℓ coStab V ( T ) has ob-jects as pairs ( n, X ) and for n ≤ m a morphism from ( n, X ) to ( m, Y ) is a map f : Σ m − n X → Y . Its homotopy category with respect to the weak equivalencesdescribed above gives the usual Spanier-Whitehead category (see also [19]).5.4.2. Coordinate Free Spanier-Whitehead Category.
Let V be as in the case ofthe coordinate free spectra and Σ : B W →
RelCat be the action given byΣ(
V, W )( X ) = Σ W X := S W ∧ X where ∧ denotes the smash product of spaces.Then ℓ coStab V ( T ) has objects as pairs ( W, X ) and for dim( W ) ≤ dim( Z ) a mor-phism from ( W, X ) to (
Z, Y ) is a triple (
V, φ, f ) where φ : V ⊕ W → Z is anisometric isomorphism and f : Σ W X → Y is a map. Such a triple ( V, φ, f ) is aweak equivalence if f is a weak equivalence. Then, following the homotopy category ℓ coStab V ( T ) with these weak equivalences gives the coordinate free version of theSpanier-Whitehead category.5.4.3. G -Equivariant Spanier-Whitehead Category. This is the G -equivariant ver-sion of coordinate free Spanier-Whitehead Category where V is chosen as in 5.2.3and Σ : B W →
RelCat be the action on G T given by Σ( V, W )( X ) = Σ W X := T ( S W , X ) where X is a G space and S W is the one point compactification of W with the induced G -action. Then ℓ coStab V ( T ) has objects as pairs ( W, X )and for dim( W ) ≤ dim( Z ) a morphism from ( W, X ) to (
Z, Y ) is a triple (
V, φ, f )where φ : V ⊕ W → Z is an equivalence class of a isometric G -isomorphisms and f : Σ W X → Y is a G -map.5.4.4. Parameterized Spanier-Whitehead Category.
This is the parameterized ver-sion of coordinate free Spanier-Whitehead category where V is chosen as in 5.2.4.Let Σ B : B W →
RelCat be the V -action on T B given by Σ B ( ζ, ξ )( t ) = Σ ξ t := S ξ ∧ B t where t : X → B be a pointed map over B (i.e., a pointed object in theover category T /B ) and S ξ is the sphere bundle over B given by the fiber-wise onepoint compactification of ξ , and ∧ B is the fiber-wise smash product. In this caseobjects are pairs ( ξ, t ) where ξ is an object in V and t : X → B a pointed map. Amorphism from ( ξ, t ) to ( ζ, z ) is a triple ( χ, m, f ) where χ is an object in V and m : ξ ⊕ χ → ζ a morphism in V and f : S χ ∧ B t → z is a pointed map over B . Equivariant Parameterized Spanier-Whitehead Category.
The description ofEquivariant Parameterized Spanier-Whitehead Category just the equivariant gen-eralization of the Parameterized Spanier-Whitehead Category above, which is thecostabilization of G T B = id B / ( G T /B ) with respect to the action given by thefibrewise suspensions as above, but equipped with G -actions.5.4.6. Diagram Spanier-Whitehead Category.
Let T , D and R be as in 5.2.6. Definea D action on Σ on T as Σ ( u, v op )( X ) := Σ v X := X ∧ T R ( u ) . Then ℓ coStab D ( T ) has objects as pairs ( v, X ) where v in D and X in T . A morphismfrom ( w, X ) to ( v, Y ) is a triple ( u, φ, f ) where φ : u ∧ D w → v is a morphisms in D and f : X ∧ T R ( u ) → Y is a morphism in T . Weak equivalences are the triplesin which φ is an isomorphism and f is a weak equivalence. Remark 5.1.
Observe that the duality between lax stabilization and oplax costa-bilization establishes a duality between stable homotopy categories and Spanier-Whitehead categories.
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