Goodwillie’s Calculus of Functors and Higher Topos Theory Mathieu Anel, Georg Biedermann, Eric Finster, Andr´e Joyal Abstract We prove a Blakers-Massey theorem for the Goodwillie tower of a homotopy functor and then reprove some delooping results. The theorem is derived from a generalized Blakers-Massey theorem in [ABFJ17]. Our main tool is fiberwise orthogonality. A new input is that the pushout product of an n-excisive map with an m-excisive map is (m + n + 1)excisive. Our proof sheds new light on Goodwillie’s constructions.

Contents 1 Introduction

2

2 Prerequisites 2.1 Topoi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Cubes, gaps and cogaps . . . . . . . . . . . . . . . . . . 2.3 Pushout product and join . . . . . . . . . . . . . . . . . 2.4 Pullback hom and external orthogonality . . . . . . . 2.5 Modalities and generalized Blakers-Massey theorems

. . . . .

4 4 4 5 6 8

3 Orthogonality conditions 3.1 Internal orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Fiberwise orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The fiberwise diagonal map . . . . . . . . . . . . . . . . . . . . . .

9 10 11 13

4 Goodwillie’s Calculus of Functors in a Higher Topos 4.1 Joins and n-excisive functors . . . . . . . . . . . . . . . . . 4.2 n-excisive maps . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The map wK and its pushout product powers . . . . . . . 4.4 Epimorphisms and Monomorphisms of Excisive Functors 4.5 Blakers-Massey theorem for the Goodwillie tower . . . .

. . . . .

16 16 19 22 28 31

5 Consequences 5.1 Delooping theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The n-homogeneous model structure is left proper . . . . . . . . .

32 32 34

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1

Introduction

Goodwillie’s calculus of homotopy functors ([Goo90], [Goo92], [Goo03]) is an advanced tool to deduce information on unstable homotopy theory from stable knowledge. The main goal of this article is to prove the following analogue of the Blakers-Massey theorem in the context of calculus. Theorem 4.5.1 Let

g

F

H ⌟

f

G

K

be a homotopy pushout square of functors. If f is a Pm -equivalence and g is a Pn -equivalence, then the induced cartesian gap map (f, g) ∶ F → G ×K H is a Pm+n+1 -equivalence. The second result is a“dual” version. Theorem 4.5.2 Let F

H ⌜

G

f g

K

be a homotopy pullback square of functors. If f is a Pm -equivalence and g is a Pn -equivalence, then the cocartesian gap map ⌊f, g⌋ ∶ G ⊔F H → K is a Pm+n+1 -equivalence. As a consequence we can rederive known delooping results in homotopy functor calculus in an easy and conceptual way. In particular, we obtain an independent proof of Goodwillie’s theorem [Goo03, Section 2] that homogeneous functors deloop. Both results rest on the companion article [ABFJ17] where generalized Blakers-Massey theorems were proved. There the language of higher topoi was adopted. We find it well suited for the calculus of homotopy functors as well, particularly because n-excisive functors to spaces form a higher topos and since the focus of this paper is to work fiberwise. Thus we will drop any reference to higher derived structures and take them for granted. When we talk about a “category” it is an “∞-category” and (co-)limits are to be interpreted as ∞-categorical (co-)limits. In particular, we will not use the terms “homotopy (co-)limit”, as we did above for the sake of introduction. The ∞-categorical situation already describes the derived, homotopy invariant setting with all higher coherences. “Isomorphism” translates to “weak equivalence”. Similarly, mapping spaces or internal hom objects are always to be taken derived. The reader 2

who finds this article easier to read by using model structure is invited to do so and should not have any difficulties in doing so. The main tool in our paper [ABFJ17] and here is the notion of modality. This is a unique factorization system whose left and right class is closed under base change. An example is the factorization of a map of spaces into an nconnected map followed by an n-truncated map. That example leads to the classical Blakers-Massey theorem. Here we observe that the factorization of a natural transformation into a Pn -equivalence followed by an n-excisive map is a modality as well. This follows since Goodwillie’s n-excisive approximation construction Pn is a left exact localization of the topos of functors. The left classes of these Goodwillie modalities for various n ≥ 0 are compatible with the pushout product in the following sense: Theorem 4.3.12(4) The pushout product of a Pm -equivalence with a Pn equivalence is a Pm+n+1 -equivalence. This statement is our new contribution to Goodwillie calculus and immediately implies the main theorems in view of the generalized Blakers-Massey theorems from [ABFJ17]. To prove this theorem we need to take a step back and develop consequently the point of working fiberwise. A modality is a factorization system satisfying fiberwise orthogonality (Def. 3.2.1): two maps are fiberwise orthogonal if each for their base changes are externally orthogonal to each other. Similarly to the pullback hom (Subsec. 2.4) that tests for external orthogonality, we introduce the fiberwise diagonal (Def. 3.3.1) of two maps that tests fiberwise othogonality. This fiberwise diagonal serves as a “parametrized right adjoint” (Prop. 3.3.6) to the pushout product where the pushout product is not viewed as a symmetric monoidal product on arrows but as a functor from two slices to the slice over the product of the base spaces. The resulting adjunction tricks enter in the proof of Theorem 4.3.12. Another observation is now used, see Subsection 4.3: in the same way as n-connected maps of spaces can be generated from the pushout product powers of the map S 0 → ˚, the Goodwillie tower can be generated by pushout product powers of the set of maps wK ∶ R˚ → RK for all K in the source category. Here, the functor RK = [K, −] is the covariant representable functor for K. For the terminal object ˚ the functor R˚ is the constant functor at ˚ if the source category is pointed finite spaces (spaces retractive over another space), or it is the identity for unpointed spaces (the slice category over a space). Given a functor F , for the map F → ˚ to be ◻n+1 externally right orthogonal to the (n + 1)-st power wK means that F sends the (n + 1)-cube P (n + 1) → S , U ↦ ⋁ K n+1−U

to a cartesian cube; in particular, the (n+1)-st cross effect of F vanishes. Clearly this does not characterize n-excisive functors. But if F → ˚ is fiberwise right ◻n+1 orthogonal to wK then F is n-excisive by Proposition 4.3.10. These new 3

generating maps are on the nose compatible with the pushout product. This finally implies Theorem 4.3.12.

2

Prerequisites

In this section we recall material from the our companion paper [ABFJ17]. In particular, we give the definition of a modality 2.5.1 and state our generalized Blakers-Massey theorems 2.5.6 and 2.5.7.

2.1

Topoi

This article is written using the language of ∞-topoi. For an outline of the theory we refer the reader to [Rez05], Joyal, and [Lur09]. A very brief overview of the essential properties taylored to our needs is given in [ABFJ17, Section 2]. Definition 2.1.1. A ∞-topos is a left exact localization of a presheaf category [C, S] for some small category C. We will drop ∞ from the notation and refer to them simply as topoi. When we speak of (co-)limits it is to be interpreted as ∞-categorical (co-)limit. Using model structures this translates into homotopy (co-)limits; In particular, the reader should be aware that “left exact localization” is to be taken in the derived sense. Spelled out in the language of model categories it means “left Bousfield localization commuting with finite homotopy limits up to weak equivalence”. Example 2.1.2. Here are a few examples: 1. The category S of spaces (as modelled by topological spaces or simplicial sets with weak homotopy equivalences) is the prime example of a topos. 2. The category [C, S] of functors to spaces is a topos. 3. As explained in Example 2.5.3, Goodwillie’s n-excisive functors to spaces form a topos.

2.2

Cubes, gaps and cogaps

Let n = {1, ⋯, n} and write P (n) for the poset of its subsets. Define P0 (n) to be the poset of non-empty subsets; let P1 (n) be the poset of proper subsets. Now consider a finitely complete category E. An n-cube in a category E is a functor X ∶ P (n) → E. The canonical map X(∅) →

lim X(U )

U ∈P0 (n)

is called the cartesian gap map or shorter the gap map. An n-cube is said to be cartesian if its gap map is an isomorphism. For example, a 2-cube is cartesian if and only if it is a pullback square. 4

For an n-cube X in a finitely cocomplete category E there also exists the canonical map colim X(U ) → X(n). U ∈P1 (n)

We will call it the cartesian gap map or shorter the gap map. An n-cube is cocartesian if its cogap map is an isomorphism. A square is cocartesian if and only if it is a pushout square. An n-cube is strongly cocartesian if if every sub-2-cube is cocartesian. To be strongly cocartesian is equivalent to demanding that only every 2-dimensional face is a pushout. Now let E be finitely complete and finitely cocomplete. Given a commutative square in E: g A C f

B

k h

D

If the square is a pushout, the gap map only depends on the maps f and g and we will denote the gap map by (f, g) ∶ A → B ×D C. If the square is a pullback, the cogap map only depends on the maps h and k and we will denote the cogap map by ⌊h, k⌋ ∶ B ∪A C → D. Remark : this convention contradict that of the other paper where notations ⌊f, g⌋ and (f, g) are used (abusely) for a square not necessarily (co)cartesian. I would keep using those notations for arbitrary squares, the abuse is harmless.

2.3

Pushout product and join

Let E be a topos. For any two maps f ∶ A → B and g ∶ X → Y in E the square A×X

1A ×g

f ×1X

B×X

A×Y f ×1Y

1B ×g

B × Y.

is cartesian, and we define the pushout product of f and g, denoted f ◻ g, to be the cocartesian gap map of the previous square: f ◻ g = A × Y ∪A×X B × X → B × Y. Let ∅ and ˚ be respectively an initial and a terminal object for E. A topos E has a strict initial object which means that any arrow C → ∅ is an isomorphism. In particular, since E has finite products, this implies ∅ × X = ∅ for all objects 5

X ∈ E. Hence, the pushout product defines a symmetric monoidal structure on the category E→ of arrows, with unit ∅ → ˚. In particular, we have f ◻ g = g ◻ f and (f ◻ g) ◻ h = f ◻ (g ◻ h). Example 2.3.1. We give some examples of pushout products that will be useful in the sequel. 1. The following formula is also useful, for any C and any map A → B in E, the maps (∅ → C) ◻ (A → B) is simply (∅ → C) ◻ (A → B) = C × A → C × B 2. For two pointed objects ˚ → A and ˚ → B in E, we have (˚ → A) ◻ (˚ → B) = A ∨ B → A × B the canonical inclusion of the wedge into the product. 3. Recall that the join of two objects A and B in E, denoted A ⋆ B, is the pushout of the diagram A ← A × B → B. Then it is easy to see that (A → ˚) ◻ (B → ˚) = A ⋆ B → ˚ 4. Since colimits commute in E with base change, the pushout product f ◻ g can be thought as the ”external” join product of the fibers of f and g. An easy computation shows that the fiber (f ∶ A → B) ◻ (g ∶ C → D) at a point (b, d) ∈ B × D is the join of the fibers of f and g. Details can be found in [ABFJ17, Rem. 2.4]. 5. For an object Z the slice category E/Z has its own pushout product denoted ◻Z . Given f ∶ A → B and g ∶ X → Y in E/Z, then f ◻Z g = (A ×Z Y ) ∪(A×Z X) (B ×Z X) → (B ×Z Y ). is the formula for the pushout product relative to Z.

2.4

Pullback hom and external orthogonality

For two objects A, B of E, we let [A, B] be the space of maps from A to B in E. For two maps A → B and X → Y in E we consider the following commutative square in S [B, X]

[B, Y ]

[A, X]

[A, Y ].

We define the external pullback hom ⟨f, g⟩ to be the cartesian gap map of the previous square: 6

⟨f, g⟩ ∶ [B, X] → [A, X] ×[A,Y ] [B, Y ]. Let ⟦A, B⟧ denote the internal hom object in E. Then we can define similarly an internal pullback hom ⟪f, g⟫ ∶ ⟦B, X⟧ → ⟦A, X⟧ ×⟦A,Y ⟧ ⟦B, Y ⟧, which is the map in C. Example 2.4.1. Given two objects A and X in a topos E. The A-diagonal of X or shortly the diagonal is the map ∆A (X) = ⟪A → ˚, X → ˚⟫ ∶ X → ⟦A, X⟧. In spaces this can be interpreted as the map that associates to an x ∈ X the constant map at x. The classical diagonal map ∆S 0 (X) ∶ X → X × X, x ↦ (x, x) is a special case. Remark 2.4.2. The external and internal pullback hom define functors ⟨−, −⟩ ∶ (E→ )

op

× E→ → S→

and

⟪−, −⟫ ∶ (E→ )

op

× E→ → E→ .

Together with the pushout product the internal pullback hom yields a closed symmetric monoidal structure on E→ . In particular, we have ⟪f ◻ g, h⟫ = ⟪f, ⟪g, h⟫⟫. Lemma 2.4.3. For f , g and h three maps in E, we have a canonical isomorphism ⟨f ◻ g, h⟩ = ⟨f, ⟪g, h⟫⟩. Proof. For any topos E there is a global sections functor [˚, −] ∶ E → S , A ↦ [˚, A] landing in spaces. This extends to a functor [˚, −]→ ∶ E→ → S→ , (A → B) ↦ [˚, A] → [˚, B] of the respective arrow categories. It is easy to see that this functor is representable for the pushout product by ∅ → ˚, i.e. that [˚, f ]→ = ⟨∅ → ˚, f ⟩. Because (E, ×, ⟦−, −⟧) is a closed symmetric monoidal category, we have [˚, ⟦B, X⟧] = [˚ × B, X] = [B, X].

7

We use this to compute that [˚, ⟪f, g⟫]→ = [˚, ⟪B, X⟫] Ð→ [˚, ⟦A, X⟧ ×⟦A,Y ⟧ ⟦B, Y ⟧] = [B, X] Ð→ [A, X] ×[A,Y ] [B, Y ] = ⟨f, g⟩ This proves ⟨∅ → ˚, ⟪f, g⟫⟩ = ⟨f, g⟩, from which we derive the result ⟨f ◻ g, h⟩ = ⟨∅ → ˚, ⟪f ◻ g, h⟫⟩ = ⟨∅ → ˚, ⟪f, ⟪g, h⟫⟫⟩ = ⟨f, ⟪g, h⟫⟩. Definition 2.4.4. Two maps f ∶ A → B and g ∶ X → Y in E are externally orthogonal or shorter orthogonal if the map ⟨f, g⟩ is an isomorphism in S. We write f ⊥ g for this relation and we say that f is left orthogonal to g and that g is right orthogonal to f . For a class S of maps in E we define S ⊥ (resp. ⊥ S) to be the class of maps that are right (resp. left) orthogonal to all maps in S. Recall that for a topos E, all slice categories E/Z are also topoi. Therefore, each E/Z has an external orthogonality relation which we will denote by ⊥Z .

2.5

Modalities and generalized Blakers-Massey theorems

Recall that a factorization system on a category E is the data of a pair (L, R) of classes of maps in E such that 1. every map f in E can be factored in f = rl where l ∈ L and r ∈ R, and 2. L⊥ = R and L = ⊥ R. Definition 2.5.1. Let E be a topos. A modality on E is a factorization system (L, R) such that the left classes L is stable by base change. Modalities were introduced in [ABFJ17]. They were conceived as an axiomatization of the n-connected/n-truncated factorization system in spaces. A central idea in this paper is that there is another family of examples. Lemma 2.5.2. Let F be a left exact localization of a topos cE. If we let L be the F -equivalences and R the F -local maps, then (L, R) forms a modality in E. Example 2.5.3. Goodwillie’s n-excisive approximation construction Pn is a left exact localization of the ∞-topos [C, S] for some small category C with finite colimits and a terminal object. Hence, the Pn -equivalences and the Pn -local maps form a modality. This example is developped in detail Subsection 4.2. Consider a commutative square Z

g

Y (2.5.4)

⌟

f

X

W

and a modality (L, R) on a topos E. 8

Definition 2.5.5. The square (2.5.4) is said to be L-cartesian if the gap map (f, g) ∶ Z → X ×W Y is in L. The square is called L-cocartesian if the cogap map ⌊f, g⌋ ∶ X ∪Z Y → W is in L. The companion paper [ABFJ17] was written to prove the following two facts. Theorem 2.5.6. Let diagram (2.5.4) be a pushout square. Suppose that ∆f ◻Z ∆g ∈ L. Then the square is L-cartesian. Theorem 2.5.7. Let diagram (2.5.4) be a pullback square. Suppose that the map f ◻ g ∈ L. Then the square is L-cartesian.

3

Orthogonality conditions

⊩

⊩

⊩

The purpose of this section is to introduce in 3.2 the condition of fiberwise orthogonality, denoted ñ. This notion extends the classical notion of orthogonality related to unique lifting properties and unique factorization systems, denoted ⊥ as well as its internalisation . Although our focus is mainly on ñ, it is convenient to formulate its properties as properties of ⊥. So we provide some recollection on the matter. The relation is introduced only for comparison purposes and to avoid any confusion between ñ and . Let us point out that our motivation for introducing fiberwise orthogonality and the fiberwise diagonal is to prove Proposition 3.3.6. This eventually leads to Theorem 4.3.12 that is our new ingredient to Goodwillie calculus that lets us prove the Blakers-Massey Theorem for the Goodwillie tower. We shall fix in the whole section a topos E. We shall denoted by ∅ and ˚ respectively the initial and terminal object of E. For sake of simplicity, all the constructions of the section will be defined in E, even though but most of them can be defined in categories with less structure (for example in any locally cartesian closed category). Notation 3.0.1. Given a map f ∶ X → Y in a topos E, it can be viewed as object f in EY . We will sometimes abuse notation and denote the corresponding object in E/Y simply by X. If another map y ∶ Z → Y is given, we will denote by fZ the base change of f along y to E/Z , i.e. fZ = y ∗ (f ) ∶ X ×Y Z → Z. In the proof of Proposition 3.3.6 a different convention is used and explained there. idZ ×f For an object Z and a map f ∶ A → B we write Z × f ∶ Z × A ÐÐÐ→ Z × B.

9

3.1

Internal orthogonality

⊩

⊩

⊩

⊩

The notion of internal orthogonality is obtained if one replaces the enrichment over spaces in the definition of external orthogonality 2.4.4 by internal hom objects. We will say that two maps f ∶ A → B and g ∶ X → Y are internally orthogonal, and write f g, if the map ⟪f, g⟫ is an isomorphism in E. Similarly we say that f is internally left orthogonal to g and that g is internally right and S analogously. orthogonal to f . As above one can define classes S Since each slice E/Z has its own internal hom objects it has an external orthogonality relation which we will denote by Z Lemma 3.1.1. The following conditions are equivalent: g

⊩

(1) f

(2) For any Z ∈ E we have (Z × f ) ⊥ g. g implies f ⊥ g.

⊩

In particular, f

Proof. By Lemma 2.4.3 there is an isomorphism ⟨∅ → Z, ⟪f, g⟫⟩ = ⟨(∅ → Z) ◻ f, g⟩ = ⟨Z × f, g⟩. Hence, (1) ⇒ (2). For the converse we compute for h ∶ A → B [Z,h]

⟨∅ → Z, h⟩ = [Z, A] ÐÐÐ→ [Z, B].

⊩

Thus, h is an isomorphism if and only if for all Z the map ⟨∅ → Z, h⟩ is an isomorphism. If we insert h = ⟪f, g⟫ into the first isomorphism above we obtain (2) ⇒ (1). Setting Z = ˚ yields the implication f g ⇒ f ⊥ g. The following lemma proves that base change commutes with internal homs. Lemma 3.1.2. For three objects A, B and C in any topos E, we have a canonical isomorphism ⟦A, B⟧ × C = ⟦A × C, B × C⟧C in E/C where ⟦−, −⟧C is the internal hom in E/C . Proof. This is proven by adjunction: let D be an object in E/C , then we have the following equivalences D → ⟦A, B⟧ × C

in E/C

D → ⟦A, B⟧

in E

A×D →B

in E

(A × C) ×C D → B × C

in E/C

D → ⟦A × C, B × C⟧C

in E/C .

10

The following lemma proves that these internal orthogonality relations are compatible with base change.

⊩

⊩

Lemma 3.1.3. For any two maps f ∶ A → B and g ∶ X → Y in E, and for any object Z ∈ E we have f g Ô⇒ fZ Z gZ Moreover, this is an equivalence if u is an effective epimorphism. Proof. Using Lemma 3.1.2, we can show that ⟨f, g⟩Z = ⟨fZ , gZ ⟩. Then if ⟨f, g⟩ is an isomorphism, so are ⟨f, g⟩Z and ⟨fZ , gZ ⟩. The last assertion follows because isomorphism decend along effective epimorphisms.

3.2

Fiberwise orthogonality

We introduce our main notion of orthogonality. Intuitively, two maps f ∶ A → B and g ∶ X → Y will be said to be fiberwise orthogonal if each fiber of f is orthogonal to each fiber of g as objects. However, in order to make this precise we need to work with ”generalized fibers”, i.e. base changes of maps but viewed as objects in slice categories. Definition 3.2.1. We will say that two maps f ∶ A → B and g ∶ X → Y are fiberwise orthogonal, which we will denote f ñ g, if, for any Z ∈ E and any maps b ∶ Z → B and y ∶ Z → Y , it is true in E/Z that ⊩

fZ

Z

gZ .

Lemma 3.1.3 proves also that fiberwise orthogonality is a property stable by base change. We list several characterizations of fiberwise orthogonality. Proposition 3.2.2. Given two maps f ∶ A → B and g ∶ X → Y in E, the following conditions are equivalent: (1) f ñ g.

⊩

(2) The base changes of f and g onto B × Y along the projections to B and Y satisfy fB×Y B ×Y gB×Y . (3) The diagonal map ∆fB×Y (gB×Y ) ∶ gB×Y → ⟦fB×Y , gB×Y ⟧B×Y (see Example 2.4.1) is an isomorphism in E/B×Y . (4) For any Z → B × Y and any T → Z we have fT ⊥Z gZ .

11

(5) For any Z ∈ E and any maps b ∶ Z → B and y ∶ Z → Y , it is true in E/Z that fZ ⊥Z gZ . (6) For any two maps Z → B and Z ′ → B we have fZ ⊥ gZ ′ . (7) For any map Z → B we have fZ ⊥ g. Proof. (1) ⇒ (2) This is obvious since (2) is a special case of (1). (2) ⇒ (1) This follows from Lemma 3.1.3 that states that orthogonality is stable by base change. (2) ⇔ (3) This is equivalent by the definition of orthogonality in EB×Y . (1) ⇔ (4) This is Lemma 3.1.1 applied to the topos E/Z . (4) ⇒ (5) Set T → Z = idZ . (5) ⇔ (6) We need to prove that for all Z and all B ← Z → Y , fZ ⊥Z gZ ⇐⇒ ∀ U → B, ∀ T → Y, fU ⊥ gT . We consider the following diagram U ×B A

U ×Y X gU

fU

U

T ×Y X

⌜

gT h

U

T

where h is arbitrary and the right square is cartesian. Because the right square is cartesian, the space of diagonal fillers of the outer square is equivalent to that of the left square. When h varies, the former condition gives fU ⊥ gT and the latter fU ⊥U gU , hence proving their equivalence. (6) ⇔ (7) Since it is clear that (6) ⇒ (7), we need to show the other implication. Let fU be the base change of f along some map U → B, and gT the base change of g along some map h ∶ T → Y , we consider the following diagram where the left square is commutative and the right square is cartesian U ×B A

T ×Y X ⌜ gT

fU

U

T

X g h

Y.

Again, because the right square is cartesian, the space of diagonal fillers of the outer square is equivalent to that of the left square, which proves (7) ⇒ (6). (7) ⇒ (4) Let us consider the following diagram T ×B A fT

k

Z ×Y X ⌜ gZ

X

Z

Y.

T 12

g

where the right square is cartesian and k is any map such that the left square is commutative. Condition (4) says that for any such k the space of fillers of the left square is contractible. Since the right square is cartesian this is equivalent to the outer square having a contractible space of fillers. But Condition (7) states that any map from fT to g, i.e. a commutative square, has a contractible space of fillers. So (7) implies (4). Example 3.2.3. Let A and X be two objects of C. The maps A → ˚ and X → ˚ are externally orthogonal, if any map A → X factors through ∗. These maps are internally orthogonal if the map X → ⟦A, X⟧ is an isomorphism. In this special case, fiberwise orthogonality is the same as internal orthogonality. The comparison between the conditions of 3.2.2(7) and 3.1.1(2) shows the difference between the fiberwise orthogonality and the internal notion. Lemma 3.2.4. A factorization system is a modality if and only if its left and right class determine each other via fiberwise orthogonality. Proof. The fiberwise orthogonality forces the left class to be closed by base change and vice versa. This is exactly what the equivalence between conditions (1) and (7) of Proposition 3.2.2 means.

3.3

The fiberwise diagonal map

We saw that the external and internal orthogonality of two maps f and g can be detected by the condition that some map (⟨f, g⟩ or ⟪f, g⟫) be an isomorphism. The same thing is true for the fiberwise orthogonality, although the construction of the corresponding map is a bit more involved. Definition 3.3.1. Take two maps f ∶ A → B and g ∶ X → Y in E; pull them back to the common target B × Y , i.e. consider the maps fB×Y = f × idY ∶ A × Y → B × Y and gB×Y = idB ×g ∶ B × X → B × Y and view them as objects over B ×Y . In the slice E/B×Y one can form the fB×Y diagonal of gB×Y already used in 3.2.2(3). We will denote this diagonal by {f, g} and name it the fiberwise diagonal map. In the notation of Example 2.4.1 we have {f, g} = ∆fB×Y (gB×Y ) = ⟪fB×Y , gB×Y ⟫B×Y , where the internal pullback hom on the right is taken in the topos E/B×Y . Explicitly, L M P A×Y Q B×X B×X P Q P Q P Ð→ , {f, g} ∶ . (idB ,g) (f,idY ) (idB ,g) Q P Q P Q P B×Y Q B×Y B×Y N OB×Y The map {f, g} will be viewed as a map in E/B×Y . 13

Remark 3.3.2. Let b ∶ ˚ → B and y ∶ ˚ → Y be points of B and Y . We denote by fb and gy the corresponding fibers of f and g. Since in a topos E colimits commute with base change, the fiber of {f, g} at (b, y) can be proven to be the diagonal map gy → ⟦fb , gy ⟧. This is one of the reasons why we call this map the fiberwise diagonal map. Proposition 3.3.3. Let f and g be maps in E, then f is fiberwise orthogonal to g, i.e. f ñ g, if and only if {f, g} is an isomorphism. Proof. This is exactly the content of 3.2.2(3). Lemma 3.3.4. For all A, C and B → C in any topos, the following square ⟦A × C, B⟧C

⟦A, B⟧

⌜ ⟦A, C⟧,

C

where the bottom map is the diagonal map, is a pullback. Proof. Using C = ⟦A × C, C⟧C at the bottom left, we can factor the square as ⟦A × C, B⟧C

⟦A, B⟧ × C

⟦A, B⟧

⟦A × C, C⟧C

⟦A, C⟧ × C

⟦A, C⟧,

Then, the right square is obviously cartesian. To prove that the left square is also cartesian we need the isomorphism ⟦A, B⟧ × C = ⟦A × C, B × C⟧C from Lemma 3.1.2. The left square is cartesian as the image of the cartesian square in E/C B

B×C

C

C ×C

by the functor ⟦A × C, −⟧C which preserves limits. Lemma 3.3.5. The square ⟦X × Z Y , Z⟧Z Y

ZX

⌜ ZY

Z X×Y

is a pullback. Hence, there is a canonical isomorphism Z X⋆Y = ⟦X × Z Y , Z⟧Z Y . 14

Proof. Setting A = X, B = Z and C = Z Y in the previous lemma we find that the square above is a pullback as claimed. Since the join is the pushout of the projections X ← X × Y → Y, the pullback of this square is canonically isomorphic to Z X⋆Y . Proposition 3.3.6. The following formula is true in any topos: {f ◻ g, h} = {f, {g, h}}. Proof. We consider first the special case where the maps are of the following form f ∶ X → ˚ , g ∶ Y → ˚ , h ∶ Z → ˚. Then the map {f ◻ g, h} becomes the X ⋆ Y -diagonal of Z {X ⋆ Y → ˚, Z → ˚} = Z → Z X⋆Y . On the other hand, the map {f, {g, h}} becomes {X → ˚, Z → Z Y } = Z → ⟦X × Z Y , Z⟧Z Y . Lemma 3.3.5 shows that these two maps are the same. This proves our claim in the special case. We prove the general case by arguing fiberwise, i.e. by viewing our maps as objects in the respective slice categories and then appealing to the special case above. We introduce the following notation (differing from 3.0.1). First, we will denote the cartesian product of two objects I and J in E by concatenation IJ. Then, for a map f ∶ X → I in a topos E, we will abuse notation and denote by X the corresponding object in E/I . If another objet J ∈ E is given, we will denote by XJ the base change of X ∈ E/I to E/IJ along the projection I × J → I, i.e. XJ is the map X × J → I × J. For two maps f ∶ X → I and g ∶ Y → J, the map f ◻ g in E corresponds to the object XJ ⋆ YI in E/IJ , where the join is also computed in E/IJ . For a third object K, it is easy to compute that (XJ ⋆ YI )K = XJK ⋆ YIK in E/IJK . Similarly, for two maps g ∶ Y → J and h ∶ Z → K, the map {g, h} is defined as the map in E/JK ⟨YK → ˚, ZJ → ˚⟩ where the pullback hom is computed in E/JK . For a third object I ∈ E, because the pullback functor E/JK → E/IJK preserve exponential, we have also (⟨YK → ˚, ZJ → ˚⟩)I = (ZJ → ⟦YK , ZJ ⟧)I = ZIJ → ⟦YIK , ZIJ ⟧ = ⟨YIK → ˚, ZIJ → ˚⟩ 15

in E/IJK . Finally, we obtain the following canonical isomorphisms: {f ◻ g, h}

viewed as a map in E/IJK

= ⟨(XJ ⋆ YI )K → ˚, ZIJ → ˚⟩

join in E/IJ , bracket in E/IJK

= ⟨XJK ⋆ YIK → ˚, ZIJ → ˚⟩

computed in E/IJK

= ⟨XJK → ˚, ⟨YIK → ˚, ZIJ → ˚⟩⟩

special case applied to the topos E/IJK

= ⟨XJK → ˚, (⟨YK → ˚, ZJ → ˚⟩)I ⟩ inside bracket computed in E/JK = {f, {g, h}}

4

viewed as a map in E/IJK .

Goodwillie’s Calculus of Functors in a Higher Topos

To apply the generalized Blakers-Massey theorem in [ABFJ17] and obtain an analogous theorem for the Goodwillie tower one needs to prove that the various modalities associated to Goodwillies approximation Pn , n ≥ 0, are compatible with the pushout product. This is achieved in Theorem 4.3.12(4). The whole section is devoted to developping the relevant machinery. As it turns out, a new way of thinking about the Goodwillie tower arises, see Remark 4.3.11. We write C for a small category with finite colimits and a terminal object ˚. Two examples are the category Fin of finite unpointed spaces and Fin˚ of finite pointed spaces. Goodwillie calculus arises from the study of certain left-exact localizations of the presheaf topos [C, S].

4.1

Joins and n-excisive functors

Here we briefly review some definitions and constructions from calculus [Goo03]. The central notion of Goodwillie Calculus is that of an n-excisive functor. Definition 4.1.1 (Goodwillie). Let F ∶ C → S be a functor. We say that F is n-excisive if for every strongly cocartesian (n + 1)-cubical diagram X in C, the composite F ○ X is cartesian. A given strongly cocartesian diagram X is completely determined by the family of maps {X(∅) → X({k})}1≤k≤n . Consequently, we may identify the category of strongly cocartesian n-cubes X such that X(∅) = K with the n-th cartesian power of the coslice category (CK/ )×n . This category clearly has a terminal object, namely the n-cube TnK determined by TnK (∅) = K and TnK ({k}) = ˚ for 1 ≤ k ≤ n. This strongly cocartesian cube has an explicit description using the following

16

Definition 4.1.2. For an object K in C and a finite set U we define the join of K with U by K K ⋆ U = colim ...

˚

˚

By assumption the category C possesses finite colimits, hence joins exist. If C happens to be a subcategory of the slice E/Y over some Y this construction yields a formula for K ⋆Y U provided that one remembers that in this case the terminal object ˚ is Y . Remark 4.1.3. We give Definition 4.1.2 to have fewer assumptions on the source category C. For categories C that also have finite products, the join is a special case of the pushout product as explained in Example 2.3.1(3). The two definitions are compatible. First, one needs to promote the set U to an object in C. Since C has a terminal object ˚ and finite colimits, the sum ∐U ˚ make sense. Now, for any K in C, K ⋆ (⊔U ˚) in the sense of 2.3.1(3) is defined. It is equal to K ⋆ U in the previous sense. The proof for the case ∣U ∣ = 2 can be obtained from the following diagram: ˚

∅

˚

K

∅

K

K

K

K

By computing the colimit first vertically then horizontally one obtains Definition 4.1.2, the other way around gives the pushout product of 2.3.1(3). For other sets U one adjusts the diagram. We remined the reader that ∅ is the initial object of C (and not necessarily empty) and ˚ is the terminal object (and not necessarily the point). If for example C is S/Y then ˚ = Y and ∐U ˚ = Y × U . Then Definition 4.1.2 gives Goodwillie’s fiberwise join [Goo03, p. 656] (K ⋆Y U ) → Y = (K → Y ) ◻ (U → ˚) = (K → Y ) ◻Y (Y × U → Y ). The discussion that follows is valid in such slice categories even though it does not explicitly show up in the notation. Remark 4.1.4. The terminal strongly cocartesian n-cube TnK with initial object K is given by TnK (U ) = K ⋆ U. This is the description used by Goodwillie.

17

Definition 4.1.5. For a functor F ∶ C → S and K in C one defines a new functor Tn F (K) =

lim U ∈P0 (n+1)

F (K ⋆ U ) =

lim U ∈P0 (n+1)

K F (Tn+1 (U ))

and a natural transformation tn F ∶ F → Tn (F ) obtained from the canonical map F (K) ≅ F (K ⋆ ∅) → lim F (K ⋆ U ) = Tn F (K). U ≠∅

Definition 4.1.6 (Goodwillie). For each homotopy functor F ∶ C → S define the new functor tn Tnk F

Pn F = colim(Tn0 F → ⋯ → Tnk F ÐÐÐÐ→ Tnk+1 F → ⋯), which comes equipped with the natural transformation pn F ∶ F → Pn F. Proposition 4.1.7. (Goodwillie [Goo03, Lemma 1.9], Rezk [Rez13]) Let X be an arbitrary strongly cocartesian (n + 1)-cube. For every F the natural transformation tn F (X) ∶ F (X) → Tn F (X) factors through a cartesian cube. Rezk’s “streamlined” proof of this fact is short, elegant, and very general. Corollary 4.1.8. (Goodwillie) For every F in [C, S] the functor Pn F is nexcisive. Proof. Let X be a strongly cocartesian (n + 1)-cube. By Propositon 4.1.7 each map tn Tnk F (X) ∶ Tnk F (X) Ð→ Tnk+1 F (X) factors through a cartesian cube. It follows that in the colimit defining Pn F each map factors through a cartesian cube, and hence the colimit is a cartesian cube. Thus, the functor Pn F sends any strongly cocartesian (n + 1)-cube to a cartesian one; Pn F is indeed n-excisive. Goodwillie [Goo03, Thm. 1.8] then shows that the map pn F ∶ F → Pn F is initial among all maps from F to n-excisive functors. However, we will not need this fact. Remark 4.1.9. It follows from Corollary 4.1.8 that both maps tn Pn F, Pn tn F ∶ Pn F → Tn Pn F are isomorphisms. In particular, Pn is idempotent. It is now an easy, but fundamental observation that Pn is a left exact localization of the topos [C, S]. 18

4.2

n-excisive maps

It is important for us to define a relative version of being n-excisive. To achieve this, our objective is to rephrase n-excisiveness in terms of representable funcK tors. The result will be, for every space K, a map γn+1 of functors such that mapping out of it into a functor F yields the map tn F (K). We finish by comK paring the various orthogonality conditions with respect to γn+1 . Definition 4.2.1. Given an object K in C, the covariant representable functor associated to K will be denoted by RK ∶ C → S , RK (X) = [K, X]. Definition 4.2.2. Given an (n + 1)-cubical diagram X ∶ P (n + 1) → C, observe that composition with the contravariant Yoneda embedding y yields a diagram Xop

y

→ [C, S] P (n + 1)op ÐÐ→ Cop Ð so that (y ○ Xop )(U ) = RX(U ) . We define ΓX =

colim RX(U ) U ∈P0 (n+1)

and observe that we have a canonical map γ X ∶ ΓX → RX(∅) , K which is in fact the cogap map of the cube y ○ X. When X = Tn+1 , then we write K K K Tn+1 K Tn+1 . , and Γn+1 instead of Γ γn+1 instead of γ

Let F ∶ C → S be a functor Recall that, by Yoneda lemma, F (X(∅)) ≅ [RX(∅) , F ] and

[ΓX , F ] = [colim RX(U ) , F ] ≃ lim [RX(U ) , F ] ≃ lim F (X(U )). U

U

U

Then, unfolding its definition, we find that the external pullback hom ⟨γ X , F → ˚⟩ ∶ [RX(∅) , F ] → [ΓX , F ] ×[ΓX ,˚] [RX(∅) , ˚] is the gap map

⟨γ X , F ⟩ ∶ F (X(∅)) → lim F (X(U )) U

K of the cube F (X). In the special case of the strongly cocartesian cube Tn+1 this construction yields a familiar objects:

[RTn+1 (∅) , F ] ≅ F (K) , [ΓK n+1 , F ] ≅ Tn F (K) K

and we can identify the pullback hom with Goodwillie’s map K tn F = ⟨γn+1 , F ⟩ ∶ F (K) → Tn F (K).

19

Remark 4.2.3. Note that 1. γ X ⊥ (F → ˚) if and only if F sends X to a cartesian cube, and that K K 2. γn+1 ⊥ (F → ˚) if and only if F sends the cube Tn+1 to a cartesian one.

Definition 4.2.4. A map f ∶ F → G in [C, S] is Pn -local if the following diagram F

pn F

Pn F

f

G

Pn f pn G

Pn G

is a pullback. We say that f is Tn -local if the analogous square for Tn is a pullback. Theorem 4.2.5. Let f ∶ F → G be a map in [C, S]. The following statements are equivalent: K (1) For all K in C we have γn+1 ⊥ f.

(2) The map f is Tn -local. (3) The map f is Pn -local. (4) For all strongly cocartesian (n + 1)-cubes X we have γX ⊥ f . K Proof. (1) ⇔ (2) A moment’s thought reveals that the map ⟨γn+1 , F ⟩ is the cartesian gap map of the commutative square

F (K)

G(K)

lim F (K ˚ U )

U ≠∅

lim G(K ˚ U ),

U ≠∅

where the limit is taken over U ∈ P0 (n + 1). Both conditions (1) and (2) assert precisely that this map is an isomorphism for all K. (2) ⇒ (3) If f is a pullback of Tn f then it is a pullback of all composites Tnk f because Tn preserves finite limits. Since finite limits commute with filtered colimits in S, f is a pullback of Pn f = colimk Tnk f , ie. f is Pn -local. (3) ⇒ (4) Now assume that f is Pn -local and let X be a strongly cocartesian

20

(n + 1)-cube. Write K = X(∅). Consider the following commutative diagram: Pn F (K) F (K)

limU ≠∅ Pn F (X(U )) limU ≠∅ F (X(U )) limU ≠∅ Pn G(X(U ))

Pn G(K)

G(K)

limU ≠∅ G(X(U ))

We need to show that the front is a pullback. The right and left faces are a pullback by assumption. The back square is trivially a pullback: both horizontal maps are isomorphisms because Pn F and Pn G are n-excisive functors. Thus, the composite diagonal square is a pullback. Hence, the front is also a pullback. K Finally, (4) ⇒ (1) is obvious because, for all K, the cubes Tn+1 are strongly cocartesian. Definition 4.2.6. A map f ∶ F → G in [C, S] is n-excisive if it satisfies one of the equivalent conditions from Theorem 4.2.5. A functor F is n-excisive if and only if the map F → ˚ is n-excisive. For any map f ∶ F → G, the induced map Pn f ∶ Pn F → Pn G is n-excisive because Pn f is obviously Pn -local: Pn Pn f ≃ Pn f by Corollary 4.1.9. As a special case one obtains K Corollary 4.2.7. For all K in C the map γn+1 is a Pn -equivalence.

Definition 4.2.8. A map w in [C, S] is a Pn -equivalence if the induced map Pn w is an isomorphism. Corollary 4.2.9. For every n ≥ 0 the pair (Pn -equivalences, n-excisive maps) is a modality. In particular: (1) The class of Pn -equivalences is closed under pullback and 2-out-of-3. (2) A map f is n-excisive if and only if w ⊥ f for all Pn -equivalences w. (3) A map w is a Pn -equivalence if and only if or w ⊥ f for all n-excisive f . Proof. By Remark 4.1.9 Pn is a left exact localization. Proposition 2.5.2 and Example 2.5.3 exhibit an associated modality. Here Pn -local maps form the right class, but they are the same as n-excisive maps by Theorem 4.2.5. Statement (1) follows directly from left exactness. Statements (2) and (3) just express the lifting property that left and right classes satisfy in a modality. Proposition 4.2.10. For a map f in [C, S] the following statements are equivalent: 21

K (1) For all K in C γn+1 ⊥ f.

f.

⊩

K (2) For all K in C γn+1

K (3) For all K in C γn+1 ñ f.

Proof. The implication (3) ⇒ (2) is proved in Proposition 3.2.2, and (2) ⇒ (1) is proved in Lemma 3.1.1. We need to show (1) ⇒ (3). If f satisfies (1) then it is n-excisive by definition. By Corollary 4.2.9 it is right orthogonal to all Pn -equivalences and they are closed under base change. In particular, it is right orthogonal to all base K changes of the maps γn+1 for all K in C by Corollary 4.2.7. This is equivalent to (3) by 3.2.2.

4.3

The map wK and its pushout product powers

K In the previous subsection we supplied a generating set of maps {γn+1 ∣ K ∈ C} for the localization Pn . Here we are going to produce another set of generators that is obviously compatible with the pushout product leading eventually to Theorem 4.3.12.

Definition 4.3.1. Let K be in C. The terminal map K → ˚ induces a canonical map wK ∶ R˚ → RK for every K. We denote the (n + 1)-fold pushout product of wK with itself by ◻n+1 wK ∶ Wn+1 (RK ) → (RK )×n+1 ,

where Wn+1 (RK ) is our notation for the source of this map. The conventions ◻1 ◻0 wK = wK and wK = (∅ → ˚) apply. Example 4.3.2. Let us give two examples. 1. If C is pointed then R˚ = ˚ and Wn+1 (RK ) is the (n + 1)-fold fat wedge of ◻n+1 RK with itself. For C = S the cofiber of the map wK is the (n + 1)-fold K objectwise smash product of R with itself. 2. If C is the category of unpointed spaces then R˚ is its identity functor. ◻n+1 We want to characterize maps that are fiberwise right orthogonal to wK in Corollary 4.3.8 explicitly. In preparation for this we define a special class of strongly cocartesian cubes. Recall that we denote the coproduct by ⊔ no matter whether the category is pointed or not.

Definition 4.3.3. Let (K0 , ⋯, Kn ) be an (n + 1)-tuple and L another object in C. Let n

κ = κ{0,⋯,n} ∶ ∐ Ki → L i=0

22

be an arbitrary map. For each inclusion ιU ∶ U ⊂ {0, ⋯, n} there is a pushout κ

n

∐i=0 Ki

L

ι∗U

`U

∐i∉U Ki ⊔ ∐i∈U ˚

κU

LU

in C defining LU . We write Lκn+1 for the strongly cocartesian (n + 1)-cube U ↦ LU . By construction, Lκn+1 (∅) = L. If κ is the identity of ∐ni=0 Ki and C = Fin∗ then the cube Lκn+1 = Lid n+1 is the cube used by Goodwillie to compute the (n + 1)-st cross effect. It is K straightforward to see that a map F → ˚ is externally orthogonal to wn+1 if and id only if the gap map of the cube F (Ln+1 ) is an isomorphism. In particular, the cross effect crn+1 F (K, ⋯, K) vanishes. This does not imply that F is n-excisive. One can work out what happens if one uses internal orthogonality, and it is still not enough. This is the reason why we introduce fiberwise orthogonality: being K fiberwise orthogonal to wn+1 for all K is equivalent to F being n-excisive! This claim will be justified by Proposition 4.3.10 below. The reason can already be K seen in the next lemma: the cube Tn+1 is among the cubes of the form Lκn+1 . Lemma 4.3.4. If K0 = ⋯ = Kn = K and κ is the (n + 1)-fold codiagonal of K K = L the cube Lκn+1 is isomorphic to the cube Tn+1 that yields the functor Tn . Proof. We need to show that the cube given by ∇ ⎛ ∐n K ⎜ ⎜ U ↦ colim ⎜ ι∗U ⎜ ⎜ ⎝ ∐i∉U K ⊔ ∐i∈U ˚

⎞ K ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

K is isomorphic to the cube Tn+1 . Here, ∇ is the codiagonal. This follows from Lemma 4.3.6 where the 2-dimensional face given by U ⊂ U ∪ {j} with ∣U ∣ = m and j ∉ U is treated.

We need to prepare for the Lemma 4.3.6. Recall that by our assumptions C has a terminal object ˚ and finite colimits. It follows that C has an initial object ∅ and admits join products with finite sets. For a natural number m let m ∶= ˚ ⊔ ⋯ ⊔ ˚ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ m times

also denote the m-fold coproduct of the terminal object of C with itself. Using this notation there are maps −+1∶m→m+1

23

in C adding another summand on the right and αK,n,m ∶ ⊔ K ⊔ m → K ⋆ m n

induced by the n-fold codiagonal K ⊔ ⋯ ⊔ K → K and the identity on m. For K in C and natural numbers m and n there is a commutative square K→˚

⊔K ⊔ m

⊔ K ⊔ (m + 1)

n

n−1

(4.3.5)

αK,n−1,m+1

αK,n,m id ⋆(1+−)

K ⋆m

K ⋆ (m + 1),

where the upper horizontal map is induced by collapsing the last K-summand to the terminal object. Lemma 4.3.6. The square (4.3.5) is a pushout. Proof. We first write each of the terms in the diagram as a colimit of a diagram of the following shape: X K

...

K

Y

˚

...

˚

with n − 1 copies of K on the left, m copies of ˚ on the right, and where the objects X and Y will change depending on the position in the square (4.3.5): • for the top left corner X = ∅ and Y = K, • for the top right corner X = ∅ and Y = ˚, • for the bottom left corner X = K and Y = K, • for the bottom right corner X = K and Y = ˚. Morphisms are identites whenever possible. Now we can compute the pushout at each of the slots and obtain the diagram K K

...

K

˚

˚

...

˚

whose colimit is K ⋆ (m + 1). Example 4.3.7. The proofs of Lemma 4.3.4 and 4.3.6 are valid in both the pointed and unpointed context. In the case C = Fin the left hand cube L∇ 2 yields the right hand cube by cobase change along ∇ ∶ K ⊔ K → K: K ⊔K

K ⊔˚

K

˚

˚⊔K

˚⊔˚

˚

ΣK

24

If C is pointed then ∅ = ˚ = m for all m. Thus, for C = Fin˚ , the cobase change along the codiagonal ∇ ∶ K ∨ K → K produces the right square from the left square: K ∨K K K ˚ K

˚

˚

ΣK

This pattern persists for higher cubes as proved above. This is a way to obtain K the cubes Tn+1 used to define n-excisiveness from the cubes Lid n+1 used to define the (n + 1)-st cross effect. Corollary 4.3.8. Let f ∶ F → G be a map in [C, S]. Then the following statements are equivalent: K (1) For all K in C we have wn+1 ñ f.

(2) For all cubes Lκn+1 , where κ∶ K ⊔ ⋯ ⊔ K → L is arbitrary, the square F (L)

∅≠U ⊂{0,⋯,n}

F (Lκn+1 (U ))

lim

G(Lκn+1 (U ))

lim

G(L)

∅≠U ⊂{0,⋯,n}

is a pullback. In particular, f is n-excisive. Proof. (1) ⇒ (2). By Proposition 3.2.2 condition (1) is equivalent to f being ◻n+1 externally right orthogonal to wK and all its base changes. From the defini◻n+1 tion of wK one sees that f yields pullback squares as above when evaluated on all the cubes Lid n+1 . External right orthogonality with respect to a base change ◻n+1 of wK along a map κ∗ ∶ RL → (RK )n means exactly that f yields pullbacks when evaluated on cubes of the form Lκn+1 . (2) ⇒ (1). Condition (2) says that f is right orthogonal to base changes of K wn+1 for all K along all maps RL → (RK )n . Since any functor is the colimit of representable ones this suffices to imply right orthogonality with respect to ◻n+1 arbitrary base changes of wK . This is condition (1). K Finally, by Lemma 4.3.4 the cubes Tn+1 are among the cubes Lκn+1 . So (2) implies that f is Tn -local, hence n-excisive by Theorem 4.2.5. For some object X and each n ≥ 2 we denote the n-fold diagonal map by ∆(n) ∶ X → X ×n . Corollary 4.3.9. We have a pullback diagram Wn+1 (RK )

ΓK n+1 K γn+1

RK

⌜ ∆

◻n+1 wK (n+1)

25

(RK )×n+1

Proof. This follows directly from Lemma 4.3.4: the corresponding suqare of representing objects is a pushout. This is formally the same as the construction of the n-th Ganea fibration K in spaces. For this reason we call the map γn+1 ∶ ΓK the n-th Ganea n+1 → R K fibration of R . Let us point out again that the external pullback hom tn F (K)

K ⟨γn+1 , F → ˚⟩ = F (K) ÐÐÐÐ→ Tn F (K)

is the map tn F (K) from [Goo03]. On the other side, in the case of C = Fin˚ , the external pullback hom ◻n+1 ⟨wK , F → ˚⟩ = F ( ⋁ K) → n+1

lim S∈P0 (n+1)

F( ⋁

K)

i∉n+1−S

yields the map whose fiber is crn+1 F (K, ⋯, K). If the first map is an isomorphism for all K, then F is n-excisive and hence its (n+1)-st cross effect vanishes. To require that the second map is an isomorphism for all K is slightly stronger than the vanishing of the cross effect, but it is still not enough to force F to be n-excisive. One needs fiberwise orthogonality. The previous lemma was a pleasant surprise to us when we discovered it. The following proposition is an even bigger surprise; it is the central fact that lets us prove a Blakers-Massey analogue for the Goodwillie tower. After our preparations on fiberwise orthogonality though it is not hard to prove anymore. Proposition 4.3.10. For a map f in [C, S] the following statements are equivalent: K (1) For all K in C we have γn+1 ⊥ f. ◻n+1 (2) For all K in C we have wK ñ f.

Proof. The implication (2) ⇒ (1) is proved in Corollary 4.3.8. Alternatively, it follows from Corollary 4.3.9 since f has to be orthogonal to all pullbacks of ◻n+1 wK . (1) ⇒ (2). Theorem 4.2.5 states that (1) implies γ X ⊥ f for any strongly cocartesian (n + 1)-cube X. In particular, condition (2) of Corollary 4.3.8 is ◻n+1 satisfied, ie. wK ñ f for arbitrary K. Remark 4.3.11. Condition (2) in the previous theorem gives a new perspective on the Goodwillie tower. The hierarchy of connected/truncated maps that yields the classical Blakers-Massey theorem and the Postnikov tower is generated by the map S 0 → ˚ in the sense that every n-truncated map is detected by orthogonality with respect to (S 0 → ˚)◻ n+1 . In the same sense the Goodwillie tower is generated by the pushout product powers of the maps wk ∶ R˚ → RK for all K in the source category C.

26

The fiberwise diagonal {f, g} of two maps f and g is defined in 3.3.1. By Proposition 3.2.2 it is an isomorphism if and only if the maps f and g are fiberwise orthogonal. A crucial role in the proof of the next theorem because is played by the formula {f ◻ g, h} = {f, {g, h}} demonstrated in Proposition 3.3.6. It allows us to use adjunction tricks for fiberwise orthogonality. The reader is invited to compare the next theorem with [ABFJ17, Cor. 3.15] where the n-connected/n-truncated modalities for n ≥ −2 are treated. Theorem 4.3.12. For s ≥ m ≥ n let f be a Pm -equivalence, g a Pn -equivalence, h an p-excisive map, and K in C. ◻n (1) The map {wK , h} is (p − n)-excisive. ◻n (2) The map f ◻ wK is a Pm+n -equivalence.

(3) The map {f, h} is (p − m − 1)-excisive. (4) The map f ◻ g is a Pm+n+1 -equivalence. Proof. Let m ≥ n, f be a Pm−n -equivalence, g a Pn−1 -equivalence, h an mexcisive map, and K in C. The assertions above follow by appropriate reindexing. For (1) we note that by Proposition 4.3.10 the map ◻ m+1 ◻ m−n+1 ◻n {wK , h} = {wK , {wK , h}}

is an isomorphism. By letting K vary over all objects in C this implies, again by ◻n Proposition 4.3.10, that {wK , h} is (m − n)-excisive. From this it immediately follows that ◻n ◻n {f ◻ wK , h} = {f, {wK , h}} is an isomorphism if f is a Pm−n -equivalence. By letting h vary over all mexcisive maps we arrive at (2). Using the other adjunction, the map ◻n ◻n {f ◻ wK , h} = {wK , {f, h}}

is still an isomorphism proving (3). Finally, for every m-excisive h the map {f ◻ g, h} = {g, {f, h}} is an isomorphism by (3). Thus, (4) holds. The compatibility of the Goodwillie tower with the pushout product stated in Theorem 4.3.12(4) is what we are really after. It will allow us to prove the Blakers-Massey analogue for the Goodwillie tower. One direct application is Example 4.3.13. Let F be m-reduced and G be n-reduced then the canonical map (˚ → F ) ◻ (˚ → G) = (F ∨ G → F × G) is a Pm+n−1 -equivalence. In the same way the map (F → ˚) ◻ (G → ˚) = (F ⋆ G → ˚) = (Σ(F ∧ G) → ˚) is a Pm+n−1 -equivalence. It follows that both F ⋆ G and F ∧ G are (m + n − 1)reduced. 27

4.4

Epimorphisms and Monomorphisms of Excisive Functors

We will write [C, S](n) for the full subcategory of n-excisive functors inside [C, S]. As already pointed out, it is a left exact localization of the presheaf topos [C, S], and hence, itself a topos. The inclusion ιn ∶ [C, S](n) → [C, S] is the right adjoint to the localization functor Pn . Our goal in this subsection is to describe the epimorphisms and monomorphisms in the topos [C, S](n) . Definition 4.4.1. In a topos E a monomorphism is a map f ∶ X → Y whose diagonal ∆f ∶ X → X ×Y X is an isomorphism. An effective epimorphism is a map that is externally left orthogonal to any monomorphism. In spaces a map is a monomorphisms if and only if it is the inclusion of a union of path components. The effective epimorphisms in spaces are exactly the maps that induce a surjection on π0 . Definition 4.4.2. In what follows, we write π0 for 0-truncation in [C, S] and (n) π0 for the 0-truncation functor in the n-excisive localization [C, S](n) . Remark 4.4.3. A left exact functor preserves monomorphisms. In particular, (n) Pn , ιn , π0 as well as π0 for all n preserve monomorphisms. It is important to note that the canonical inclusion ιn ∶ [C, S](n) → [C, S] does not preserve 0-truncation, except when n = 0 (because in this case, it is the inclusion of spaces as constant functors, where for higher n, one must apply the localization Pn again). On the other hand, since ιn preserves limits, it preserves n-truncated objects for any n. A consequence, which will be used below, is that discrete, n-excisive functors take values in sets. Here is a general characterization of monomorphisms and epimorphisms in a topos. Proposition 4.4.4. Let f ∶ X → Y be a morphism in a topos. (1) The map f is a monomorphism if and only if π0 f is a monomorphism and the square X π0 X ⌜ f π0 f Y

π0 Y

is a pullback. (2) The map f is an epimorphism if and only if π0 f is an epimorphism. Lemma 4.4.5. Let F be a 1-excisive functor which is discrete, i.e. which takes values in sets. Then F is in fact constant. More specifically F (K) = F (˚) for all finite pointed spaces K. 28

Georg: Give reference!

Proof. Note that for any K ∈ C, ΣK is a pointed object. Consequently, we have a factorization of the identity map on F (˚). F (˚)

F (ΣK)

F (˚)

Since each of these objects is a set, we conclude that the map F (˚) → F (ΣK) is an injective map of sets, and hence a monomorphism in S. By the definition of monomorphism, then, the square F (˚)

F (˚) ⌜

F (˚)

F (ΣK)

is a pullback. On the other hand, since F is 1-excisive, for any object K we have a pullback square: F (K)

F (˚) ⌜

F (˚)

F (ΣK)

hence the canonical map F (K) → F (˚) must be an equivalence by the uniqueness of pullbacks. Lemma 4.4.6. For F ∈ [C, S](1) , we have π 0 F ≃ π 0 P0 F (1)

(1)

Proof. Note that π0 F is discrete by definition, and thus in light of the remarks (1) above takes values in sets. The previous lemma then asserts that π0 F is in fact constant, hence the equivalence. Proposition 4.4.7. Let f ∶ F → G be an n-excisive monomorphism. Then f is 0-excisive. Proof. The proof is by induction on n ≥ 1. For n = 1, we are to show that every 1-excisive monomorphism is in fact 0-excisive. So let f ∶ F → G be a 1-excisive monomorphism. We claim first that, without loss of generality, we may assume that F and G are in fact 1-excisive functors. This is because every 1-excisive map sits in a pullback square of the form: F f

P1 F

⌜

P1 f

G

P1 G

29

But since P1 preserves monomorphisms and 0-excisive maps are closed under pullback, then if we can show P1 f is 0-excisive, it follows that f is as well. But of course, P1 f has the desired form: it is a monomorphism between 1-excisive functors. Now let f ∶ F → G be a monomorphism and 1-excisive map between 1excisive functors. Consider the cube: F

P0 F

(1)

≃

π0 F G

π 0 P0 F

P0 G

(1)

≃

π0 G

π 0 P0 G

(1)

All of the vertical maps are monomorphisms since P0 , π0 and π0 preserve them. Both the left and the right face are pullbacks by Proposition 4.4.4. By Lemma 4.4.6, the front two horizontal maps are in fact isomorphisms. Consequently, the back face is a pullback, which says that f is 0-excisive. For the inductive step, let f ∶ F → G be (n+1)-excisive and a monomophism. Note that the functor {wK , −} preserves monomorphisms. So, for all K, {wK , f } is a monomorphism. But it is also n-excisive by Theorem 4.3.12(1). By the induction hypothesis, it is then 0-excisive. This, in turn, shows that f is 1excisive. Then the case n = 1 implies that f is also 0-excisive. Remark 4.4.8. A consequence of the previous proposition is that the result of Lemma 4.4.6 is in fact true for all n. That is, for F ∈ [C, S](n) , we have π 0 F ≃ π 0 P0 F (n)

Theorem 4.4.9. Let f ∶ F → G be a morphism of [C, S](n) . Then: (1) The map f is monomorphism if and only if P0 f is a monomorphism and the square F f

⌜

G

P0 F P0 f

P0 G

is a pullback. 30

(2) The map f is epimorphism if and only if P0 f is epimorphism. Remark 4.4.10. We invite the reader to admire the symmetry between the statements of Proposition 4.4.4 and Theorem 4.4.9. Proof. (1 ⇒). This is immediate since P0 preserves monomorphisms and the pullback expresses just the statement that f is 0-excisive. (1 ⇐). Monomorphisms are always stable by pullback. (2 ⇒). The functor P0 preserves epimorphisms because it is cocontinuous. (2 ⇐). Note that f is an epimorphism if and only if it is orthogonal to every monomorphism in [C, S](n) . So let g ∶ H → K be such a monomorphism and consider a lifting problem as follows: F

H g

f

G

K

Note that since g is a monomorphism, it is enough to show that a lift exists, as its uniqueness is automatic. Now apply the functor P0 to obtain P0 F

P0 H ∃!

P0 f

P0 g

P0 G

P0 K

Observe that the left map is an epimorphism by assumption. Since P0 preserves monomorphisms, this square has a unique lift. But now composition of our lift with the map p0 G ∶ G → P0 G yields the lift shown in the diagram: F f

H

P0 H

g⌜

∃!

G

P0 g

K

P0 K

On the other hand, Proposition 4.4.7 asserts that the right hand square is a pullback. Hence we have an induced unique lift to the original problem. This shows that f is an epimorphism.

4.5

Blakers-Massey theorem for the Goodwillie tower

Theorem 4.5.1 (Blakers-Massey theorem for Goodwillie Calculus). Let F

g

H ⌟

f

G

K

31

be a pushout square of functors. If f is a Pm -equivalence and g is a Pn equivalence, then the induced map (f, g) ∶ F → G ×K H is a Pm+n+1 -equivalence. Proof. If a map h is k-excisive then its diagonal ∆h is also k-excisive because Pk is left exact. Theorem 4.3.12(4) then implies that ∆f ◻ ∆g is a Pm+n+1 equivalence: ∆f ◻ ∆g is in the left class of the modality associated to Pm+n+1 . Now we apply the Theorem 2.5.6 and learn that (f, g) is in the same left class. Hence, the gap map is a Pm+n+1 -equivalence. Theorem 4.5.2 (“Dual” Blakers-Massey theorem for Goodwillie Calculus). Let F

H ⌜

f g

G

K

be a pullback square of functors. If f is a Pm -equivalence and g is a Pn equivalence, then the cogap map ⌊f, g⌋ ∶ G ⊔F H → K is a Pm+n+1 -equivalence. Proof. By Theorem 4.3.12(4) the map f ◻ g is a Pm+n+1 -equivalence. By Theorem 2.5.7 the same holds for the cogap ⌊f, g⌋.

5 5.1

Consequences Delooping theorems

There is a canonical map qn F ∶ Pn F → Pn−1 F with an explicit model exhibited in [Goo03, p. 664]. Theorem 5.1.1. For a functor F , let us denote by C the pushout of the following diagram Pn F P0 F qn F

⌟

Pn−1 F

c

C.

in [C, S]. Then by applying Pn the induced square Pn F qn F

P0 F ⌜

Pn c

Pn−1 F

Pn C. 32

becomes a pullback square in [C, S](n) , and in turn in [C, S]. The left vertical map becomes a surjection in [C, S](n) . In particular, for a reduced functor F the map qn F is a principal fibration. Georg: Please, check

Proof. The top horizontal map is a P0 -equivalence and the left vertical map is a Pn−1 -equivalence. By Theorem 4.5.1 the gap map is a Pn -equivalence. This says that the square is a pullback after applying Pn as claimed. The inclusion of [C, S](n) to [C, S] preserves pullbacks. Using Theorem 4.4.9 repeatedly, observe that the map qn F is a surjection in [C, S](n) , and consequently, so is the right vertical map c. So their composite Pn c is a surjection. For a reduced functor P0 F = F (˚) = ˚. It follows that qn F is a principal fibration. Corollary 5.1.2. (Arone-Dwyer-Lesh [ADL08, Thm. 4.2]) For every n-reduced functor F the canonical map P2n−1 F → ΩP2n−1 ΣF is an isomorphism. If F is also (2n − 1)-excisive it is infinitely deloopable. Proof. The isomorphism follows by applying Theorem 4.5.1 to the pushout square F ˚ ⌟ ΣF.

˚ The second statement is now obvious. A special case of the previous fact is

Corollary 5.1.3. (Goodwillie [Goo03, Section 2]) If F is n-homogeneous then F ≅ ΩPn ΣF. Hence, it is infinitely deloopable. The corollary is independent of Goodwillie’s result in [Goo03, Section 2]. Of course, we use the fact ([Goo03, Lemma 1.9], [Rez13]) that the map tn F ∶ F → Tn F factors through a cartesian cube, and that hence Pn F is indeed n-excisive, but only that. Theorem 5.1.4. The category of n-homogeneous functors over a fixed base is stable for n ≥ 1. Proof. One needs to prove that a commutative square is a pushout if and only if it is a pullback. This follows directly from Theorems 4.5.1 and 4.5.2.

33

this proof!

5.2

The n-homogeneous model structure is left proper

In [BR14] several model structures around Goodwillie’s calculus of homotopy functors are developed. One of them is the n-homogeneous model structure on S∗ -enriched functors from a suitable pointed category C to pointed simplicial sets S∗ given in [BR14, Section 6]. In fact, the target category can be more general but that will lead to far. A map f ∶ F → G between homotopy functors is an n-homogeneous equivalence if the induced map Dn f is an objectwise weak equivalence. The cofibrations in this model structure are projective cofibrations and Pn−1 -equivalences satisfying an additional technical property that is not important here. The n-homogeneous model structure was obtained by right Bousfield localization from a proper model structure and hence itself right proper. The question whether it is left proper was left open. Lemma 5.2.1. The n-homogeneous model structure from [BR14] is left proper. Proof. Consider a pushout F

f

g

G

k

H ⌟

h

K

where g is an n-homogeneous cofibration and f an n-homogeneous equivalence. The map Pn−1 g, and hence Pn−1 h, is an objectwise equivalence. Applying Pn−1 yields therefore a pullback square. Since all functors in [BR14] are S∗ -enriched they are reduced. Hence f is a P0 -equivalence. Thus, Theorem 4.5.1 shows that applying Pn to the square yields a pullback. Therefore, applying Dn yields a pullback. But the n-homogeneous model structure is stable either by Theorem 5.1.4 or by [BR14, Cor. 6.18]. So the objectwise equivalence Dn f pushes out to an objectwise equivalence Dn h because the objectwise model structure is left proper by [BR14, Thm. 3.19]. So the n-homogeneous model structure is left proper.

References [ABFJ17] Mathieu Anel, Georg Biedermann, Eric Finster, and Andr´e Joyal. A generalized Blakers-Massey theorem. unpublished, 2017. [ADL08] Gregory Z. Arone, William G. Dwyer, and Kathryn Lesh. Loop structures in Taylor towers. Algebr. Geom. Topol., 8(1):173–210, 2008. [BR14] G. Biedermann and O. R¨ondigs. Calculus of functors and model categories, II. Algebr. Geom. Topol., 14(5):2853–2913, 2014. [Goo90] T. G. Goodwillie. Calculus I: The First Derivative of Pseudoisotropy Theory. K-Theory, 4, 1990. [Goo92] T. G. Goodwillie. Calculus II: Analytic Functors. K-Theory, 5, 1992. 34

[Goo03] T. G. Goodwillie. Calculus III: Taylor Series. Geometry and Topology, 7, October 2003. [Lur09] Jacob Lurie. Higher Topos Theory. Number 170 in Annals of Mathematics Studies. Princeton University Press, Princeton and Oxford, 2009. [Rez05] Charles Rezk. Toposes and homotopy toposes. http://www.math.uiuc.edu/∼rezk, 2005.

unpublished,

[Rez13] C. Rezk. A streamlined proof of Goodwillie’s n-excisive approximation. Algebr. Geom. Topol., 13(2):1049–1051, 2013.

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