Goodwillie's Calculus of Functors and Higher Topos Theory
aa r X i v : . [ m a t h . A T ] S e p Goodwillie’s Calculus of Functorsand Higher Topos Theory
Mathieu Anel ∗ , Georg Biedermann † , Eric Finster ‡ ,and Andr´e Joyal § Abstract
We develop an approach to Goodwillie’s Calculus of Functors using thetechniques of higher topos theory. Central to our method is the introduc-tion of the notion of fiberwise orthogonality, a strengthening of ordinaryorthogonality which allows us to give a number of useful characterizationsof the class of n -excisive maps. We use these results to show that thepushout product of a P n -equivalence with a P m -equivalence is a P m + n + -equivalence. Then, building on our previous work [ABFJ17], we prove aBlakers-Massey type theorem for the Goodwillie tower of functors. Weshow how to use the resulting techniques to rederive some foundationaltheorems in the subject, such as delooping of homogeneous functors. Contents n -excisive modality . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Cubical Diagrams and Orthogonality . . . . . . . . . . . . . . . . . 223.3 A Characterization of n -excisive maps . . . . . . . . . . . . . . . . 28 ∗ SPHERE, UMR 7219, Univ. Paris Diderot. [email protected] † LAGA (UMR7539), Institut Galil´ee, Univ. Paris 13; new address: Universidad del Norte,Departamento de Matem´aticas y Estad´ıstica, Barranquilla, Colombia, [email protected] ‡ IRIF, Univ. Paris Diderot. ericfi[email protected] § CIRGET, UQ `AM. [email protected]
A Truncated and connected maps of n -excisive functors 35 Goodwillie’s calculus of homotopy functors [Goo90, Goo92, Goo03] is a powerfultechnique in homotopy theory for approximating possibly complicated functorsby simpler ones using a generalized notion of excision. In particular, applied tothe identity functor on the category of spaces, it produces a filtration interpolat-ing between stable and unstable homotopy which has proved extremely usefulin calculations.In this article, we revisit some of the foundations of the subject from thepoint of view of higher topos theory. In particular, we will show that many ofthe fundamental results can be deduced from the following Blakers-Massey typetheorem, which we feel is of independent interest.
Theorem 3.4.1
Let
F HG K gf ⌟ be a homotopy pushout square of functors. If f is a P m -equivalence and g is a P n -equivalence, then the induced cartesian gap map ( f, g ) ∶ F → G × K H is a P m + n + -equivalence.The second result is a“dual” version. Theorem 3.4.2
Let
F HG K ⌜ fg be a homotopy pullback square of functors. If f is a P m -equivalence and g is a P n -equivalence, then the cocartesian gap map ⌊ f, g ⌋ ∶ G ⊔ F H → K is a P m + n + -equivalence.We show how to rederive known delooping results in homotopy functor cal-culus in an easy and conceptual way as consequences. In particular, we obtaina new proof of Goodwillie’s Lemma 2.2 [Goo03] that homogeneous functorsdeloop, independent of the material of [Goo03, Section 2]2oth of these results rest on the material of the companion article [ABFJ17]where a very general version of the Blakers-Massey theorem was proved. There,the language of higher topoi was adopted, and we find it equally well suited forthe calculus of homotopy functors, particularly because n -excisive functors tospaces form a higher topos. Indeed many of the results of this article arise fromworking systematically fiberwise, a method very much encouraged by the topos-theoretic point of view. Given this general framework, it will thus be convenientto drop any reference to higher derived structures and take them for granted.When we talk about a “category”, we mean “ ∞ -category” and all (co-)limits areto be interpreted as ∞ -categorical (co-)limits. In particular, we will not use theterms “homotopy (co-)limit”, as was done above for the sake of introduction.The ∞ -categorical machinery already describes a derived, homotopy invariantsetting with all higher coherences. We will fequently say “isomorphism” forwhat is perhaps more commonly called “weak equivalence”. Similarly, mappingspaces or internal hom objects are always to be taken derived. The reader whofinds this article easier to read by using model structures should not have anydifficulties in doing so.The main tool of our paper [ABFJ17] was the notion of modality , and itwill be equally important here. A modality is a unique factorization systemwhose left and right class are closed under base change. An example is thefactorization of a map of spaces into an n -connected map followed by an n -truncated map. Application of our generalized Blakers-Massey theorem to thisexample leads to the classical version of the theorem. Here we observe thatthe factorization of a natural transformation into a P n -equivalence followed byan n -excisive map is a modality in a presheaf topos. That this is the case isa consequence of the fact that Goodwillie’s n -excisive approximation P n is, infact, a left exact localization of the topos of functors. The left classes of these n -excisive modalities for various n ≥ Theorem 3.3.4(4)
The pushout product of a P m -equivalence with a P n -equi-valence is a P m + n + -equivalence.This fact immediately implies the main theorems by means of the general-ized Blakers-Massey theorems from [ABFJ17]. It also yields that the smash orjoin of an m -reduced functor with an n -reduced functor is ( m + n − ) -reduced,see Example 3.3.5. In order to prove this theorem we need to take a step backand develop systematically a fiberwise approach, which is to say, concentrateour attention on constructions which are compatible with base change. Indeed,one characterization of a modality is a factorization system whose left and rightclasses determine each other via fiberwise orthogonality , a notion which we in-troduce in Definition 2.4.6. Briefly, two maps are fiberwise orthogonal if all oftheir base changes are externally orthogonal to each other. As is the case in thetheory of ordinary factorization systems, this condition can be reformulated assay that a certain map, which we term the fiberwise diagonal (Definition 2.5.1) isan isomorphism. The resulting adjunction tricks exploited in Proposition 2.5.4lead us to Theorem 3.3.1 where we prove that the n -excisive modalities are3enerated by pushout product powers of certain explicit generating maps.From here, Theorems 3.4.1 and 3.4.2 which provide analogues of the Blakers-Massey theorems for the Goodwillie tower are easily deduced. This allows usin Theorem 3.5.2 to find a classifying map for the map P n F → P n − F in theGoodwillie tower and reprove delooping results (Corollaries 3.5.3 and 3.5.5) forfunctors whose derivatives live only in a certain range; in particular homoge-neous functors are infinitely deloopable.To justify a portion of the result in 3.5.2, and because we feel it is of inde-pendent interest, Appendix A is included. In Theorem A.0.5 we give a charac-terization of monomorphisms and covers (effective epimorphisms) in the topos [ C , S ] ( n ) of n -excisive functors. As far as we know this is the first time n -excisivefunctors are studied in detail as a topos and we wish to advertise this as a fruitfulline of thought.Finally, a few remarks are in order about the overall placement of our resultsin the larger landscape of studies of the Goodwillie Calculus. Indeed, by nowthere are many versions of Goodwillie-style filtrations which appear in a numberof different contexts. What is traditionally known as the homotopy calculus andconcerns excisive properties of functors (defined by their behavior on certaincubical diagrams) can be developed in a very general setting as is done, forexample, model category theoretically in [Kuh07] or ∞ -categorically in [Lur16].From the point of view of this theory, the results of this article are somewhat re-stricted: our arguments require that the functors under study have as codomainan ∞ -topos. In particular, this means we can not immediately apply our resultsto functors taking values in stable categories, such as, for example, spectra. Itremains for future work to understand to what extent our techniques might beapplied to the stable case.From another point of view, however, our results can be seen as providinga generalization of the homotopy calculus. For example, when we restrict tofunctors with values in a topos, our Theorem 3.3.1 leads to a completely inter-nal characterization of the construction of the Goodwillie tower which makesno mention of cubical diagrams. In upcoming work we will show how thesetechniques can be applied to give a uniform treatment of other varieties ofGoodwillie calculus such as the orthogonal calculus [Wei95] where the approxi-mation scheme is not necessarily defined in terms of the behavior of a functorwith respect to limits and colimits.This last point perhaps best illustrates the philosophy of our approach. In-deed, while the theory of higher topoi has applications to Goodwillie calculus,providing streamlined and conceptual proofs of its main results, the reverse isalso true: we can use the Goodwillie calculus as tool in the study of higher topoithemselves. We feel that much remains to be done in developing this point ofview. Acknowledgment:
The first author has received funding from the Eu-ropean Research Council under the European Community’s Seventh Frame-work Programme (FP7/2007-2013 Grant Agreement n ○ In this section we recall material from the our companion paper [ABFJ17]. Inparticular, we give the definition of a modality 2.6.1 and state our generalizedBlakers-Massey theorems 2.6.7 and 2.6.8.
This article is written using the language of higher topoi. For an outline of thetheory we refer the reader to [Rez05, Joy08, Lur09]. A very brief overview ofthe essential properties tailored to our needs is given in [ABFJ17, Section 2].We will now drop ∞ from the notation and refer to them simply as topoi. Wewrite S for the category of spaces. We will denote the category of space-valuedfunctors C → S on a small category C by [ C , S ] . A functor C → S is a presheafon C op . Definition 2.1.1.
A topos is an accessible left exact localization of a presheafcategory [ C , S ] for some small category C .The reader should be aware that “left exact localization” is to be taken inthe derived sense. Spelled out in the language of model categories it means“left Bousfield localization commuting with finite homotopy limits up to weakequivalence”. This is in line with the general approach in this article thateverything should be interpreted in ∞ -categorical terms. We recall that whenwe speak of (co-)limits, the corresponding notions in the language of modelstructures are homotopy (co-)limits. Example 2.1.2.
The primary examples of topoi of interest to us here are:1. The category S of spaces (as modelled by topological spaces or simplicialsets with weak homotopy equivalences) is the prime example of a topos.2. The category [ C , S ] of functors to spaces is a topos.3. The full subcategory [ C , S ] ( n ) ⊂ [ C , S ] of n -excisive functors, which, asexplained in Example 2.6.5, is itself a topos.We recall that within a topos colimits are preserved by base change.5 .2 Cubes, gaps and cogaps Let n = { , ⋯ , n } and write P ( n ) for the poset of its subsets. Define P ( n ) tobe the poset of non-empty subsets; let P ( n ) be the poset of proper subsets.Now consider a finitely complete category E . An n -cube in a category E is afunctor X ∶ P ( n ) → E . We will refer to the canonical map X ( ∅ ) → lim U ∈ P ( n ) X ( U ) as the cartesian gap map or simply the gap map for brevity. An n -cube is saidto be cartesian if its gap map is an isomorphism. For example, a 2-cube iscartesian if and only if it is a pullback square.For an n -cube X in a finitely cocomplete category E there also exists thecanonical map colim U ∈ P ( n ) X ( U ) → X ( n ) . which we will call it the cocartesian gap map or briefly, cogap map . An n -cubeis cocartesian if its cogap map is an isomorphism. A square is cocartesian if andonly if it is a pushout square.An n -cube is called strongly cartesian (resp. strongly cocartesian ) if all its2-dimensional subcubes are cartesian (resp. cocartesian). Definition 2.2.1.
The external cartesian product of two cubical diagrams X ∶ P ( m ) → E and Y ∶ P ( n ) → E is a cubical diagram X ⊠ Y ∶ P ( m + n ) = P ( m ) × P ( n ) → E defined by putting ( X ⊠ Y )( A, B ) = X ( A ) × Y ( B ) for every A ∈ P ( m ) and B ∈ P ( n ) . The external coproduct X ⊞ Y ∶ P ( m + n ) = P ( m ) × P ( n ) → E is defined by putting ( X ⊞ Y )( A, B ) = X ( A ) ⊔ Y ( B ) . The external cartesian (co)product of two strongly (co)cartesian cubes isstrongly (co)cartesian. Every map f ∶ A → B defines a 1-cube f ∶ P ( ) → E .The n -cube f ⊠ ⋯ ⊠ f n ∶ P ( n ) → E is strongly cartesian and the n -cube f ⊞ ⋯ ⊞ f n ∶ P ( n ) → E is strongly cocartesian for any sequence of maps { f i ∶ K i → L i } ni = . In particular,the square A ∨ B BA
16s cocartesian for any pair of objects in a pointed category.Now let E be finitely complete and finitely cocomplete. Given a commutativesquare in E : A CB D gf kh
We will denote the gap map by ( f, g ) ∶ A → B × D C. The cogap map of the square will be denoted by ⌊ h, k ⌋ ∶ B ∪ A C → D. Strictly speaking these maps depend on the whole square. In practice the re-maining maps will always be clear from the context.
Let E be a topos. For any two maps u ∶ A → B and v ∶ S → T in E the square A × S A × TB × S B × T. u × S A × v u × T B × v is cartesian, and we define the pushout product of u and v , denoted u ◻ v , to bethe cocartesian gap map of the previous square: u ◻ v = A × T ⊔ A × S B × S → B × T. Let 0 and 1 be respectively an initial and a terminal object for E . A topos E has a strict initial object which means that any arrow C → E has finite products, this implies 0 × X = X ∈ E . The pushout product defines a symmetric monoidal structure on thecategory E → of arrows, with unit 0 →
1. In particular, we have u ◻ v = v ◻ u and ( u ◻ v ) ◻ w = u ◻ ( v ◻ w ) . Example 2.3.1.
We give some examples of pushout products that will be usefulin the sequel.1. The iterated pushout product f ◻ ⋯ ◻ f n of a sequence of maps { f i ∶ K i → L i } ni = is the cogap map of n -cube f ⊠ ⋯ ⊠ f n .2. For any map A → B in E and any object C , the map ( → C ) ◻ ( A → B ) is simply ( → C ) ◻ ( A → B ) = C × A → C × B
7. For two pointed objects 1 → A and 1 → B in E , we have ( → A ) ◻ ( → B ) = A ∨ B → A × B the canonical inclusion of the wedge into the product.4. Recall that the join of two objects A and B in E , denoted A ⋆ B , is thepushout of the diagram A ← A × B → B . One sees immediately that ( A → ) ◻ ( B → ) = A ⋆ B → .
5. The fiberwise join X ⋆ B Y of two maps f ∶ X → B and g ∶ Y → B is thepushout of the diagram X ← X × B Y → Y . It is the pullback of the map f ◻ g along the diagonal B → B × BX ⊔ X × B Y Y ( X × B ) ⊔ X × Y ( B × Y ) B B × B. f ⋆ B g ⌜ f ◻ g The name fiberwise join is justified by the fact that for b ∈ B , we have theidentification fib b ( X ⋆ B Y ) = ( fib b f ) ⋆ ( fib b g ) since colimits are stable by base change in the topos. More generally, theiterated fiberwise join X ⋆ B ⋯ ⋆ B X n of a sequence of maps f i ∶ X i → B (1 ≤ i ≤ n ) with codomain B is the the pullback of the map f ◻ ⋯ ◻ f n along the diagonal B → B n .6. Since colimits in E commute with base change, the pushout product f ◻ g can be thought as the “external” join product of the fibers of f and g . Aneasy computation shows that the fiber of the map ( f ∶ A → B ) ◻ ( g ∶ C → D ) at a point ( b, d ) ∈ B × D is the join of the fibers of f and g . Details canbe found in [ABFJ17, Rem. 2.4].7. For an object Z the slice category E / Z has its own pushout product de-noted ◻ Z . Given f ∶ A → B and g ∶ X → Y in E / Z , the correspondingformula reads f ◻ Z g = ( A × Z Y ) ∪ ( A × Z X ) ( B × Z X ) → ( B × Z Y ) . We will make use of these observations in Section 3 in order to relate the cal-culus of strongly cocartesian diagrams in a category C with that of orthogonalityin the presheaf category [ C , S ] .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 u ∶ A → B and f ∶ X → Y in E we consider the following8ommutative square in S [ B, X ] [
B, Y ][ A, X ] [
A, Y ] . We define the external pullback hom ⟨ u, f ⟩ to be the cartesian gap map of theprevious square: ⟨ u, f ⟩ ∶ [ B, X ] → [ A, X ] × [ A,Y ] [ B, Y ] . Let ⟦ A, B ⟧ denote the internal hom object in E . Then we can define similarlyan internal pullback hom ⟪ u, f ⟫ ∶ ⟦ B, X ⟧ → ⟦ A, X ⟧ × ⟦ A,Y ⟧ ⟦ B, Y ⟧ , which is a map in E . Example 2.3.2.
The internal pullback hom has a number of useful specialcases:1. The category of spaces S is cartesian closed and we have ⟪ u, f ⟫ = ⟨ u, f ⟩ for any pair of maps u, f ∈ S .2. If S = ⊔ E , then ⟪ S → , X → ⟫ = X → X × X is the diagonal map of X . Similarly, for any map f ∶ X → Y the map ⟪ S → , f ⟫ = X → X × Y X is the diagonal map ∆ f of f .3. More generally, for any object A in E , the map∆ A ( X ) = ⟪ A → , X → ⟫ ∶ X → ⟦ A, X ⟧ is the A -diagonal of X . Similarly, for any map f ∶ X → Y the map ⟪ A → , f ⟫ defines the A -diagonal of f .It is useful to keep in mind that the global section functor Γ ∶= [ , − ] ∶ E → S takes the object ⟦ A, B ⟧ to the space [ A, B ] :Γ (⟦ A, B ⟧) = [ , ⟦ A, B ⟧] = [ A, B ] . Since Γ commutes with all limits, one hasΓ (⟪ f, g ⟫) = [ , ⟪ f, g ⟫] = ⟨ f, g ⟩ . Z ∈ E , the slice topos E / Z has an internal hom and an internalpullback hom that we will denote respectively by ⟦ − , − ⟧ Z and ⟪ − , − ⟫ Z . The basechange u ∗ ∶ E / Z → E / T along a map u ∶ T → Z , preserves cartesian products andinternal homs. For two objects A , B in E / Z , we have a canonical isomorphism u ∗ ⟦ A, B ⟧ Z = ⟦ u ∗ A, u ∗ B ⟧ T . We leave to the reader the proof of the following lemma asserting that the sameformula is true for the internal pullback hom.
Lemma 2.3.3.
For two maps f ∶ A → B and g ∶ X → Y in E / Z , and any map u ∶ T → Z we have a canonical isomorphism in E / T : u ∗ ⟪ f, g ⟫ Z = ⟪ u ∗ f, u ∗ g ⟫ T . Remark 2.3.4.
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 closedsymmetric monoidal structure on E → . In particular, we have ⟪ f ◻ g, h ⟫ = ⟪ f, ⟪ g, h ⟫⟫ . We have also the relation ⟨ f ◻ g, h ⟩ = ⟨ f, ⟪ g, h ⟫⟩ . In this section, we define and compare three notions of orthogonality for maps:the external orthogonality ⊥ and internal orthogonality ⊩ , which will be relatedto the internal and external pullback hom, and the new fiberwise orthogonality ñ (Definition 2.4.6), which will be related to a variation of the pullback hom inSection 2.5.Although our focus is mainly on ñ , it is convenient to formulate its propertiesas properties of ⊥ . So we provide some recollection on the matter. The rela-tion ⊩ is introduced only for comparison purposes and to avoid any confusionbetween ñ and ⊩ .Let us point out that our motivation for introducing fiberwise orthogonalityand the fiberwise diagonal is to prove Proposition 2.5.4. This eventually leadsto Theorem 3.3.4 that is our new ingredient to Goodwillie calculus that lets usprove the Blakers-Massey Theorem for the Goodwillie tower. Definition 2.4.1.
Two maps f ∶ A → B and g ∶ X → Y in E are externallyorthogonal or simply orthogonal if the map ⟨ f, g ⟩ is an isomorphism in S . Wewrite f ⊥ g for this relation and we say that f is externally left orthogonal to g and that g is externally right orthogonal to f .10nfolding the definitions, one immediately verifies that if f ⊥ g then forevery commutative square A XB Y. f h gkd the space of diagonal fillers is contractible, that is to say, a digonal filler existsand is unique up to homotopy.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 . Definition 2.4.2.
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 internallyright orthogonal to f .For an object Z and a map f ∶ A → B we write Z × f ∶ Z × A id Z × f ÐÐÐ→ Z × B .We leave to the reader the proof of the following lemma. Lemma 2.4.3.
The following conditions are equivalent: (1) f ⊩ g (2) For any Z ∈ E we have ( Z × f ) ⊥ g .In particular, f ⊩ g implies f ⊥ g . Since each slice topos E / Z has its own internal hom objects it has an internalorthogonality relation which we will denote by ⊩ Z . The following lemma provesthat these internal orthogonality relations are compatible with base change. Lemma 2.4.4.
For any two maps f ∶ A → B and g ∶ X → Y in E , and for anyobject Z ∈ E we have f ⊩ g Ô⇒ Z ∗ f ⊩ Z Z ∗ g where Z ∗ is the base change functor along the map Z → . Moreover, theconverse is true if the map Z → is a cover.Proof. If u ∶ Z →
1, then by Lemma 2.3.3, we have a canonical isomorphism u ∗ ⟪ f, g ⟫ = ⟪ u ∗ f, u ∗ g ⟫ Z . This proves that f ⊩ g ⇒ u ∗ f ⊩ Z u ∗ g . The converse is true since the functor u ∗ is conservative when u is a cover.The following proposition lists several equivalent properties that will be usedto define fiberwise orthogonality. In order to facilitate the reading, we employthe following convention in the proofs which follow: given a map f ∶ A → B anda map u ∶ Z → B , we denote by f Z the map u ∗ f ∶ A × B Z → Z . The point of thisnotation is to make u implicit, remembering only the new base. The contextwill make clear along which map the base is changed.11 roposition 2.4.5. Given two maps f ∶ A → B and g ∶ X → Y in E , thefollowing conditions are equivalent: (1) For any Z ∈ E and any maps b ∶ Z → B and y ∶ Z → Y , it is true in E / Z that f Z ⊩ Z g Z . (2) The base changes of f and g onto B × Y along the projections to B and Y satisfy f B × Y ⊩ B × Y g B × Y . (3) The diagonal map in E / B × Y ∆ f B × Y ( g B × Y ) ∶ g B × Y → ⟦ f B × Y , g B × Y ⟧ B × Y is an isomorphism (see Example 2.3.2.3 ). (4) For any Z → B × Y and any T → Z we have f T ⊥ Z g Z . (5) For any Z ∈ E and any maps b ∶ Z → B and y ∶ Z → Y , it is true in E / Z that f Z ⊥ Z g Z . (6) For any two maps Z → B and Z ′ → Y we have f Z ⊥ g Z ′ . (7) For any map Z → B we have f Z ⊥ g .Proof. (1) ⇒ (2) This is obvious since (2) is a special case of (1).(2) ⇒ (1) This follows from Lemma 2.4.4 that states that orthogonality isstable by base change.(2) ⇔ (3) This is equivalent by the definition of orthogonality in E B × Y .(1) ⇔ (4) This is Lemma 2.4.3 applied to the topos E / Z .(4) ⇒ (5) Set T → Z = id Z .(5) ⇔ (6) We need to prove that for all Z and all B ← Z → Y , f Z ⊥ Z g Z ⇐⇒ ∀ U → B, ∀ T → Y, f U ⊥ g T . We consider the following diagram U × B A U × Y X T × Y XU U T f U g U ⌜ g T h where h is arbitrary and the right square is cartesian. Because the right squareis cartesian, the space of diagonal fillers of the outer square is equivalent to thatof the left square. When h varies, the former condition gives f U ⊥ g T and thelatter f U ⊥ U g U , hence proving their equivalence.126) ⇔ (7) Since it is clear that (6) ⇒ (7), we need to show the other im-plication. Let f U be the base change of f along some map U → B , and g T thebase change of g along some map h ∶ T → Y , we consider the following diagramwhere the left square is commutative and the right square is cartesian U × B A T × Y X XU T Y. f U g T ⌜ gh Again, because the right square is cartesian, the space of diagonal fillers of theouter square is equivalent to that of the left square, which proves (7) ⇒ (6).(7) ⇒ (4) Let us consider the following diagram T × B A Z × Y X XT Z Y. f T k g Z ⌜ g where the right square is cartesian and k is any map such that the left square iscommutative. Condition (4) says that for any such k the space of fillers of theleft square is contractible. Since the right square is cartesian this is equivalentto the outer square having a contractible space of fillers. But Condition (7)states that any map from f T to g , i.e. a commutative square, has a contractiblespace of fillers. So (7) implies (4). Definition 2.4.6.
We will say that two maps f ∶ A → B and g ∶ X → Y are fiberwise orthogonal if they satisfy the equivalent properties of Proposition 2.4.5.We will denote this relation by f ñ g and say that f is fiberwise left orthogonal to g , and that g is fiberwise right orthogonal to f .The intuitive idea behind this relation is that any fiber of f is orthogonalto any fiber of g in the external sense. This is the meaning of Condition (6)where “any fiber” has to be understood as “any pullback over an arbitrarybase”. Another way to understand fiberwise orthogonality is to say that it isthe stabilization by base change of the relation f ⊥ g , which is the meaning ofCondition (5).Condition (7) helps to see that the relation f ñ g is stronger than the relation f ⊩ g since, by Lemma 2.4.3, the latter only requires that Z × f is orthogonalto g for every object Z ∈ E . Thus, f ñ g ⇒ f ⊩ g ⇒ f ⊥ g. Remark 2.4.7.
We have the following immediate observations:1. If f is fiberwise left orthogonal to g , then every base change f ′ of f is leftorthogonal to every base change g ′ of g . Moreover, f ñ g ⇒ f ′ ñ g ′ .13. The map A → A is fiberwise left orthogonal to a map f ∶ X → Y if and only if it is internally left orthogonal to f . In particulartwo objects A and X are fiberwise orthogonal ( A → ) ñ ( X → ) if andonly if they are internally orthogonal ( A → ) ⊩ ( X → ) . We saw that the external and internal orthogonality of two maps f and g can bedetected 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 constructionof the corresponding map is a bit more involved. Definition 2.5.1.
Take two maps f ∶ A → B and g ∶ X → Y in E ; pull themback to the common target B × Y , i.e. consider the maps f B × Y = f × id Y ∶ A × Y → B × Y and g B × Y = id B × g ∶ B × X → B × Y and view them as objects over B × Y . In the slice E / B × Y one can form the f B × Y -diagonal of g B × Y already used in 2.4.5.(3). We will denote this diagonalby { f, g } and name it the fiberwise diagonal map . { f, g } = ∆ f B × Y ( g B × Y ) = ⟪ f B × Y , g B × Y ⟫ B × Y , where the internal pullback hom on the right is taken in the topos E / B × Y .Explicitly, { f, g } ∶ B × XB × Y ( id B ,g ) Ð→ LPPPPPPPN A × YB × Y ( f, id Y ) , B × XB × Y ( id B ,g ) MQQQQQQQO B × Y . Remark 2.5.2.
Let b ∶ → B and y ∶ → Y be points of B and Y . We denoteby f b and g y the corresponding fibers of f and g . Since in a topos E colimitscommute with base change, the fiber of { f, g } at ( b, y ) can be proven to be thediagonal map g y → ⟦ f b , g y ⟧ . This is one of the reasons why we call this map the fiberwise diagonal map.
Proposition 2.5.3.
Let f and g be maps in E . Then f ñ g if and only if { f, g } is an isomorphism.Proof. This is exactly the content of 2.4.5(3).We now arrive at our key technical result.14 roposition 2.5.4.
The following formula is true in any topos: { f ◻ g, h } = { f, { g, h }} . For the proof of this proposition we need the following two auxiliary lemmas.
Lemma 2.5.5.
For all A , C and B → C in any topos, the following square ⟦ A × C, B ⟧ C ⟦ A, B ⟧ C ⟦ A, C ⟧ , ⌜ where ⟦ − , − ⟧ C is the internal hom in E / C and where the bottom map is thediagonal 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 use first the fact that thebase change E → E / C preserves internal homs; this shows that ⟦ A × C, B × C ⟧ C = ⟦ A, B ⟧ × C . Then the left square is cartesian as the image of the cartesian squarein E / C B B × CC C × C by the functor ⟦ A × C, − ⟧ C which preserves limits. Lemma 2.5.6.
The square ⟦ X × ⟦ Y, Z ⟧ , Z ⟧ ⟦ Y,Z ⟧ ⟦ X, Z ⟧⟦ Y, Z ⟧ ⟦ X × Y, Z ⟧⌜ is a pullback. Hence, there is a canonical isomorphism ⟦ X ⋆ Y, Z ⟧ = ⟦ X × ⟦ Y, Z ⟧ , Z ⟧ ⟦ Y,Z ⟧ . Proof.
Setting A = X , B = Z and C = ⟦ Y, Z ⟧ in the previous lemma we find thatthe square above is a pullback as claimed. Since the join is the pushout of theprojections X ← X × Y → Y, the pullback of this square is canonically isomorphic to ⟦ X ⋆ Y, Z ⟧ .15 roof of Proposition 2.5.4. We consider first the special case where the mapsare 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 → ⟦ X ⋆ Y, Z ⟧ . On the other hand, the map { f, { g, h }} becomes { X → , Z → ⟦ Y, Z ⟧} = Z → ⟦ X × ⟦ Y, Z ⟧ , Z ⟧ ⟦ Y,Z ⟧ . Lemma 2.5.6 shows that these two maps are the same. This proves our claimin the special case.We prove the general case by arguing fiberwise, i.e. by viewing our maps asobjects in the respective slice categories and then appealing to the special caseabove. We introduce the following convenient notation. First, we will denotethe 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 thecorresponding object in E / I . If another object J ∈ E is given, we will denote by X J the base change of X ∈ E / I to E / IJ along the projection I × J → I , i.e. X J is the map X × J → I × J .For two maps f ∶ X → I and g ∶ Y → J , the map f ◻ g in E corresponds tothe object X J ⋆ Y I in E / IJ , where the join is also computed in E / IJ . For a third object K , it is easyto compute that ( X J ⋆ Y I ) K = X JK ⋆ Y IK in E / IJK .Similarly, for two maps g ∶ Y → J and h ∶ Z → K , the map { g, h } is definedas the map in E / JK ⟪ Y K → , Z J → ⟫ where the pullback hom is computed in E / JK . For a third object I ∈ E , becausethe pullback functor E / JK → E / IJK preserves exponentials, we have also ( ⟪ Y K → , Z J → ⟫) I = ( Z J → ⟦ Y K , Z J ⟧) I = Z IJ → ⟦ Y IK , Z IJ ⟧ = ⟪ Y IK → , Z IJ → ⟫ in E / IJK . Finally, we obtain the following canonical isomorphisms: { f ◻ g, h } viewed as a map in E / IJK = ⟪( X J ⋆ Y I ) K → , Z IJ → ⟫ join in E / IJ , bracket in E / IJK = ⟪ X JK ⋆ Y IK → , Z IJ → ⟫ computed in E / IJK = ⟪ X JK → , ⟪ Y IK → , Z IJ → ⟫⟫ special case applied to the topos E / IJK = ⟪ X JK → , (⟪ Y K → , Z J → ⟫) I ⟫ inside bracket computed in E / JK = { f, { g, h }} viewed as a map in E / IJK . .6 Modalities and generalized Blakers-Massey theorems Given a class of maps M of E , we write M ⊥ for the class of maps which areexternally right orthogonal to every map of M . Similarly, the class ⊥ M denotesthe class of maps externally left orthogonal to every map of M .Recall that a factorization system on a category E is the data of a pair ( L , R ) of classes of maps in E such that1. every map f in E can be factored in f = rl where l ∈ L and r ∈ R , and2. L ⊥ = R and L = ⊥ R .In a factorization system, the right class is always stable by base change. Definition 2.6.1.
Let E be a topos. A modality on E is a factorization system ( L , R ) such that the left class L is also stable by base change. Proposition 2.6.2.
A factorization system ( L , R ) is a modality if and only ifthe stronger orthogonality property L ñ R holds.Proof. The equivalence is given by Proposition 2.4.5.(7) which states exactlythat the left class L is stable by base change.An important source of modalities on a topos E are provided by the accessibleleft exact localizations of E . (These are, in fact, exactly the subtopoi of E , thoughwe will not have occasion to use this observation.) To recall the construction,let F ′ ∶ E → E ′ be a functor with fully-faithful right adjoint i ∶ E ′ → E . As i isfully-faithful, it is convenient to work with the associated endofunctor F = i ○ F ′ ,identifying E ′ with its corresponding reflective subcategory in E . We now havethe following standard definitions: Definition 2.6.3.
Let F ∶ E → E be as above.1. A map f ∶ A → B is said to be F -local if the square A F ( A ) B F ( B ) f F ( f ) is cartesian.2. A map f ∶ A → B is an F -equivalence if F ( f ) is an isomorphism. Lemma 2.6.4.
Let F be a left exact localization of a topos E . If we let L bethe class of F -equivalences and R the class of F -local maps, then ( L , R ) formsa modality on E . roof. Given a map f ∶ A → B , one may produce directly a factorization f = v ○ u by first applying F and defining C , u and v by forming the pullback as in thefollowing diagram A C F ( A ) B F ( B ) uf v ⌜ F ( f ) The map v is F -local by construction, and one immediately checks that u is an F -equivalence using the idempotence of F . That the class of F -equivalences isstable by base change is clear from the fact that F preserves finite limits.To check orthogonality we use the following observation. Let g ∈ R be any F -local map. Note that for any map f we have ⟨ f, g ⟩ = ⟨ f, F g ⟩ = ⟨ F f, F g ⟩ , where the first equality comes from the fact that g is a base change of F g andthe second equality comes from the universal property of the localization F .It follows that if f is an F -equivalence, so that F ( f ) is an isomorphism, then ⟨ F f, F g ⟩ = ⟨ f, g ⟩ is an isomorphism. This shows that L ⊆ ⊥ R and L ⊥ ⊇ R .Now let f ∶ A → B ∈ ⊥ R . We must show that f is an F -equivalence. Considerthe diagram A C F ( A ) B B F ( B ) uf ⌜ v F ( f ) ∃ ! h where C , u and v are defined by pullback. Since v is F -local by construction,we have f ⊥ v . Hence we obtain a unique lift h . One can easily check that F ( h ) provides an inverse to F ( f ) which shows that F ( f ) is an equivalence. Thisshows that L = ⊥ R .Finally, let g ∶ X → Y ∈ L ⊥ . Factor g as g = v ○ u as in the diagram X C F ( X ) Y Y F ( Y ) ug ⌜ v F ( g ) so that v is F -local and u is an F -equivalence by construction. We would like toshow that the map u is an isomorphism, as this implies that g is F -local. Butnow, we have a unique lift h in the following diagram X XC Y u gv ∃ ! h u ⊥ g by assumption and one readily checks that h is the required inverse.This shows that R = L ⊥ and completes the proof. Example 2.6.5.
Goodwillie’s n -excisive approximation construction P n is aleft exact localization of the ∞ -topos [ C , S ] for some small category C withfinite colimits and a terminal object. Hence, the P n -equivalences and the P n -local maps form a modality. This example is developed in detail Section 3, seeDefinition 3.1.5.Let ( L , R ) on a topos E and suppose we are give a commutative square Z YX W f g kh (1)
Definition 2.6.6.
The square (1) 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 ⌊ h, k ⌋ ∶ X ∪ Z Y → W is in L .Given a map f ∶ X → Y , its diagonal ∆ f is the map∆ f ∶ X → X × Y X induced by pulling back f along itself. In particular ∆ ( X → ) is the classicaldiagonal X → X × X .In [ABFJ17], the following two facts were proven about this situation: Theorem 2.6.7 (Blakers-Massey [ABFJ17, Thm. 4.0.1]) . Let Diagram (1) bea pushout square. Suppose that ∆ f ◻ ∆ g ∈ L . Then the square is L -cartesian. Theorem 2.6.8 (“Dual” Blakers-Massey [ABFJ17, Thm. 3.6.1]) . Let Dia-gram (1) be a pullback square. Suppose that the map h ◻ k ∈ L . Then the squareis L -cocartesian. We will now revisit the Goodwillie n-excisive localization from the perspectiveof topos theory. Our approach here is not the most general possible. In [BR14] areasonably general framework for Goodwillie calculus in the language of modelcategories is developed. In [Heu15], the author constructs Goodwillie approx-imations of arbitrary categories. Here, however, we are particularly interestedin functor categories, and more specifically, those valued in spaces.19 .1 The n -excisive modality All of our arguments can be carried out in the presheaf topos [ C , S ] where C isa category with finite colimits and a terminal object, and hence we will workin that level of generality. We note that the standard examples of finite spaces( F in ) and finite pointed spaces ( F in ∗ ) fall into this category. Moreover, theclass of such categories is closed under slicing and taking pointed objects. Itincludes in particular the source categories used by Goodwillie to construct theGoodwillie tower of a functor at a fixed object.We stress that the target category is unpointed spaces because we rely ontopos-theoretic arguments. No pointed category can be a non-trivial topos.However, our main results 3.3.4, 3.4.2 and 3.4.1 are still valid for functors withvalues in pointed spaces. This follows from the observation that a naturaltransformation of functors to pointed spaces is n -excisive if and only it is still n -excisive after forgetting the basepoint, and analogously for P n -equivalences.Let us fix in this section a category C as above, writing 1 and 0 for the termi-nal and initial objects respectively. Recall that the starting point for Goodwilliecalculus is the following Definition 3.1.1.
A functor F ∶ C → S is n -excisive if it carries strongly co-cartesian ( n + ) -cubes in C to cartesian cubes in S .In order to provide examples of n -excisive functors, Goodwillie introducesthe following construction. Given a functor F ∶ C → S , define a new functor T n F by the formula: T n F ( K ) ∶= lim U ∈ P ( n + ) F ( K ⋆ U ) There is a natural map t n F ∶ F → T n F determined at an object K by the cartesian gap map of the cube U ↦ F ( K ⋆ U ) . Remark 3.1.2.
While we do not require that the category C admits finiteproducts, the above formula nonetheless makes sense in our setting. Indeed, as C admits finite coproducts, it admits a tensoring over the category of finite setsby setting K ⊗ U = ∐ U K Since C has a terminal object, we can regard U as an object of C by consideringthe object 1 ⊗ U . One can easily check that this makes K ⊗ U into a productin C , so that one can define the join using the usual formula. Equivalently, onemay define K ⋆ U directly by the colimit: K ⋆ U = colim ⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩ K . . . ⎫⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎭ with U copies of the terminal object appearing in the diagram. When C is takento be F in or F in ∗ , this definition coincides with the standard one.20ith this construction in hand, we now iterate, defining a functor P n F asthe colimit of the induced sequence P n F ∶= colim { F → T n F → T n F → ⋯ } We summarize the relevant facts about this construction with the following
Proposition 3.1.3 (Goodwillie [Goo03]) . Let F ∈ [ C , S ] .1. P n F is n -excisive.2. The functor P n ∶ [ C , S ] → [ C , S ] commutes with finite limits.3. The canonical map F → P n F is universal for maps from F to n -excisivefunctors. In particular, the functor P n is idempotent.Proof. The proofs appearing in [Goo03], as well as Rezk’s streamlined version[Rez13] are sufficiently general to go through in our setting with only minormodifications. Indeed, for a translation of these arguments into the language of ∞ -categories, the reader may consult [Lur16][Section 6.1.1].Let us write [ C , S ] ( n ) for the full subcategory of n -excisive functors. Theprevious proposition can be summarized by asserting that the functor P n ∶ [ C , S ] → [ C , S ] ( n ) is a left exact localization [Lur09, Prop. 5.2.7.4]. In particular, [ C , S ] ( n ) is itselfa topos [Lur09, Prop. 6.1.0.1]). Remark 3.1.4.
The functor P n ∶ [ C , S ] → [ C , S ] commutes with filtered colimits.Since colimits in the localization [ C , S ] ( n ) are computed by reflecting the colimitsof the ambient topos [ C , S ] down to [ C , S ] ( n ) via P n , the functor P n viewed astaking values in [ C , S ] ( n ) actually commutes with all colimits: P n colim i ∈ I F i = P n colim i ∈ I P n F i . The right hand side is the colimit in [ C , S ] ( n ) .As is the case for any left exact localization, the functor P n determines twoclasses of maps via Definition 2.6.3. Moreover, according to Lemma 2.6.4, thesetwo classes of maps form a modality. Definition 3.1.5.
We refer to the modality( P n -equivalences, P n -local maps)as the n -excisive modality .Since the Generalized Blakers-Massey theorem of [ABFJ17] applies to an arbitrary modality on a topos, we may apply the result already at this point,using nothing but the left-exactness of the functor P n . The statement obtainedis the following: 21 roposition 3.1.6. Let
F HG K gf ⌟ be a pushout square of functors. Suppose that ∆ f ◻ ∆ g is a P n -equivalence.Then so is the cartesian gap map ( f, g ) ∶ F → G × K H We think the reader will agree that the statement in its current form isnot entirely satisfactory: supposing that f is a P i -equivalence and g is a P j -equivalence, we would like a determination of n in terms of i and j . In thefollowing sections, we will develop the tools to make such a calculation usingthe calculus of orthogonality developed above. The final result is the following. Theorem 3.1.7.
Let f be a P i -equivalence and g a P j -equivalence. Then themap ∆ f ◻ ∆ g is a P i + j + -equivalence. In order to prove Theorem 3.1.7, we are going to examine how the notion ofcubical diagram in C is transformed by the Yoneda embedding y ∶ C op → [ C , S ] . We will see that there is a close connection between strongly cocartesian cubicaldiagrams in C and fiberwise join products in [ C , S ] , leading to a number of usefulcharacterizations of the classes of P n -equivalences and P n -local maps. Fromhere, the calculus of orthogonality, and in particular the adjunction formula ofProposition 2.5.4 ultimately lead to the desired result. In the discussion whichfollows, we write R K = C ( K, − ) = y ( K ) for the representable functor determined by an object K ∈ C . For a map k ∶ K → L we write R k = y ( k ) ∶ C ( L, − ) → C ( K, − ) for the induced map. We recall for later use that the Yoneda embedding pre-serves all limits and hence sends colimits in C to limits in [ C , S ] .Now let X be a cubical diagram in C and let us put K = X ( ∅ ) . We denote byˆ X the cubical diagram obtained by composition with the (contravariant) Yonedaembedding. That is, ˆ X = y ○ X . The cocartesian gap map of this cube takes theform Γ ( X ) ∶ colim U ≠∅ R X ( U ) → R K The interest in this map arises from the following elementary observation:22 emma 3.2.1.
Let F ∈ [ C , S ] be a functor. Then Γ ( X ) ⊥ ( F → ) ⇐⇒ F ○ X is cartesianProof. Unfolding the definition of the pullback hom ⟨ Γ ( X ) , F → ⟩ (and ignoringthe trivial factors) we find ⟨ Γ ( X ) , F → ⟩ ∶ [ R X ( ∅ ) , F ] → [ colim U R X ( U ) , F ] But of course [ R X ( ∅ ) , F ] = F ( X ( ∅ )) and [ colim U R X ( U ) , F ] = lim U [ R X ( U ) , F ] = lim U F ( X ( U )) by Yoneda. Hence this is the map ⟨ Γ ( X ) , F → ⟩ ∶ F ( X ( ∅ )) → lim U F ( X ( U )) which is an isomorphism exactly if the cube F ○ X is cartesian. Corollary 3.2.2.
A functor F ∈ [ C , S ] is n -excisive if and only if, for everystrongly cocartesian ( n + ) cube X , we have Γ ( X ) ⊥ F → . In view of the previous corollary, it is natural to extend the definition of n -excisiveness to maps so that a functor is n -excisive if and only if the map F → Definition 3.2.3.
A map f ∶ F → G of functors is said to be n -excisive if forall strongly cocartesian ( n + ) -cubes X we have Γ ( X ) ⊥ f .For convenience we note that f is n -excisive if and only if for all stronglycocartesian X as above, the square F ( X (∅)) lim U ≠∅ F ( X ( U )) G ( X (∅)) lim U ≠∅ G ( X ( U )) is a pullback.The following construction is a useful source of strongly cocartesian dia-grams. The reader may wish to compare [BJM15, Example 2.8] where a similarconstruction is considered. Construction 3.2.4.
Let { k i ∶ K i → L i } ni = be a family of maps in C . For U ⊆ n , define σ U ( k i ) = ⎧⎪⎪⎨⎪⎪⎩ K i i ∉ UL i i ∈ U { k i } is a n -cubical diagram K defined by the formula K ( U ) = ⊔ ≤ i ≤ n σ U ( k i ) where for U ⊆ V , the induced map K ( U ) → K ( V ) is given by K ( U ↪ V ) = ⎧⎪⎪⎨⎪⎪⎩ k i i ∈ V ∖ U id σ V ( i ) otherwise Lemma 3.2.5.
For any family of maps { k i ∶ K i → L i } ni = the cubical diagram K is strongly cocartesian.Proof. In the notation of Definition 2.2.1, we have K = k ⊞ ⋯ ⊞ k n . Lemma 3.2.6. Γ ( K ) = R k ◻ ⋯ ◻ R k n Proof.
We have ˆ K ∶= y ○ ˆ K = R k ⊠ ⋯ ⊠ R k n , since the Yoneda functor takescoproduct to product. Hence the cocartesian gap map of ˆ K is equal to R k ◻⋯ ◻ R k n . Example 3.2.7.
Suppose the category C is pointed , that is, that the initial andterminal objects coincide in C . It will be convenient in this case to write ∨ forthe coproduct in C in order to make contact with the traditional notation. Inparticular, we have K ∨ = K for all objects K ∈ C .Now consider a family of objects { K i } ni = in C . Applying Construction 3.2.4to the collection of maps { K i → } ni = we find that the resulting cube may bemore simply described as K ( U ) = ⋁ i ∉ U K i Now let F ∶ C → S be a functor. Unraveling the definition shows that thepullback hom ⟨ Γ ( K ) , F ⟩ is the map F ( ⋁ i K i ) → lim U ≠∅ F ( ⋁ i ∉ U K i ) The fiber of this map is what Goodwillie refers to as the n -th cross-effect , writing ( cr n F )( K , . . . , K n ) . It follows immediately from these considerations that wehave Γ ( K ) ⊥ ( F → ) for every family { K i } ni = of objects of C if and only if F is of degree ( n − ) in the sense of [BJM15, Definition 3.21]; that is, cr n F vanishes.It is well known that to be of degree n is strictly weaker than to be n -excisive.Nonetheless, we will show below that we can recover the notion of n -excisivenessfrom the cubical diagrams K by replacing the external orthogonality relation ⊥ by the stronger fiberwise orthogonality relation ñ . It is exactly this observationwhich motivates the introduction of this stronger notion.24onstruction 3.2.4 turns out to be quite general: in fact, as we now show, every cubical diagram can be obtained from it after a single cobase change. Tomake this precise, suppose we are given a strongly cocartesian cube X ∶ P ( n ) → C . Let us put K = P ( ∅ ) and K i = P ({ i }) . The functorial action of X gives usmaps k i ∶= X ( ∅ → { i }) ∶ K → K i . Applying Construction 3.2.4, we obtain a new cubical diagram which, in thiscase, we will denote by X ◻ (the notation being inspired by Lemma 3.2.6 above).Unwinding the definition, we find that X ◻ ( ∅ ) = K ⊔ n so that the codiagonal ∇ ∶ K ⊔ n → K provides a canonical map X ◻ ( ∅ ) → X ( ∅ ) . Lemma 3.2.8.
The strongly cocartesian cube X is obtained from X ◻ by cobasechange along the codiagonal map ∇ ∶ K ⊔ n → K Proof.
The lemma asserts that for any U ⊆ n , the square K ⊔ ⋯ ⊔ K K ⊔ ≤ i ≤ n σ U ( i ) X ( U ) is a pushout. But since X is strongly cocartesian, we have that X ( U ) = colim ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ K i K K i K i ⋮⋮ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ i k ∈ U and one easily sees that this coincides with the pushout above by a simplecofinality argument.It is immediate from the previous lemma and the fact that the Yonedaembedding sends colimits in C to limits in [ C , S ] that the corresponding cube ofrepresentable functors ˆ X is obtained from ˆ X ◻ by base change along the n -folddiagonal map ∆ ∶ R X ( ∅ ) → ( R X ( ∅ ) ) × n = R X ◻ ( ∅ ) It will be convenient in what follows to introduce special notation for the co-cartesian gap maps of these two cubes. We will use this notation exclusively in25he case where the given cubical diagram X is known to be strongly cocartesian.In this case, the cocartesian gap map of the cube ˆ X will be denoted γ X ∶ Γ X → R X ( ∅ ) where Γ X ∶= colim U ≠∅ R X ( U ) . For ˆ X ◻ , on the other hand, we will write w X ∶ W X → ( R X ( ∅ ) ) × n with W X defined by the analogous colimit for the cube X ◻ . Some justificationfor this special notation will be given in Remark 3.2.11 below. For now weobserve Lemma 3.2.9.
For any strongly cocartesian cube X in C , the square Γ X W X R X ( ∅ ) ( R X ( ∅ ) ) × nγ X ⌜ w X ∆ is a pullback in [ C , S ] .Proof. Immediate since colimits in [ C , S ] are stable by base change.Combining Lemma 3.2.6 with the definition of the fiberwise join, we deduceimmediately that Lemma 3.2.10.
For any strongly cocartesian cubical diagram X in C , the co-cartesian gap map γ X of the cube of representable functors ˆ X is given by theexpression γ X = R k ⋆ R K ⋯ ⋆ R K R k n where k i = X ( ∅ ↪ { i }) ∶ K → K i . The above discussion has an important special case, which we now describe.Note that a given strongly cocartesian diagram X is completely determined bythe family of maps { f i ∶ X ( ∅ ) → X ({ i })} ≤ i ≤ 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 ( C K / ) × n . As C has a terminal object, this category clearly has one as well, an n -cube which we will denote by T Kn and which is determined by T Kn ( ∅ ) = K and T Kn ({ i }) = ≤ i ≤ n . More generally, the reader can easily check that wehave T Kn ( U ) = K ⋆ U in the sense of Remark 3.1.2. Applying Yoneda as in the proof of 3.2.1 we findthat ⟨ γ T Kn , F ⟩ = ( t n F )( K ) ∶ F ( K ) → lim U F ( K ⋆ U )
26s Goodwillie’s map t n F introduced in the previous section. As these distin-guished strongly cocartesian cubes play a central role in the theory and areentirely determined by the given object K , it will be convenient to use theabbreviation γ Kn ∶ Γ Kn → R K and w Kn ∶ W Kn → ( R K ) × n for the maps γ T Kn and w T Kn constructed above. Note that by construction w Kn = ( w K ) ◻ n where w K = R K → ∶ R → R K . In this case, then the statement of Lemma 3.2.9asserts that the square Γ Kn W Kn R K ( R K ) × nγ Kn ⌜ w Kn ∆ (2)is a pullback for any K ∈ C . Remark 3.2.11.
The pullback diagram (2) is analogous to a well-known con-struction in classical homotopy theory. For a pointed space ( X, x ) , the n -fold fat wedge of X , denoted W n ( X ) may be defined as the iterated pushout product W n ( X ) X × nw n ∶= ⎛⎜⎜⎜⎜⎝ X x ⎞⎟⎟⎟⎟⎠ ◻ n . Note that it comes equipped with a canonical inclusion w n into the n -fold prod-uct as shown. The pullback of this map along the diagonal X → X × n is knownas the n -th Ganea fibration , and denoted Γ n ( X ) .Γ n ( X ) W n ( X ) X X × nγ n w n ∆ Recall from Subsection 2.3 that the pullback of an n -fold pushout product alongthe diagonal map is called the n -fold fiberwise join. Thus the map γ n mayalternatively be described as γ n = ⋆ X ⋯ ⋆ X γ n has the descriptionfib x γ n = ( Ω X ) ⋆ n
27s is well known.In fact, this construction makes sense in any topos. Returning to the situ-ation at hand, when the category C is pointed, we find that the representablefunctor R is in fact the terminal functor in [ C , S ] . Hence for any object K ∈ C ,the terminal map K → R K with a canonicalbase point R → R K Examining the pullback diagram (2) above, we find that it is exactly the n -thGanea fibration of the representable R K as calculated in the topos [ C , S ] , whichis the justification for the notation introduced above. From this perspective,Theorem 3.3.1 (4) below may be read as saying that the Goodwillie localizationof the functor category [ C , S ] is obtained by inverting the n -th Ganea fibrationof the representable R K for all K ∈ C .Let us also point out that diagram (2) is well-defined and still a pullbackeven if C is not pointed. n -excisive maps We now proceed to give a number of characterizations of the class of n -excisivemaps as defined above. The reader will perhaps not be surprised to learn thatthey coincide with the P n -local maps determined by the localization functor P n ∶ [ C , S ] → [ C , S ] , though this is not a priori obvious. Furthermore, character-ization (2) in the following theorem provides the main tool for establishing thecompatibility of P n -equivalences with the pushout product. Theorem 3.3.1.
Let f ∶ F → G be a map in [ C , S ] . The following statementsare equivalent: (1) For every family of maps { h i ∶ K i → L i } ni = in C we have R h ◻ ⋯ ◻ R h n ñ f (2) For all K ∈ C we have w Kn + ñ f . (3) For every family of maps { h i ∶ K → K i } ni = in C we have R K ⋆ R K ⋯ ⋆ R K R K n ⊥ f (4) For all K ∈ C we have γ Kn + ⊥ f . (5) The map f is P n -local. (6) The map f is n -excisive.Proof. We will begin with the equivalences ( ) ⇔ ( ) ⇔ ( ) ⇔ ( ) . ( ) ⇒ ( ) This is the special case K i = i ) ⇒ ( ) Examining the definition, we find that ⟨ γ Kn + , F ⟩ is the cartesiangap map of the commutative square F ( K ) lim U ≠∅ F ( K U ) G ( K ) lim U ≠∅ G ( K U ) , Hence if γ Kn + ⊥ f , this square is a pullback. Recognizing the right vertical mapas T n ( f ) , it follows that f is a pullback of T n ( f ) . But then it is a pullback of allcomposites T kn f because T n preserves finite limits. Since finite limits commutewith filtered colimits in S , f is a pullback of P n f = colim k T kn f , ie. f is P n -local. ( ) ⇒ ( ) Now assume that f is P n -local and let X be a strongly cocartesian ( n + ) -cube. Write K = X (∅) . Consider the following commutative diagram: P n F ( K ) lim U ≠∅ P n F ( X ( U )) F ( K ) lim U ≠∅ F ( X ( U )) P n G ( K ) lim U ≠∅ P n G ( X ( U )) G ( K ) lim U ≠∅ G ( X ( U )) We need to show that the front face is a pullback (see Definition 3.2.3) . Theright and left faces are a pullbacks by assumption. The back square is triviallya pullback: both horizontal maps are isomorphisms because P n F and P n G are n -excisive functors by Proposition 3.1.3. Thus, the composite diagonal squareis a pullback. Hence, the front is also a pullback. ( ) ⇒ ( ) According to Lemma 3.2.10, the cocartesian gap map of anystrongly cocartesian diagram can be expressed in this form. Hence if f is n -excisive, it is orthogonal to such a map by definition.We now treat statements ( ) and ( ) . ( ) ⇒ ( ) This is the special case where h i = K → i . ( ) ⇒ ( ) We have seen above that there is a pullback squareΓ Kn + W Kn + R K ( R K ) × n + γ Kn + ⌜ w Kn + ∆ for any K ∈ C . But by the Definition 2.4.6 of fiberwise orthogonality, or moreprecisely by Proposition 2.4.5(7), f is orthogonal to any pullback of the map w Kn + , in particular γ Kn + as claimed. 29 ) ⇒ ( ) Lemma 3.2.6 identifies the map R h ◻ ⋯ ◻ R h n as the cocartesian gap map of the strongly cocartesian cube determined byConstruction 3.2.4 and so the relation R h ◻ ⋯ ◻ R h n ⊥ f holds by definition.It remains to show that f is orthogonal to any base change of this map. Butsince we already know ( ) ⇒ ( ) , f is P n -local. The result now follows since P n -equivalences are stable by base change. Remark 3.3.2.
It is not possible to replace the fiberwise orthogonality relation ñ it items (1) and (2) with the weaker external orthogonality relation ⊥ . Indeed,as pointed out in Example 3.2.7, the latter notion detects functors which are ofdegree n , a strictly weaker condition. Remark 3.3.3.
In [ABFJ17] we deduce the classical Blakers-Massey theoremfrom our generalized version by using the fact that the n -connected/ n -truncatedmodalities are generated by pushout product powers of the map S →
1. The-orem 3.3.1(2) states that in the same sense the Goodwillie tower, that is the n -excisive modalities, are generated by the pushout product powers of the maps w K ∶ R → R K for all K in C .We can now prove the main result of this section. Recall the fiberwise di-agonal { f, g } of two maps f and g as defined in 2.5.1. By Proposition 2.5.3it is an isomorphism if and only if the maps f and g are fiberwise orthogo-nal. A crucial role in the proof of the next theorem is played by the formula { f ◻ g, h } = { f, { g, h }} demonstrated in Proposition 2.5.4. It allows us to useadjunction tricks for fiberwise orthogonality. The reader is invited to comparethe next theorem with [ABFJ17, Cor. 3.15] where the n -connected/ n -truncatedmodalities for n ≥ − Theorem 3.3.4.
Let f be a P m -equivalence, g a P n -equivalence and h a p -excisive map. Then: (1) The map { w Kn , h } is ( p − n ) -excisive (2) The map f ◻ w Kn is a P n + m -equivalence (3) The map { f, h } is ( p − m − ) -excisive (4) The map f ◻ g is a P m + n + -equivalenceProof. (1). It is immediate from Lemma 3.2.6 that w Kp + = w Kp − n + ◻ w Kn Therefore, by Proposition 2.5.4 we have { w Kp + , h } = { w Kp − n + , { w Kn , h }} K ∈ C . The map on the left is an isomorphism by the assumption that h is p -excisive, and hence so is the one on the right. Theorem 3.3.1 (2) then givesthe desired result.(2). If k is any ( n + m ) -excisive map, we have { f ◻ w Kn , k } = { f, { w Kn , k }} But the map { w Kn , k } is m -excisive by (1).(3). Again by Theorem 3.3.1 (2), it suffices to check that the map { w Kp − m , { f, h }} is an isomorphism for any K ∈ C . But { w Kp − m , { f, h }} = { w Kp − m ◻ f, h } and since w Kp − m ◻ f is p -excisive by (2), the right map is an isomorphism.(4). Let k be any ( m + n + ) -excisive map. Then since { f ◻ g, k } = { f, { g, k }} and the map { g, k } is ( m + n + ) − n − = m excisive, the result follows.The compatibility of the Goodwillie tower with the pushout product statedin Theorem 3.3.4(4) is what we are really after. It will allow us to prove theBlakers-Massey analogue for the Goodwillie tower. One direct application is Example 3.3.5.
Recall that a functor F is m -reduced if the map F → P m − -equivalence. Let F be m -reduced and G be n -reduced. Then the map ( → F ) ◻ ( → G ) = ( F ∨ G → F × G ) is a P m + n − -equivalence. Taking the cofiber it follows that F ∧ G is ( m + n − ) -reduced because, as a left class, P n -equivalences are stable by cobase change.Similarly, the map ( F → ) ◻ ( G → ) = ( F ⋆ G → ) is a P m + n − -equivalence, i.e. F ⋆ G is ( m + n − ) -reduced. Theorem 3.4.1 (Blakers-Massey theorem for Goodwillie Calculus) . Let
F HG K gf ⌟ be a pushout square of functors. If f is a P m -equivalence and g is a P n -equivalence, then the induced map ( f, g ) ∶ F → G × K H is a P m + n + -equivalence. roof. If a map h is k -excisive then its diagonal ∆ h is also k -excisive because P k is left exact. Theorem 3.3.4(4) then implies that ∆ f ◻ ∆ g is a P m + n + -equivalence: ∆ f ◻ ∆ g is in the left class of the modality associated to P m + n + .Now we apply the Theorem 2.6.7 and learn that ( f, g ) is in the same left class.Hence, the gap map is a P m + n + -equivalence. Theorem 3.4.2 (“Dual” Blakers-Massey theorem for Goodwillie Calculus) . Let
F HG K ⌜ fg be a pullback square of functors. If f is a P m -equivalence and g is a P n -equivalence, then the cogap map ⌊ f, g ⌋ ∶ G ⊔ F H → K is a P m + n + -equivalence.Proof. By Theorem 3.3.4(4) the map f ◻ g is a P m + n + -equivalence. By Theo-rem 2.6.8 the same holds for the cogap ⌊ f, g ⌋ . In this section, we rederive some of the fundamental delooping results of [Goo03].To begin, let us recall the following definition:
Definition 3.5.1.
Let f ∶ X → Y be a map in a topos E . We say that f isa principal fibration if there exists an object B ∈ E , map h ∶ Y → B and cover b ∶ ↠ B such that the square X Y B f ⌜ bh is cartesian.Fix a functor F ∶ C → S . It is easily checked that if F is k -excisive, then itis n -excisive for any n ≥ k . Hence the universal property of P n implies that forany k ≤ n we have a canonical map q n,k ∶ P n F → P k F . They form the Goodwillietower of F . The map q n,k is a P k -equivalence by construction. We will use theabbreviation q n = q n,n − ∶ P n F → P n − F. Now let us denote by C the pushout of the following diagram P n F P FP n − F C q n, q n F c ⌟ (3)32n [ C , S ] . We obtain an induced map ⌊ q n − , , id P F ⌋ ∶ C → P F so that we mayregard the above diagram as living in the slice category [ C , S ] / P F . Note that c is a P n − -equivalence because, as a left class of a factorization system, P n − -equivalences are closed by cobase change. Clearly id P F is a P n − -equivalence.So C → P F is also a P n − -equivalence. More vaguely stated, C is pointed and n -reduced relative to the constant functor P F .Applying P n to the square (3) one obtains the induced square P n F P FP n − F P n C q n F P n c (4)in [ C , S ] ( n )/ P F . By Remark 3.1.4 this is still a pushout in [ C , S ] ( n )/ P F , but more istrue. Theorem 3.5.2.
The square (4) is cartesian and the map q n ∶ P n F → P n − F is a principal fibration in the topos [ C , S ] ( n )/ P F .Proof. To see that the square is cartesian, it suffices to work in the ambienttopos [ C , S ] , since the forgetul functors [ C , S ] ( n )/ P F → [ C , S ] ( n ) → [ C , S ] preservesand reflects pullbacks. The map q n, is a P -equivalence and q n is a P n − -equivalence. Hence, applying Theorem 3.4.1, we find that the cartesian gapmap P n F → P n − F × P n C P F is a P n -equivalence. But since the source andtarget of this map are n -excisive, the gap map is an isomorphism. Thus thesquare 4 is cartesian as claimed.Unwinding the definition of principal fibration given above in the slice cat-egory [ C , S ] ( n )/ P F , we find that it remains to verify that the map P n c is a cover.In fact, it suffices to check the statement in the category [ C , S ] ( n ) . By Propo-sition A.0.5 it suffices to check that the map P n c is a P -equivalence. But it iseven a P n − -equivalence since c is, as we have already observed.If the category C is pointed, there exists a canonical map F ( ) → F ( X ) forany X ∈ C . This induces maps P F → F → P n F . We define D n F , the n -thhomogeneous layer of F , by the pullback square D n F P n FP F P n − F. Corollary 3.5.3.
The functor P n C is a delooping of D n F in the categories [ C , S ] / P F and [ C , S ] ( n )/ P F in the sense that D n F = Ω P F P n C . Proof.
We have the following commutative diagram D n F P n F P FP F P n − F P n C. ⌜ ⌜ where both squares are pullbacks. We deduce that D n F = Ω P F P n C whereΩ P F denotes the loop functor in the category [ C , S ] / P F . Moreover, as everyobject in the above diagram is n -excisive, we may regard the diagram as livingin the subcategory [ C , S ] ( n )/ P F , and since the inclusion [ C , S ] ( n )/ P F ↪ [ C , S ] / P F isfully-faithful and preserves limits, the second assertion follows as well. Theorem 3.5.4.
Consider a cocartesian square
F HG K gf ⌟ in [ C , S ] where f and g are P n -equivalences and F, G and H are ( n + ) -excisive.Then the induced square F HG P n + K gf is cartesian in [ C , S ] .Proof. By Theorem 3.4.1 the gap ( f, g ) ∶ F → G × K H is a P n + -equivalence. The comparison map G × K H → G × P n + K H induced by K → P n + K is also a P n + -equivalence. So their composition isa P n + -equivalence between ( n + ) -excisive functors. Hence it is an isomor-phism. Corollary 3.5.5. (Arone-Dwyer-Lesh [ADL08, Thm. 4.2])
For every n -reducedfunctor F the canonical map P n − F → Ω P n − Σ F is an isomorphism. If F is also ( n − ) -excisive it is infinitely deloopable. Proof.
The isomorphism follows by applying Theorem 3.4.1 to the pushoutsquare F
11 Σ F. ⌟ Since the class of P n − -equivalence is stable by colimits in E → , the functor Σ F is n -reduced when F is. The theorem may then be iterated by taking P n − Σ F in place of F . Theorem 3.5.6.
Consider a cartesian square
F HG K ⌜ hk in [ C , S ] where h and k are P n -equivalences and G, H and K are ( n + ) -excisive. Then the cogap map ⌊ h, k ⌋ is a P n + -equivalence and the square iscocartesian in [ C , S ] ( n ) .Proof. By Theorem 3.4.2 the cogap map ⌊ h, k ⌋ ∶ G ⊔ F H → K is a P n + -equivalence. Now note that F , as a limit of ( n + ) -excisive functors,is ( n + ) -excisive. Hence P n + ( G ⊔ F H ) is the pushout in the category [ C , S ] ( n ) .So the cogap map in [ C , S ] ( n ) P n + ( G ⊔ F H ) → K is a P n + -equivalence between ( n + ) -excisive functors. Hence it is an isomor-phism. A Truncated and connected maps of n -excisivefunctors Recall that every topos admits a factorization system consisting of the monomor-phisms and covers (whose definition we will recall momentarily). The goal ofthis appendix is to describe this modality in the topos [ C , S ] ( n ) of n -excisivefunctors. Definition A.0.1.
A map f ∶ X → Y in a topos E a monomorphism if itsdiagonal map ∆ f ∶ X → X × Y X is an isomorphism. A map is a cover if it is left orthogonal to every monomor-phism. 35 xample A.0.2. In the category of spaces S , a map is a monomorphism if andonly if it is the inclusion of a union of path components. The covers in spacesare exactly the maps that induce a surjection on the set of path components.Before stating the next result, recall that an object X in a category C is called discrete if the space C ( K, X ) is discrete for every object K ∈ C . Moreover,[Lur09, Proposition 5.5.6.18] shows that if the category C is presentable (forexample, if C is a topos) then the inclusion of the full subcategory E ↪ E ofdiscrete objects admits a left adjoint τ ∶ E → E which we will refer to as 0 -truncation functor . A posteriori, we may characterize the discrete objects X ∈ E as those for which the canonical map X → τ X is an isomorphism. Example A.0.3.
Here are examples of discrete objects.1. If E = S , then τ is the functor sending a space X to its set π X of pathcomponents regarded as a discrete space.2. If E = [ C , S ] it is not hard to see that ( τ F )( K ) = π ( F ( K )) for every F ∈ E and every K ∈ C , so that a functor F is discrete if and onlyif it takes values in discrete spaces.The following folklore proposition asserts that monomorphisms and coversin a topos E are essentially determined by their restriction to discrete objects. Proposition A.0.4.
Let f ∶ X → Y be a morphism in a topos. (1) The map f is a monomorphism if and only if τ f is a monomorphism andthe square X τ XY τ Y f ⌜ τ f is a pullback. (2) The map f is a cover if and only if τ f is a cover. We will prove the following result which characterizes monomorphisms andcovers in the topos of n -excisive functors: Theorem A.0.5.
Let f ∶ F → G be a map in [ C , S ] ( n ) . Then: (1) The map f is monomorphism if and only if P f is a monomorphism andthe square F P FG P G f ⌜ P f is a pullback. The map f is a cover if and only if P f is a cover. Remark A.0.6.
We invite the reader to observe the similarity between Propo-sition A.0.4 and Theorem A.0.5. The category [ C , S ] ( ) is equivalent to thecategory of spaces and the functor P is equivalent to the evaluation functor F ↦ F ( ) . It follows from the theorem that a map f ∶ X → Y in [ C , S ] ( n ) is acover if and only if the map f ( ) ∶ X ( ) → Y ( ) is a cover in the category ofspaces.We begin with some generalities: let us suppose that we have a left exactlocalization P ∶ E → F with fully faithful right adjoint i ∶ F ↪ E . We will write τ E and τ F for the 0-truncation functors of E and F respectively. Now, it followsfrom the fact that P preserves colimits that P also commutes with 0-truncation.That is P τ E ≃ τ F P (5)On the other hand, the inclusion i does not , in general, preserve colimits andhence when we identify F with a full subcategory of E , we must distinguishbetween these two distinct operations.Specializing to the case at hand, the following notation will be convenient: Definition A.0.7.
In what follows, we write τ for 0-truncation in [ C , S ] and τ ( n ) for the 0-truncation functor in the n -excisive localization [ C , S ] ( n ) . Remark A.0.8.
The case n = i ∶ [ C , S ] ( ) ↪ [ C , S ] admits both a left and right adjoint givenrespectively by left and right Kan extension along the inclusion of the terminalobject 1 ↪ C . As a consequence, the 0-truncation functors τ ( ) and τ do coincide on the essential image of i , which can be identified with the constant functors. Lemma A.0.9.
A discrete functor F ∶ C → S is -excisive if and only it isconstant.Proof. If F ∶ C → S is 1-excisive, then the following square is cartesian for every K ∈ C : F ( K ) F ( ) F ( ) F ( Σ K )⌜ But the map F ( Σ K ) → F ( ) is a left inverse of the map F ( ) → F ( Σ K ) ,since the map Σ K → → Σ K . Hence the map F ( ) → F ( Σ K ) is monic, since F ( Σ K ) is discrete by hypothesis. It follows thatthe map F ( K ) → F ( ) is invertible since the square above is cartesian.The preceding lemma allows us to calculate the action of the 0-truncationfunctor τ ( ) in the category of 1-excisive functors. The result asserts that the0-truncation of a 1-excisive F is constant with value the 0-truncation of thespace F ( ) . 37 emma A.0.10. For F ∈ [ C , S ] ( ) , we have τ ( ) F = τ P F Proof.
Note that the functor τ ( ) F is both discrete and 1-excisive by definition.According to Lemma A.0.9, then, it is constant. The functor τ P F is alsoconstant and hence, to show that the two agree, it suffices to show they agreeafter evaluation at 1 ∈ C . But we have τ ( ) ( F )( ) = P τ ( ) ( F )( ) = τ ( ) P ( F )( ) = τ P ( F )( ) Where the first equality is by the definition of P , the second is an applicationof 5 to the localization P ∶ [ C , S ] ( ) → [ C , S ] ( ) , and the last follows fromRemark A.0.8. Proposition A.0.11.
Every n -excisive monomorphism is -excisive.Proof. The proof is by induction on n ≥
0. The case n = n =
1. So let f ∶ F → G be a 1-excisive monomorphism. The map P f is monic, since the functor P is a left exact localization. The following squareis cartesian, since f is 1-excisive. F P FG P G f ⌜ P f Hence it suffices to show that P f is 0-excisive. Hence we may suppose, withoutloss of generality, that F and G are in fact 1-excisive functors. Consider thecube: F P Fτ ( ) F τ P FG P Gτ ( ) G τ P G ≃≃ Let us show that the back face is a pullback. All of the vertical maps aremonomorphisms since the functors P , τ and τ ( ) preserve them. Both the leftand the right face are pullbacks by Proposition A.0.4. By Lemma A.0.10, thefront two horizontal maps are in fact isomorphisms, since F and G are 1-excisive.Consequently, the back face is a pullback, which says that f is 0-excisive.38or the inductive step, let f ∶ F → G be a ( n + ) -excisive monomorphism.Note that the functor { w K , −} preserves monomorphisms. So, for all K , { w K , f } is a monomorphism. But it is also n -excisive by Theorem 3.3.4.(1). By theinduction hypothesis, it is then 0-excisive. This, in turn, shows that f is 1-excisive. Then the case n = f is also 0-excisive. Proof of Theorem
A.0.5 . (1 ⇒ ). This is immediate since P preserves monomor-phisms and the pullback expresses just the statement that f is 0-excisive.(1 ⇐ ). Monomorphisms are always stable by pullback.(2 ⇒ ). The functor P preserves covers because it is a localization.(2 ⇐ ). Note that f is a cover if and only if it is orthogonal to every monomor-phism in [ C , S ] ( n ) . So let g ∶ H → K be such a monomorphism and consider alifting problem as follows: F HG K f g
Note that since g is a monomorphism, it is enough to show that a lift exists, asits uniqueness is automatic. Now apply the functor P to obtain P F P HP G P K P f P g ∃ ! Observe that the left map is a cover by assumption. Since P preserves monomor-phisms, this square has a unique lift. But now composition of our lift with themap p G ∶ G → P G yields the lift shown in the diagram: F H P HG K P K f g ⌜ P g ∃ ! On the other hand, Proposition A.0.11 asserts that the right hand square is apullback. Hence we have an induced unique lift to the original problem. Thisshows that f is a cover. References [ABFJ17] Mathieu Anel, Georg Biedermann, Eric Finster, andAndr´e Joyal. A generalized Blakers-Massey theorem. https://arxiv.org/abs/1703.09050 , 2017.[ADL08] Gregory Z. Arone, William G. Dwyer, and Kathryn Lesh. Loop struc-tures in Taylor towers.
Algebr. Geom. Topol. , 8(1):173–210, 2008.39BJM15] Kristine Bauer, Brenda Johnson, and Randy McCarthy. Cross ef-fects and calculus in an unbased setting.
Transactions of the AmericanMathematical Society , 367(9):6671–6718, 2015.[BR14] Georg Biedermann and Oliver R¨ondigs. Calculus of functors and modelcategories, II.
Algebr. Geom. Topol. , 14(5):2853–2913, 2014.[Goo90] Thomas G. Goodwillie. Calculus I: The First Derivative of Pseu-doisotropy Theory.
K-Theory , 4, 1990.[Goo92] Thomas G. Goodwillie. Calculus II: Analytic Functors.
K-Theory , 5,1992.[Goo03] Thomas G. Goodwillie. Calculus III: Taylor Series.
Geometry andTopology , 7, October 2003.[Heu15] Gijs Heuts. Goodwillie approximations to higher categories. arXiv:1510.03304 , October 2015.[Joy08] Andre Joyal. Notes on quasicategories. , 2008.[Kuh07] Nicholas J Kuhn. Goodwillie towers and chromatic homotopy: anoverview.
Proceedings of the Nishida Fest (Kinosaki 2003) , 10:245–279, 2007.[Lur09] Jacob Lurie.
Higher topos theory , volume 170 of
Annals of MathematicsStudies . Princeton University Press, Princeton, NJ, 2009.[Lur16] Jacob Lurie. Higher algebra. ,2016.[Rez05] Charles Rezk. Toposes and homotopy toposes. , 2005.[Rez13] Charles Rezk. A streamlined proof of Goodwillie’s n -excisive approxi-mation. Algebr. Geom. Topol. , 13(2):1049–1051, 2013.[Wei95] Michael Weiss. Orthogonal calculus.