Ambidexterity §4

Home , Diagonal functor, Diagonal morphism, Kan extension

Ambidexterity §4

Peter J. Haine November 12, 2018

1 Motivations

1.1 Notation. We write Spc for the ∞-category of spaces or ∞-groupoids, Cat∞ for the ∞-category of ∞-categories, Sp for the ∞-category of spectra, and Sp퐾(푛) for the ∞-category of 퐾(푛)-local spectra (where 퐾(푛) is a Morava 퐾-theory).

Recall that the goal of [1] is to show that for any 휋-finite space 푋, prime number 푝, and Morava 퐾-theory 퐾(푛) (at 푝), the functors

colim푋, lim푋 ∶ Fun(푋, Sp퐾(푛)) → Sp퐾(푛) are (canonically) equivalent. If we let 푓∶ 푋 → ∗ denote the unique morphism, then:

• The diagonal functor Sp퐾(푛) → Fun(푋, Sp퐾(푛)) is identified with the functor ⋆ 푓 ∶ Fun(∗, Sp퐾(푛)) → Fun(푋, Sp퐾(푛)) given by precomposition with 푓. ⋆ • The functor colim푋 is identified with the left adjoint 푓! of 푓 , given by left Kan extension along 푓.

⋆ • The functor lim푋 is identified with the right adjoint 푓⋆ of 푓 , given by right Kan extension along 푓. Rephrasing our problem, we’re interested in studying the assignment

푓!

⊥ 푓⋆ 푋 ↦ Fun(∗, Sp퐾(푛)) Fun(푋, Sp퐾) . ⊥

푓⋆ and determining when there is a natural equivalence 푓! ⥲ 푓⋆.

(1) To encode all the relevant functoriality in 푋, we should adopt the relative point of view and determine which morphisms 푓∶ 푋 → 푌 in Spc have the property that 푓! ⥲ 푓⋆.

1 (2) For setting up the general theory, it is not really relevant that we are working with local systems of 퐾(푛)-local spectra; we might as well consider any functor that assigns ⋆ a morphism 푓∶ 푋 → 푌 in Spc a chain of three adjunctions 푓! ⊣ 푓 ⊣ 푓⋆. (3) The perspective we’ve taken is to assume that to any morphism 푓∶ 푋 → 푌 in ⋆ Spc, we have three adjoints 푓! ⊣ 푓 ⊣ 푓⋆, and to try to construct an equivalence Nm푓 ∶ 푓! ⥲ 푓⋆. We can equivalently not assume the existence of the extreme right adjoint 푓⋆ (or, alternatively, the extreme left adjoint 푓!), and try to exhibit 푓! as a right adjoint to 푓⋆. That is, we might as well restrict our attention to functors ladj ladj Spc → Cat∞ , where Cat∞ is the ∞-category of ∞-categories and left adjoint functors (see Notation 2.1).

ladj (4) The fact that we’re considering functors to Cat∞ with source the ∞-category of spaces isn’t particularly relevant for setting up the general theory; we can replace the ∞-category of spaces with an essentially arbitrary index ∞-category 푿.

There is one non-obvious fact particular to our situation that is relevant to the general theory (Proposition 1.5). First we recall the colimit formula for left Kan extensions.

1.2 Recollection (comma ∞-categories). The comma ∞-category 푋 ↓푌 푍 associated to two functors 푓∶ 푋 → 푌 and 푔∶ 푍 → 푌 is the universal ∞-category fitting into a lax-commutative diagram

푋 ↓푌 푍 푋 푓 ⟸

푍 푔 푌 .

Explicitly, 푋 ↓푌 푍 can be computed as the iterated pullback

1 (1.3) 푋 ↓푌 푍 ≔ 푋 ×Fun({0},푌) Fun(훥 , 푌) ×Fun({1},푌) 푍 . Notice that since all morphisms are invertible in an ∞-groupoid, if 푌 is an ∞- groupoid, then the comma ∞-category 푋 ↓푌 푍 is simply given by the pullback

푋 ↓푌 푍 ≔ 푋 ×푌 푍 . (This can also be seen from the formal description (1.3) by noting that an ∞-category 푌 is an ∞-groupoid if and only if evaluation at 0 or 1 defines an equivalence of ∞- categories Fun(훥1, 푌) ⥲ 푌.) 1.4 Recollection (colimit formula for left Kan extensions). Let 퐶 be an ∞-category with colimits and let 퐿푋 ∶ 푋 → 퐶 and 푓∶ 푋 → 푌 be functors. Then for each object 푦 ∈ 푌, the value of the left Kan extension Lan푓 퐿푋 of 퐿푋 along 푓 at 푦 is given by the colimit

퐿푋 (Lan푓 퐿푋)(푦) ≃ colim ( 푋 ↓푌 {푦} 푋 퐶 ) , where 푋 ↓푌 {푦} is the comma ∞-category.

2 1.5 Proposition ([1, Proposition 4.3.3]). Let 퐶 be an ∞-category with colimits and

̄ ′ 푓 ′ 푋 ⌟ 푌 (1.6) ̄푔 푔 푋 푌 푓 a pullback square in Spc. Then the associated square

푓̄ Fun(푋′, 퐶) ! Fun(푌′, 퐶)

⋆̄푔 푔⋆ Fun(푋, 퐶) Fun(푌, 퐶) 푓! commutes. Proof. Note that by the universal property of the left Kan extension, for any local system 퐿푋 ∶ 푋 → 퐶 we have a natural transformation

휃∶ Lan푓(퐿̄ 푋 ∘ ̄푔)→ (Lan푓 퐿푋) ∘ 푔 ,

̄ ⋆ ⋆ i.e., a natural transformation 푓! ̄푔 (퐿푋) → 푔 푓!(퐿푋). We want to show that for each 푦′ ∈ 푌′, the morphism

′ ′ ′ 휃(푦 )∶ Lan푓(퐿̄ 푋 ∘ ̄푔)(푦 ) → (Lan푓 퐿푋)(푔(푦 )) is an equivalence. Since 퐶 has all colimits, by the pointwise formula for left Kan extensions (Recollection 1.4), for all 푦′ ∈ 푌′ we have

′ ′ ′ ′ ̄푔 퐿푋 Lan푓(퐿̄ 푋 ∘ ̄푔)(푦 ) ≃ colim ( 푋 ↓푌′ {푦 } 푋 푋 퐶 ) .

Since 푌′ is an ∞-groupoid,

′ ′ ′ ′ 푋 ↓푌′ {푦 } ≃ 푋 ×푌′ {푦 } (Recollection 1.2). Since the square (1.6) is a pullback square and 푌 is an ∞-groupoid, we see that ′ ′ ′ ′ 푋 ×푌′ {푦 } ≃ 푋 ×푌 {푔(푦 )} ≃ 푋 ↓푌 {푔(푦 )} (Recollection 1.2). Hence

′ ′ 퐿푋 Lan푓(퐿̄ 푋 ∘ ̄푔)(푦 ) ≃ colim ( 푋 ↓푌 {푔(푦 )} 푋 퐶 )

′ ≃ Lan푓(퐿푋)(푔(푦 )) .

Moreover, this equivalence is induced by 휃.

3 1.7 Remark. For an ∞-category 퐶 with colimits, there is a version of Proposition 1.5 involving the (−)⋆ right adjoints that says that given the pullback square (1.6) in Spc there’s a natural equivalence ⋆ ⋆̄ 푓 푔⋆ ⥲⋆ ̄푔 푓 of functors Fun(푌′, 퐶) → Fun(푋, 퐶). The proof is the same as Proposition 1.5; one just uses the limit formula for right Kan extensions. ⋆ ̄ ⋆ The identification 푔 푓! ≃ 푓! ̄푔 in Proposition 1.5 is a Beck–Chevalley condition, which we examine in the next section.

2 Beck–Chevalley morphisms

ladj 2.1 Notation. Wewrite Cat∞ ⊂ Cat∞ for the subcategory with objects any ∞-category and morphisms functors 퐶 → 퐷 which are left adjoints. We usually write 푓! ∶ 퐶 → 퐷 for a left adjoint and 푓⋆ ∶ 퐷 → 퐶 for its corresponding right adjoint. In this case, we write ⋆ ⋆ 휂푓 ∶ id퐶 → 푓 푓! and 휀푓 ∶ 푓!푓 → id퐷 ⋆ for the unit and counit of the adjunction 푓! ⊣ 푓 , respectively. 2.2 Definition. Consider a commutative square 휎

푓̄ 퐶′ ! 퐷′

(2.3) !̄푔 푔! 퐶 퐷 푓!

ladj in Cat∞ . The Beck–Chevalley morphism associated to the square 휎 is the composite natural transformation

⋆ ⋆ 푓̄ ̄푔 휂 휀 ̄푔 푓 ̄ ⋆ ! 푓 ̄ ⋆ ⋆ ∼ ̄ ⋆ ⋆̄ 푓 ! ⋆ BC(휎)∶ 푓! ̄푔 푓! ̄푔 푓 푓! 푓!푔 푓 푓! 푔 푓! , where the middle equivalence comes from the identification of right adjoints

⋆̄푔 푓⋆ ⥲ 푔⋆푓⋆̄ .

The Beck–Chevalley morphism is depicted diagrammatically as

푓̄ 퐶′ ! 퐷′

⟹ ⋆̄푔 푔⋆ BC(휎) 퐶 퐷 . 푓!

We say that the square (2.3) satisfies the Beck–Chevalley condition if the Beck– ̄ ⋆ ⋆ Chevalley morphism BC(휎)∶ 푓! ̄푔 → 푔 푓! is an equivalence.

4 2.4 Remark. As in Remark 1.7, there is a Beck–Chevalley morphism for the (−)⋆ right adjoints: it is a natural transformation

⋆ ⋆̄ (2.5) 푓 푔⋆ →⋆ ̄푔 푓 .

This is the basechange morphism that one often sees in algebraic geometry, for example, in the smooth and proper basechange theorems for étale cohomology (see [3, Chapter vi Corollary 2.3 & Theorem 4.1]). An alternative approach to ambidexterity is to dispose of the (−)! adjoints and in- stead work with the (−)⋆ adjoints and the other Beck–Chevalley morphism (2.5). 2.6 Example. By giving a more careful proof of Proposition 1.5, we can conclude that for any ∞-category 퐶 with colimits and pullback square

̄ ′ 푓 ′ 푋 ⌟ 푌 ̄푔 푔 푋 푌 푓 in Spc, the induced square

푓̄ Fun(푋′, 퐶) ! Fun(푌′, 퐶)

!̄푔 푔! Fun(푋, 퐶) Fun(푌, 퐶) 푓! satisfies the Beck–Chevalley condition. (See also [2].) 2.7 Observation. Please observe that given commutative squares

퐶′ 퐷′ 퐸′

퐶 퐷 퐸 ,

ladj 휎 and 휏 (from left to right) in Cat∞ , the Beck–Chevalley morphism of the outer rectangle is equivalent to natural transformation given by the horizontal composite of the Beck–Chevalley morphisms

퐶′ 퐷′ 퐸′

⟹ ⟹ BC(휎) BC(휏) 퐶 퐷 퐸 .

5 Similarly, given commutative squares

퐶″ 퐷″

퐶′ 퐷′

퐶 퐷 ,

′ ladj 휎 and 휎 (from top to bottom) in Cat∞ , the Beck–Chevalley morphism of the outer rectangle is equivalent to natural transformation given by the vertical composite of the Beck–Chevalley morphisms 퐶″ 퐷″

⟹ BC(휎′)

퐶′ 퐷′

⟹ BC(휎)

퐶 퐷 . Given our generalizations and reformulations of our goal from §1, we are interested ladj in considering functors 푿 → Cat∞ sending pullback squares to squares satisfying the ⋆ Beck–Chevalley condition, and exhibiting the left adjoint 푓! of 푓 as a right adjoint of 푓⋆ for a certain class of morphisms 푓∶ 푋 → 푌 in 푿. ladj 2.8 Definition. Let 푿 be an ∞-category with pullbacks. A functor 퐶∶ 푿 → Cat∞ is a Beck–Chevalley functor if for every pullback square

̄ ′ 푓 ′ 푋 ⌟ 푌 ̄푔 푔 푋 푌 푓 in 푿, the induced square ̄ 푓! 퐶푋′ 퐶푌′

!̄푔 푔!

퐶푋 퐶푌 푓! ladj in Cat∞ satisfies the Beck–Chevalley condition. 2.9 Example. For any ∞-category 퐶 with colimits, the functor ladj Fun(−, 퐶)∶ Spc → Cat∞ , where the functoriality is in the (−)! adjoints, is a Beck–Chevalley functor.

6 ladj 2.10 Remark. Beck–Chevalley functors 푿 → Cat∞ are classified by functors 푞∶ 퐶 → 푿 that are both cartesian and cocartesian fibrations and for each pullback square in 푿 ladj the associated square in Cat∞ satisfies the Beck–Chevalley condition. Such fibrations are called Beck–Chevalley fibrations [1, Definition 4.1.3].

3 The definition of ambidexterity & basic properties

In this section we define ambidexterity of morphisms in an ∞-category 푿 with pull- ladj backs with respect to a Beck–Chevalley functor 퐶∶ 푿 → Cat∞ . This definition isa rather intricate inductive definition; to warm up we recall how to define truncatedness in an ∞-category with pullbacks as well as the universal property of the counit of an adjunction 3.1 Recollection ([HTT, Lemma 5.5.6.15]). Let 푿 be an ∞-category with finite limits (e.g., 푿 = Spc). For each integer 푛 ≥ −2 we define the class of 푛-truncated morphisms of 푿 inductively as follows. A morphism 푓∶ 푋 → 푌 in 푿 is (−2)-truncated if 푓 is an equivalence. Now suppose that 푛-truncated morphisms in 푿 have been defined for some integer 푛 ≥ −2. Then we say that a morphism 푓∶ 푋 → 푌 in 푿 is (푛 + 1)-truncated if the diagonal morphism 훿푓 ∶ 푋 → 푋 ×푌 푋 is 푛-truncated. Note that if 푿 = Spc is the ∞-category of spaces, then a morphism 푓 is 푛-truncated in the sense just defined if and only if the fibers of 푓 are 푛-truncated spaces, so the notion just defined agrees with the classical notion of truncatedness. 3.2 Recollection (universal property of the (co)unit [HTT, Definition 5.2.2.7 & Propo- sition 5.2.2.8]). Given functors between ∞-categories 푓∶ 퐶 ⇄ 퐷 ∶ 푔, the functor 푓 is left adjoint to 푔 if and only if there exists a natural transformation 휂∶ id퐶 → 푔푓 such that for every pair of objects 푐 ∈ 퐶 and 푑 ∈ 퐷, the composite

휂(푐)⋆ Map퐷(푓(푐), 푑) Map퐶(푔푓(푐), 푔(푑)) Map퐶(푐, 푔(푑)) is an equivalence in Spc. Dually, 푓 is left adjoint to 푔 if and only if there exists a natural transformation 휀∶ 푓푔 → id퐷 such that for every pair of objects 푐 ∈ 퐶 and 푑 ∈ 퐷, the composite

휀(푑)⋆ Map퐶(푐, 푔(푑)) Map퐷(푓(푐), 푓푔(푑)) Map퐶(푓(푐), 푑) is an equivalence in Spc.

⋆ ⋆ Our approach to exhibiting a left adjoint 푓! ⊣ 푓 as a right adjoint to 푓 is to define ⋆ a counit transformation that exhibits 푓! as right adjoint to 푓 . 3.3 Construction (definition of ambidexterity [1, Construction 4.1.8]). Let 푿 be an ∞- ladj category with pullbacks and 퐶∶ 푿 → Cat∞ a Beck–Chevalley functor. We define the following data for each integer 푛 ≥ −2:

(푎푛) A class of morphisms in 푿 which we call 푛-ambidextrous morphisms (with respect to the Beck–Chevaley functor 퐶).

7 (푏푛) For each 푛-ambidextrous morphism 푓∶ 푋 → 푌 in 푿, a natural transformation 휇(푛) ∶ id → 푓푓⋆ (well-defined up to equivalence) that exhibits 푓 as a right 푓 퐶푌 ! ! adjoint to 푓⋆.

A morphism 푓∶ 푋 → 푌 in 푿 is (−2)-ambidextrous if 푓 is an equivalence. In this (−2) −1 ⋆ case, we let 휇푓 ≔ 휀푓 be an inverse to the counit 휀푓 ∶ 푓!푓 ⥲ id퐶 of the adjunction ⋆ 푌 푓! ⊣ 푓 . Assume that (푎푛) and (푏푛) have been defined for some integer 푛 ≥ −2. Let 푓∶ 푋 → 푌 be a morphism in 푿, and write 훿푓 ∶ 푋 → 푋 ×푌 푋 for the diagonal map, which fits into a commutative diagram

푋

훿푓 (3.4) 푋 ×푌 푋 pr 푋 ⌟ 1 pr2 푓 푋 푌 . 푓

Write 휎 for the pullback square in the diagram (3.4), so that the Beck–Chevalley mor- ⋆ ⋆ phism BC(휎)∶ pr1,! pr2 → 푓 푓! is an equivalence. We say that 푓 is weakly (푛 + 1)- ambidextrous if the diagonal 훿푓 is 푛-ambidextrous. If 푓 is weakly (푛 + 1)-ambidextrous, we define a natural transformation

휈(푛+1) ∶ 푓⋆푓 → id 푓 ! 퐶푋 as the composite natural transformation

(푛) ⋆ pr1,! 휇훿 pr2 ⋆ ∼ ⋆ 푓 ⋆ ⋆ 푓 푓! pr pr pr 훿푓,!훿 pr ≃ id퐶 ∘ id퐶 = id퐶 . BC(휎)−1 1,! 2 1,! 푓 2 푋 푋 푋

We say that a morphism 푓∶ 푋 → 푌 is (푛 + 1)-ambidextrous if the following condition is satisfied: for every pullback diagram

̄ ′ 푓 ′ 푋 ⌟ 푌 ̄푔 푔 푋 푌 푓

̄ (푛+1) ⋆̄ ̄ in 푿, the map 푓 is weakly (푛 + 1)-ambidextrous and 휈푓̄ ∶ 푓 푓! → id퐶 ′ is the counit ⋆̄ ̄ 푋 for an adjunction 푓 ⊣ 푓!. If 푓 is (푛 + 1)-ambidextrous, we let 휇(푛+1) ∶ id → 푓푓⋆ denote a compatible unit 푓 퐶푌 ! for the adjunction ⋆ 푓 ∶ 퐶푌 ⇄ 퐶푋 ∶ 푓! (푛+1) determined by 휈푓 .

8 3.5 Definition ([1, Definition 4.1.1]). Let 푿 be an ∞-category with pullbacks and let ladj 퐶∶ 푿 → Cat∞ be a Beck–Chevalley functor. We say that a morphism 푓∶ 푋 → 푌 in 푿 is (weakly) ambidextrous (with respect to the Beck–Chevalley functor 퐶) if 푓 is (weakly) 푛-ambidextrous for some integer 푛 ≥ −2. The following basic properties are easily deduced from the definitions. 3.6 Proposition ([1, Proposition 4.1.10]). Let 푿 be an ∞-category with pullbacks, 퐶∶ 푿 → ladj Cat∞ a Beck–Chevalley functor, and 푓∶ 푋 → 푌 a morphism in 푿. Then: (3.6.1) If 푓 is weakly 푛-ambidextrous for some integer 푛 ≥ −2, then 푓 is 푛-truncated. (3.6.2) For each integer 푛 ≥ −2, the class of 푛-ambidextrous morphisms are stable under pullback. (3.6.3) For each integer 푛 ≥ −1, the class of weakly 푛-ambidextrous morphisms are stable under pullback. (3.6.4) Let −1 ≤ 푚 ≤ 푛 be integers. If 푓 is weakly 푚-ambidextrous, then 푓 is weakly 푛-ambidextrous. Moreover, the natural transformations 휈(푚), 휈(푛) ∶ 푓⋆푓 → id 푓 푓 ! 퐶푋 agree up to homotopy. (3.6.5) Let −2 ≤ 푚 ≤ 푛 be integers. If 푓 is 푚-ambidextrous, then 푓 is 푛-ambidextrous. Moreover, the natural transformations 휇(푚), 휇(푛) ∶ id → 푓푓⋆ agree up to ho- 푓 푓 퐶푌 ! motopy. (3.6.6) Let −1 ≤ 푚 ≤ 푛 be integers. If 푓 is (weakly) 푛-ambidextrous, then 푓 is (weakly) 푚-ambidextrous if and only if 푓 is 푚-truncated. ladj 3.7 Notation. Let 푿 be an ∞-category with pullbacks, 퐶∶ 푿 → Cat∞ a Beck–Chevalley functor, and 푓∶ 푋 → 푌 a morphism in 푿 that is weakly ambidextrous. Then we simply (푛) write 휈푓 for 휈푓 for some integer 푛 ≥ −2 such that 푓 is weakly 푛-ambidextrous (so that 휈푓 is well-defined up to homotopy). If 푓 is ambidextrous, we write 휇푓 for a compatible unit of 휈푓. 3.8 Reformulation (the norm map [1, Remark 4.1.12]). Let 푿 be an ∞-category with ladj pullbacks, 퐶∶ 푿 → Cat∞ a Beck–Chevalley functor, and 푓∶ 푋 → 푌 a morphism ⋆ ⋆ in 푿. If 푓 ∶ 퐶푌 → 퐶푋 admits a right adjoint, we denote the right adjoint to 푓 by 푓⋆ ∶ 퐶푋 → 퐶푌. We then have an equivalence (3.9) Map (푓⋆푓, id ) ≃ Map (푓, 푓 ) . Fun(퐶푋,퐶푋) ! 퐶푋 Fun(퐶푋,퐶푌) ! ⋆ If 푓 is weakly ambidextrous, we let Nm ∶ 푓 → 푓 denote the image of 휈 ∶ 푓⋆푓 → id 푓 ! ⋆ 푓 ! 퐶푋 under the equivalence (3.9). We call Nm푓 the norm map associated to 푓. We can reformulate the definition of ambidexterity as follows. A weakly ambidextrous morphism 푓∶ 푋 → 푌 is ambidextrous if and only if for every pullback square

̄ ′ 푓 ′ 푋 ⌟ 푌

푋 푌 푓

9 in 푿, the following conditions are satisfied: (3.8.1) The morphism 푓̄ is weakly ambidextrous. ⋆̄ ̄ (3.8.2) The functor 푓 ∶ 퐶푌′ → 퐶푋′ admits a right adjoint 푓⋆. ̄ ̄ (3.8.3) The norm map Nm푓̄ ∶ 푓! → 푓⋆ is an equivalence.

4 Naturality properties of the norm

The goal of this section is to establish two naturality properties of the construction 푓 ↦ 휇푓 (or, equivalently, 푓 ↦ Nm푓). 4.1 Proposition ([1, Proposition 4.2.1]). Let 푿 be an ∞-category with pullbacks, 퐶∶ 푿 → ladj Cat∞ a Beck–Chevalley functor, and let 휏 be a pullback diagram ̄ ′ 푓 ′ 푋 ⌟ 푌 ̄푔 푔 푋 푌 푓 in 푿. Then: (4.1.1) If 푓 is weakly ambidextrous, then 푓̄ is weakly ambidextrous and the diagram

⋆̄ ⋆̄ ̄ ⋆ 푓 BC(휏) ⋆̄ ⋆ ∼ ⋆ ⋆ 푓 푓! ̄푔 푓 푔 푓! ̄푔 푓 푓!

⋆ ⋆ 휈푓̄ ̄푔 ̄푔 휈푓 ⋆̄푔 ⋆̄푔

commutes up to homotopy. (4.1.2) If 푓 is ambidextrous, then 푓̄ is ambidextrous and the diagram

푔⋆ 푔⋆

⋆ ⋆ 휇푓̄푔 푔 휇푓 푓̄푓⋆̄ 푔⋆ ∼ 푓̄ ⋆̄푔 푓⋆ 푔⋆푓푓⋆ ! ! BC(휏)푓⋆ ! commutes up to homotopy. 4.2 Reformulation (Proposition 4.1 in terms of norms [1, Remark 4.2.3]). In the situation of Proposition 4.1, assume that 푓 and 푓̄ are weakly ambidextrous and that the ⋆ ⋆̄ ̄ functors 푓 and 푓 admit right adjoints 푓⋆ and 푓⋆. Then we can reformulate assertion (4.1.1) as follows: the morphism

푔⋆ Nm ̄ ⋆ BC(휏) ⋆ 푓 ⋆ ⋆̄ ⋆ 푓! ̄푔 ∼ 푔 푓! 푔 푓⋆ 푓 ̄푔 .

⋆ us homotopic to Nm푓̄ ̄푔 , where the last morphism is the Beck–Chevalley morphism involving the (−)⋆ adjoints (Remark 2.4).

10 Proof of Proposition 4.1. The statement that 푓 is (weakly) ambidextrous implies that 푓̄ is (weakly) ambidextrous is immediate from the definitions. Recall that if 푓 is weakly ambidextrous, then 푓 is 푛-truncated for some integer 푛 ≥ −2 (3.6.1). We prove both (4.1.1) and (4.1.2) simultaneously by induction on the truncatedness of 푓, which we denote by 푛. If 푛 = −2, then 푓 is an equivalence, hence 푓̄ is an equivalence, and (4.1.1) and (4.1.2) are obvious. So assume that 푛 ≥ −1 and that 푓 (and hence 푓)̄ is weakly ambidextrous. Thus we have a pullback square 휌 훿 ′ 푓̄ ′ ′ 푋 ⌟ 푋 ×푌′ 푋 ̄푔 휋

푋 푋 ×푌 푋 , 훿푓 where the morphism 휋 is induced by 푔 and ,̄푔 and 훿푓 and 훿푓̄ are (푛 − 1)-truncated and ambidextrous by the assumption that 푓 is weakly ambidextrous. Let 휎 denote the pullback square pr 푋 × 푋 1 푋 푌 ⌟ pr2 푓 푋 푌 , 푓 let 휎̄ denote the pullback square

′ ′ pr1 ′ 푋 × ′ 푋 푋 푌 ⌟ ̄ pr2 푓 푋′ 푌′ , 푓̄ and let 휉 denote the pullback square

′ ′ pr1 ′ 푋 × ′ 푋 푋 푌 ⌟ 휋 ̄푔 푋 × 푋 푋 . 푌 pr1

11 Now consider the diagram

−1 휇훿 ⋆̄ ̄ ⋆ BC(휎)̄ ⋆ ⋆ 푓̄ ⋆ ⋆ ⋆ ∼ ⋆ 푓 푓! ̄푔 ∼ pr1,!pr2 ̄푔 pr1,!훿푓,!̄ 훿푓̄pr2 ̄푔 ̄푔

≀

BC(휏) ≀ ⋆ ⋆ ⋆ ≀ pr1,!훿푓,!̄ 훿푓̄휋 pr2

휇 훿푓̄ ≀

(4.3) ⋆̄ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ∼ ⋆ 푓 푔 푓! pr1,!휋 pr2 pr1,!훿푓,!̄ 휋 훿푓 pr2 ̄푔

≀ BC(휌) 휇 훿푓

≀ ⋆ ⋆ ⋆ BC(휉) ≀ pr1,!휋 훿푓,!훿푓 pr2

≀ BC(휉)

⋆ ⋆ ∼ ⋆ ⋆ ⋆ ⋆ ⋆ ∼ ⋆ ̄푔 푓 푓! −1 ̄푔 pr1,! pr2 휇 ̄푔 pr1,! 훿푓,!훿푓 pr2 ̄푔 , BC(휎) 훿푓 where the long composites at the top and bottom of the diagram are the definitions of ⋆ ⋆ 휈푓̄ ̄푔 and ̄푔 휈푓, respectively, and morphisms labeled with ‘∼’ and no other decorations are given by identification of adjoints. Our goal is to show that the outer rectangle of(4.4) commutes up to homotopy; we do this by showing that each sub-diagram commutes up to homotopy. The diagrams in the middle column of(4.4) commute up to homotopy by the inductive hypothesis (for (4.1.2)). The upper-right diagram in (3.4) commutes because we’re just identifying adjoints. The lower-right diagram in (3.4) commutes because Beck– Chevalley morphisms compose horizontally (Observation 2.7) and we have a commutative diagram 훿 ̄ ′ 푓 ′ ′ pr1 ′ 푋 푋 × ′ 푋 푋 ⌟ 푌 ⌟ ̄푔 휋 ̄푔

푋 푋 ×푌 푋 pr 푋 , 훿푓 1 where the long composites on the top and bottom are the identity on 푋′ and 푋, respectively. To see that the left third rectangle in (3.4) commutes up to homotopy, it suffices to show that the diagram

12 ⋆̄ ̄ ⋆ BC(휎)̄ ⋆ ⋆ 푓 푓! ̄푔 ∼ pr1,!pr2 ̄푔

BC(휏) ≀ ≀

(4.4) ⋆̄ ⋆ ⋆ ⋆ 푓 푔 푓! pr1,!휋 pr2

≀ ≀ BC(휉)

⋆̄푔 푓⋆푓 ∼ ⋆̄푔 pr pr⋆ ! BC(휎) 1,! 2 commutes up to homotopy. This again follows from the fact that Beck–Chevalley morphisms compose vertically (Observation 2.7) and the fact that the outer rectangles of the two diagrams

pr ′ ′ 1 ′ ′ ′ pr1 ′ 푋 × ′ 푋 푋 푋 × ′ 푋 푋 푌 ⌟ 푌 ⌟ ̄푔 ̄ 휋 pr2 푓 pr ̄ 푋 × 푋 1 푋 and ′ 푓 ′ 푌 ⌟ 푋 ⌟ 푌 pr2 푓 ̄푔 푔 푋 푌 푋 푌 푓 푓 are the same. Now we prove the iductive step for (4.1.2). Assume that 푓 is ambidextrous so that ⋆ ⋆ the natural transformations 휈 ∶ 푓 푓 → id and 휈 ̄ ∶ 푓̄ 푓̄ → id are counits of 푓 ! 퐶푋 푓 ! 퐶푋′ ⋆ ⋆̄ ̄ adjunctions 푓 ⊣ 푓! and 푓 ⊣ 푓!. Then by the universal property of the unit 휇푓, the composite map

−∘푓 Map (푔⋆, 푔⋆푓푔⋆) ! Map (푔⋆푓, 푔⋆푓푓⋆푓) Fun(퐶푌,퐶푌′ ) ! Fun(퐶푋,퐶푌′ ) ! ! !

(푔⋆푓휈 )∘− 훼 ! 푓 Map (푔⋆푓, 푔⋆푓) Fun(퐶푋,퐶푌′ ) ! !

⋆ ⋆ is an equivalence. Moreover, by the triangle identity, the natural transformation 푔 휇푓 ∶ 푔 → ⋆ ⋆ ⋆ ⋆ 푔 푓!푓 corresponds to the identity 푔 푓! → 푔 푓! under the equivalence 훼. Thus proving that the diagram appearing in (4.1.2) commutes up to homotopy is equivalent to showing that the composite

⋆ ⋆ 휇 ̄푔 푓 ⋆ 푔 푓휈 ⋆ 푓 ! ̄ ⋆̄ ⋆ ̄ ⋆ ⋆ BC(휏)푓 푓! ⋆ ⋆ ! 푓 ⋆ 푔 푓! 푓!푓 푔 푓! ≃ 푓! ̄푔 푓 푓! ∼ 푔 푓!푓 푓! 푔 푓!

13 is homotopic to the identity. To prove this, consider the diagram

휇 ̄ 휈 ⋆ 푓 ̄ ⋆̄ ⋆ ∼ ⋆ ⋆ BC(휏) ⋆ ⋆ 푓 ⋆ 푔 푓! 푓!푓 푔 푓! 푓! ̄푔 푓 푓! ∼ 푔 푓!푓 푓! 푔 푓!

BC(휏) ≀ BC(휏) ≀ 휈푓 ∼ BC(휏)

푓̄ ⋆̄푔 푓̄푓⋆̄ 푓̄ ⋆̄푔 푓̄ ⋆̄푔 . ! 휇푓̄ ! ! 휈푓̄ !

The left-hand square and right-hand triangle obviously commute, and the middle square commutes by the inductive step for (4.1.1). To show that the top composite is homotopic to the identity, it suffices to show that the bottom composite is homotopic to the identity: this is true by the triangle identity sicne 휇푓̄ and 휈푓̄ are a compatible unit and counit. 4.5 Corollary (adjoint formulation of Proposition 4.1 [1, Corollary 4.2.6]). Let 푿 be ladj an ∞-category with pullbacks, 퐶∶ 푿 → Cat∞ a Beck–Chevalley functor, and let 휏 be a pullback diagram ̄ ′ 푓 ′ 푋 ⌟ 푌

푋 푌 푓 in 푿. Then: (4.5.1) If 푓 is weakly ambidextrous, then 푓̄ is weakly ambidextrous and the diagram

̄ ⋆̄ ̄ BC(휏)푓! ⋆ ̄ ∼ ⋆ !̄푔 푓 푓! 푓 푔!푓! 푓 푓! !̄푔

!̄푔 휈푓̄ 휈푓 !̄푔

!̄푔 !̄푔

commutes up to homotopy. (4.5.2) If 푓 is ambidextrous, then 푓̄ is ambidextrous and the diagram

푔! 푔!

푔!휇푓̄ 휇푓푔! ̄ ⋆̄ ∼ ⋆̄ ⋆ 푔!푓!푓 푓! !̄푔 푓 푓!푓 푔! 푓! BC(휏)

commutes up to homotopy. The proof of the following proposition is similar to the proof of Proposition 4.1, though a little more involved. 4.6 Proposition ([1, Proposition 4.2.2]). Let 푿 be an ∞-category with pullbacks, 퐶∶ 푿 → ladj Cat∞ a Beck–Chevalley functor, and suppose we are given morphisms 푓∶ 푋 → 푌 and 푔∶ 푌 → 푍 in 푿. Then:

14 (4.1.1) If 푓 and 푔 are weakly ambidextrous, then 푔푓 is weakly ambidextrous and 휈푔푓 is homotopic to the composition

⋆ 푓 휈푔푓! 휈푓 (푔푓)⋆(푔푓) ≃ 푓⋆푔⋆푔 푔 푓⋆푓 id . ! ! ! ! 퐶푋

commutes up to homotopy.

(4.1.2) If 푓 and 푔 are ambidextrous, then 푔푓 is ambidextrous and 휇푔푓 is homotopic to the composition

⋆ 휇푔 푔!휇푓푔 id 푔 푔⋆ 푔 푓푓⋆푔⋆ ≃ (푔푓) (푔푓)⋆ . 퐶푍 ! ! ! !

4.7 Reformulation (Proposition 4.6 in terms of norms [1, Remark 4.2.4]). In the situation of Proposition 4.6, assume that 푓 and 푔 are weakly ambidextrous and that the ⋆ ⋆ ⋆ functors 푓 and 푔 admit right adjoints 푓⋆ and 푔⋆. Then (푔푓) is left adjoint to (푔푓)⋆ ≔ 푔⋆푓⋆ and we can reformulate assertion (4.6.1) as follows: the norm Nm푔푓 ∶ (푔푓)! → (푔푓)⋆ is given by the composite

Nm푔 푓! 푔⋆ Nm푓 (푔푓)! ≃ 푔!푓! 푔⋆푓! 푔⋆푓⋆ ≃ (푔푓)⋆ .

5 Ambidexterity for local systems

So far we have considered ambidexterity for arbitrary Beck–Chevalley functors 푿 → ladj Cat∞ . We now specify to the case of 퐶-valued local systems, i.e., 푿 = Spc and we con- ladj sider the Beck–Chevalley functor Fun(−, 퐶)∶ Spc → Cat∞ , where 퐶 is an ∞-category with colimits (recall Proposition 1.5). 5.1 Definition ([1, Definition 4.3.4]). Let 퐶 be an ∞-category with colimits. • A space 푋 is weakly 퐶-ambidextrous if the unique morphism 푓∶ 푋 → ∗ is weakly 퐶-ambidextrous (with respect to the Beck–Chevalley functor Fun(−, 퐶)∶ Spc → ladj Cat∞ ). • A space 푋 is 퐶-ambidextrous if 푋 is weakly 퐶-ambidextrous and the natural trans- ⋆ ⋆ formation 휈푓 ∶ 푓 푓! → idFun(푋,퐶) is the counit of an adunction (so that 푓 ⊣ 푓!). 5.2 Proposition ([1, Proposition 4.3.5]). Let 퐶 be an ∞-category with colimits and 푓∶ 푋 → 푌 and 푓∶ 푋 → 푌 a morphism in Spc. Then: (5.2.1) The morphism 푓 is ambidextrous if and only if 푓 is 푛-truncated for some integer 푛 ≥ −2 and each fiber 푋푦 of 푓 is 퐶-ambidextrous. (5.2.2) The morphism 푓 is weakly ambidextrous if and only if 푓 is 푛-truncated for some integer 푛 ≥ −2 and each fiber 푋푦 of 푓 is weakly 퐶-ambidextrous. 5.3 Corollary ([1, Corollary 4.3.6]). Let 퐶 be an ∞-category with colimits and 푓∶ 푋 → 푌 a morphism between truncated spaces. If 푌 is 퐶-ambidextrous and each fiber 푋푦 of 푓 is 퐶-ambidextrous, then 푋 is 퐶-ambidextrous.

15 The proof of Proposition 5.2 uses the following fact to reduce to proving the claim in a special case. 5.4 Lemma ([1, Lemma 4.3.8]). Let 퐶 be an ∞-category with colimits and 푋 a space. Then Fun(푋, 퐶) is generated under colimits by objects of the form 푥!푐, where 푥∶ ∗ → 푋 is a point of 푋 and 푐 ∈ 퐶 ≃ Fun(∗, 퐶). Lemma 5.4 is also used in the proof of the following proposition. 5.5 Proposition ([1, Proposition 4.3.9]). Let 퐶 be an ∞-category with colimits and 푋 a truncated space. Let 푓∶ 푋 → ∗ denote the unique morphism. Then 푋 is 퐶-ambidextrous if and only if the following conditions are satisfied:

′ ′ (5.5.1) 푋 is weakly 퐶-ambidextrous (i.e., Map푋(푥, 푥 ) is 퐶-ambidextrous for all 푥, 푥 ∈ 푋).

⋆ (5.5.2) The pullback functor 푓 admits a right adjoint 푓⋆.

(5.5.3) The functor 푓⋆ preserves colimits. Proof. If 푋 is 퐶-ambidextrous, then (5.5.1) is obvious and (5.5.2) and (5.5.3) follow from ⋆ ⋆ the fact that the left adjoint 푓! of 푓 is also right adjoint to 푓 . ⋆ Now assume that (5.5.1)–(5.5.3) are satisfied. Using (5.5.2), the counit 휈푓 ∶ 푓 푓! → idFun(푋,퐶) corresponds to the norm map Nm푋 ∶ 푓! → 푓⋆ (Reformulation 3.8), and our goal is to show that Nm푓 is an equivalence. That is, we want to show that for every local system 퐿∶ 푋 → 퐶, the natural transformation

Nm푓(퐿)∶ 푓!퐿 → 푓⋆퐿 is an equivalence in 퐶. By asumption (5.5.3), the collection of objects 퐿 ∈ Fun(푋, 퐶) for which Nm푓(퐿) is an equivalence is closed under colimits, so by Lemma 5.4 we are reduced to the case where 퐿 = 푥!푐 for 푥∶ ∗ → 푋 a point and 푐 ∈ 퐶. By assumption (5.5.1), for every point 푥 ∈ 푋 the morphsim 푥∶ ∗ → 푋 is 퐶-ambidextrous. By Reformulation 4.7, the composite

Nm푓(퐿) 푓⋆ Nm푥(퐿) 푐 ≃ (푓푥)!푐 ≃ 푓!퐿 푓⋆퐿 푓⋆푥⋆푐 ≃ (푓푥)⋆푐 ≃ 푐 is homotopic to the identity. Since 푥∶ ∗ → 푋 is 푋-ambidextrous, the norm Nm푥 is an equivalence, hence Nm푓 is an equivalence as well. 5.6 Remark. Note that Proposition 5.5 gives a characterization of (weakly) 퐶-ambidextrous morphisms that doesn’t explicitly mention the natural transformations 휈푓 or 휇푓.

6 Semiadditivity & ambidexterity of Eilenberg–MacLane spaces

6.1 Definition ([1, Definition 4.4.1]). Let 푛 ≥ −2 be an integer. A space 푋 is a finite 푛-type if 푋 is 휋-finite and 푛-truncated.

16 6.2 Definition ([1, Definition 4.4.2]). Let 퐶 be an ∞-category with colimits and 푛 ≥ −2 be an integer. We day that 퐶 is 푛-semiadditive if every finite 푛-type is 퐶-ambidextrous. 6.3 Examples. (6.3.1) Since a space 푋 is a finite (−2)-type if and only if 푋 is contractible, every ∞- category with colimits is (−2)-semiadditive. (6.3.2) Since the only (−1)-types are contractible and empty spaces, an ∞-category with colimits 퐶 is (−1)-semiadditive if and only if is 퐶-ambidextrous if and only if 퐶 is pointed. (6.3.3) Since a space 푋 is a finite 0-type if and only if 푋 is equivalent to a finite set, an ∞-category with colimits is 0-semiadditive if and only if 퐶 is semiadditive in the usual sense, i.e., the finite products in 퐶 are also finite coproducts. (6.3.4) Any stable ∞-category with colimits is 0-semiadditive. 6.4 Notation ([1, Notation 4.4.15]). Let 퐶 be a 0-semiadditive ∞-category with colimits and 푛 ≥ 0 be an integer. Write [푛]∶ id퐶 → id퐶 for the natural transformation determined by the composite

푐 훥 푐×푛 ≃ 푐⊔푛 훻 푐 of the diagonal and codiagonal for each object 푐 ∈ 퐶. 6.5 Proposition ([1, Proposition 4.4.16]). Let 퐶 be a 0-semiadditive ∞-category with limits and assume that there exists a prime number 푝 with the following property: (6.5.1) For every integer 푛 ≥ 1 which is relatively prime to 푝, the natural transformation [푛]∶ id퐶 → id퐶 is an equivalence Then 퐶 is 1-semiadditive if and only if the Eilenberg–MacLane space 퐾(퐙/푝, 1) is 퐶-ambidextrous. Proof. The fact that the 1-semiadditivity of 퐶 implies that 퐾(퐙/푝, 1) is 퐶-ambidextrous is immediate from the definition of 1-semiadditivity. For the other direction, suppose that 퐾(퐙/푝, 1) is 퐶-ambidextrous. Let 푋 be a finite 1-type; our goal is to show that 푋 is 퐶-ambidextrous. By applying Corollary 5.3 to the map 푋 → 휋0(푋), we can reduce to the case where 푋 is connected, so that 푋 ≃ 퐵퐺 for some finite group 퐺. Now we reduce to the case where 퐺 is a 푃-group Let 푃 ⊂ 퐺 be a 푝-Sylow subgroup, and consider the maps 푔∶ 퐵푃 → 퐵퐺 induced by the inclusion 푃 ↪ 퐺 and 푓∶ 퐵퐺 → ∗. Then 푔 is equivalent to a covering space with finite fibers, hence is 퐶-ambidextrous since 퐶 is 0-semiadditive. We want to show that Nm푓 ∶ 푓! → 푓⋆ is an equivalence. Let 퐿 ∈ Fun(퐵퐺, 퐶) and let 훼 denote the composite

⋆ ⋆ 훼∶ 퐿 푔⋆푔 퐿 ≃ 푔!푔 퐿 퐿 , where the middle equivalence comes from the fact that 푔 is ambidextrous. We claim that 훼 is an equivalence. To prove this, it suffices to show that for every point 푥 ∈ 퐵퐺, the

17 morphism 푥⋆훼∶ 푥⋆퐿 → 푥⋆퐿 is an equivalence. Unwinding the definitions, we see that 푥⋆훼 is given by the morphism [#(퐺/푃)]∶ 푥⋆퐿 → 푥⋆퐿. Since 푃 is a 푝-Sylow subgroup of 퐺, the number #(퐺/푃) is relatively prime to 푝, so that [#(퐺/푃)] is an equivalence ⋆ by assumption (6.5.1). Since 훼 is an equivalence, we see that 퐿 is a retract of 푔!푔 퐿. ⋆ Hence to see that Nm푓(퐿) is an equivalence, it suffices to prove that Nm푓(푔!푔 퐿) is ′ ′ an equivalence. We may therefore assume that 퐿 = 푔!퐿 for some 퐿 ∈ Fun(퐵푃, 퐶). Consider the composite

Nm (푔 퐿′) 푓 Nm (퐿′) ′ ′ 푓 ! ′ ⋆ 푔 ′ Nm푓푔(퐿 )∶ (푓푔)!퐿 푓⋆푔!퐿 ∼ (푓푔)⋆퐿 , where the second morphism is an equivalence since 푔 is 퐶-ambidextrous. By the 2-of- 3 property we see that to prove that Nm푓 is an equivalence, it suffices to prove that ′ Nm푓푔(퐿 ) is an equivalence, so we can replace 퐺 by 푃 and assume that 퐺 is a 푝-group. With this reduction, we now proceed by induction on the cardinality of the 푝-group 퐺. If 퐺 is trivial, there is nothing to prove. For the induction step, we can choose a normal subgroup 푁 ◃ 퐺 of order 푝. It follows from the inductive hypothesis that 퐵(퐺/푁) is 퐶- ambidextrous. We have a fiber sequence

퐾(퐙/푝, 1) ≃ 퐵푁 퐵퐺 퐵(퐺/푁) , so an application of Corollary 5.3 and the assumption that 퐾(퐙/푝, 1) is 퐶-ambidextrous show that 퐵퐺 is 퐶-ambidextrous, completing the proof. 6.6 Proposition ([1, Proposition 4.4.17]). Let 퐶 be a 0-semiadditive ∞-category with limits and 푝 be a prime number. If the natural transformation [푝]∶ id퐶 → id퐶 is an equivalence, then for every finite 푝-group 퐺, the Eilenberg–MacLane space 퐵퐺 is 퐶-ambidextrous.

Proof. As in the proof of Proposition 6.5, we can reduce to the case where 퐺 = 퐙/푝. Consider the maps 푔∶ ∗ → 퐵퐺 given by any point and 푓∶ 퐵퐺 → ∗, so that 푔 is equivalent to a covering space with finite fibers (namely the projection 퐸퐺 → 퐵퐺). We need to show that Nm푓 ∶ 푓! → 푓⋆ is an equivalence. Let 퐿 ∈ Fun(퐵퐺, 퐶) and let 훼 denote the composite ⋆ ⋆ 훼∶ 퐿 푔⋆푔 퐿 ≃ 푔!푔 퐿 퐿 , where the middle equivalence comes from the fact that 푔 is ambidextrous. As in the proof of Proposition 6.5, we see that for each 푥 ∈ 퐵퐺, the map 푥⋆훼∶ 푥⋆퐿 → 푥⋆퐿 is homotopic ⋆ ⋆ ⋆ to [푝]∶ 푥 퐿 → 푥 퐿, hence an equivalence by assumption. Thus 퐿 is a retract of 푔!(푔 퐿). ⋆ ⋆ Thus it suffices to show that Nm푓 induces as equivalence 푓!푔⋆(푔 퐿) ⥲ 푓⋆푔!(푔 퐿). Set ′ ⋆ 퐿 ≔ 푔 퐿. Since 푓푔 = id∗ the composite map

Nm Nm ′ ′ 푓 ′ 푔 ′ ′ Nm푓푔 ∶ 퐿 ≃ (푓푔)!퐿 푓⋆푔!퐿 (푓푔)⋆퐿 ≃ 퐿 is an equivalence. By the 2-of-3 property, we are reduced to proving that Nm푔 induces an ′ ′ equivalence 푓⋆푔!퐿 ⥲ 푓⋆푔⋆퐿 . This follows from the assumption that 퐶 is 0-semiadditive.

18 6.7 Corollary ([1, Corollary 4.4.18]). Let 퐶 be a 0-semiadditive ∞-category with limits. If for each integer 푛 ≥ 1 the natural transformation [푛]∶ id퐶 → id퐶 is an equivalence, then 퐶 is 1-semiadditive. 6.8 Proposition ([1, Proposition 4.4.20]). Let 퐶 be a stable ∞-category with limits and colimits and let 푝 be a prime number such that the natural transformation [푝]∶ id퐶 → id퐶 is an equivalence. Then the Eilenberg–MacLane spaces 퐾(퐙/푝, 푚) are 퐶-ambidextrous for 푚 ≥ 1. 6.9 Corollary ([1, Corollary 4.4.21]). Let 퐶 be a stable ∞-category with limits and col- 0 imits. Assume that for each object 푐 ∈ 퐶, the endomorphism ring Ext퐶(푐, 푐) is a 퐐-algebra. Then 퐶 is 푛-semiadditive for every integer 푛 ≥ −2.

6.10 Example ([1, Example 4.4.22]). Let 푅 be an 퐸1-ring spectrum with the property that 휋0(푅) is a 퐐-vector space. Then the ∞-category of left 푅-module spectra is 푛-semiadditive for every integer 푛 ≥ −2. 6.11 Corollary ([1, Corollary 4.4.23]). Let 퐶 be a stable ∞-category with limits and colimits and 푝 a prime number. Assume that for each object 푐 ∈ 퐶, the endomorphism 0 ring Ext퐶(푐, 푐) is a 퐙(푝)-module. Then 퐶 is 푛-semiadditive if and only if the Eilenberg– MacLane spaces 퐾(퐙/푝, 푚) are 퐶-ambidextrous for 1 ≤ 푚 ≤ 푛.

References

HTT J. Lurie, Higher topos theory, Annals of Mathematics Studies. Princeton, NJ: Princeton University Press, 2009, vol. 170, pp. xviii+925, isbn: 978-0-691-14049-0; 0-691-14049- 9.

1. M. J. Hopkins and J. Lurie, Ambidexterity in 퐾(푛)-local stable homotopy theory, Preprint available at math.harvard.edu/~lurie/papers/Ambidexterity.pdf, Dec. 2013. 2. Math overflow question 286886: Pointwise evaluation of the Beck–Chevalley map in ∞- categories, MO:286886, Nov. 2017. 3. J. S. Milne, Étale cohomology, Princeton Mathematical Series. Princeton University Press, Princeton, N.J., 1980, vol. 33, pp. xiii+323, isbn: 0-691-08238-3.