3.7 Higher Segal objects in the additive or stable context
3.7.2 Goodwillie calculus in stable ∞ -categories
Homotopy coherent theorems of Dold–Kan type 109/118 Given a small pointed categoryQand a natural numberk∈N, we define the full subcategories P?≤k(Q) :=
_
j∈I
qj
|I| ≤k, qj ∈Q
⊂ P?(Q) and P≤k(Q) :=
a
j∈I
qj
|I| ≤k, qj ∈Q
⊂ P(Q)
spanned by those objects which are the wedge/coproduct of at most k representables. We denote by P?<∞(Q) :=S
k∈NP?≤k(Q) the full subcategory of P?(Q) spanned by all finite wedges of representables.
Example 3.7.2.3. Consider the pointed category{1}+ obtained by freely adding a basepoint to the terminal category {1}. Explicitly, {1}+ is the category
0 1
where the composite 0→ 1→ 0 is the identityId0. It follows from Remark 3.1.1.6 (using that {1}+ is self-dual) that there is a canonical equivalence
P?({1}+)−−→' S? (3.7.2)
which sends a presheafX: {1}op+ →Sto the pointed space?=X(0)→ X(1), hence in particular the representables 0,1∈ {1}+ to the pointed spaces 0 = ptand S0, respectively. It follows that the equivalence (3.7.2) restricts to equivalences
P?<∞(Q)−−→' Fin? and P?≤k({1}+)−−→' Fin≤k?
becauseFin? andFin≤k? are precisely the full subcategories ofS? spanned by wedges of finitely
many/at most k many copies ofS0. ♦
Lemma 3.7.2.4. Let Z be an ∞-category and let C be a presentable ∞-category. For each object x∈Z, denote byx!:C→Fun(Z,C) the left Kan extension functor along x: {?} →Z.
(1) The functor x!is given explicitly onX ∈Cby
x!(X)'MapZ(x,−)⊗X.
where⊗:S×C→Cis the canonical tensoring ofCoverS(see [Lur09, Proposition 4.8.1.15]).
(2) The functor categoryFun(Z,C)is generated under colimits by the functorsx!(X)forx∈Z
and X∈C.
Proof. Lemma 3.7.2.4 is precisely the content of the first two paragraphs in the proof of [Lur17, Theorem 6.1.5.6], where it is stated for the specific ∞-categoryZ=P≤n(C) but proved in a way
that works for allZ.
Recall that a functor is called finitary if it preserves filtered colimits. We will need the following theorem, which classifies finitaryk-excisive functors in presentable stable∞-categories.
Theorem 3.7.2.5. [Lur17, Theorem 6.1.5.6]. LetZbe a small∞-category andDa presentable stable ∞-category. Fix a natural number k ∈ N and let F: P(Z) → D be a functor. The following are equivalent:
(1) The functor F is a left Kan extension of its restriction to P?≤k(Z)
(2) The functor F is k-excisive and preserves filtered colimits.
More specifically, we need the following pointed version.
Corollary 3.7.2.6. LetQbe a small pointed∞-category andDa presentable stable∞-category.
Fix a natural number k∈Nand letF:P?(Q)→Dbe a functor. The following are equivalent:
(1) The functor F is a left Kan extension of its restriction to P?≤k(Q)
(2) The functor F is k-excisive and preserves filtered colimits.
Proof. We first prove that (1) implies (2), mimicking the proof of [Lur17, Theorem 6.1.5.6]. It is enough to show (2) wheneverF is a functorq!(X) :P?(Q)→Dfor someq∈ P?≤k(Q)andX∈D, because these are the left Kan extensions of the homonymous functorsq!(X) : P?≤k(Q)→Dwhich by Lemma 3.7.2.4 generate under colimits the∞-categoryFun(P?≤k(Q),D). Ifqis actually repre-sentable (i.e., lies inQ⊂ P?≤k(Q)) then it follows from Lemma 3.7.2.1 thatMapP?(Q)(q,−)(which by the Yoneda lemma is just evaluation atq) preserves filtered colimits and pushouts. Ifq=W
qj
is the wedge of at mostkrepresentables then if follows thatMapP?(Q)(q,−)'Q
jMapP?(Q)(qj,−) preserves filtered colimits (because in S filtered colimits commute with products) and sends strongly coCartesian [k]-cubes to coCartesian [k]-cubes (because in S the product of at most k strongly coCartesian [k]-cubes is coCartesian7)). It follows that q!(X) is finitary and k-excisive, because− ⊗X:S→Dpreserves all colimits and because coCartesian cubes inDare Cartesian by stability.
For the converse we must show that each finitary k-excisive functor P?(Q) → Dis the left Kan extension of some functor P?≤k(Q)→D. Since the localization functor L sends coproducts to wedges, it induces a commutative square
P≤k(Q) P(Q)
P?≤k(Q) P?(Q)
L L
to which we applyFun(−,D) to obtain the following commutative square of adjoint pairs:
Fun(P≤k(Q),D) Fun(P(Q),D)
Fun(P?≤k(Q),D) Fun(P?(Q),D)
LKE
⊥
Res
L! a L? i?=L! a L?=i?
LKE
⊥
Res
(3.7.3)
For each finitaryk-excisive functorF:P?(Q)→D, the functorL?F:P(Q)→Dis again finitary and k-excisive (because L preserves colimits); hence we can apply Theorem 3.7.2.5 to obtain a functor g: P?≤k(Q) →D whose left Kan extension along P≤k(Q) ,→ P(Q) is L?F. Then by the commutativity of (3.7.3), the functorF 'L!L?F is the left Kan extension alongP?≤k(Q),→ P?(Q)
of L!g.
Denote by
Excdf(P?(Q),D)⊂Funf(P?(Q),D)
the full subcategory generated under colimits by the finitary k-excisive functors for all k ∈N. We call the functors in Excdf(P?(Q),D) coanalytic.
Corollary 3.7.2.7. With Q and D as in Corollary 3.7.2.6, restriction and left Kan extension along P?<∞(Q),→ P?(Q) give rise to an equivalence
Excdf(P?(Q),D)←−→' Fun(P?<∞(Q),D).
which for eachk∈Nrestricts to an equivalence
Exckf(P?(Q),D)←−→' Fun(P?≤k(Q),D). (3.7.4)
7) This follows from [Lur17, Lemma 6.1.5.8] using the fact that the cartesian product in S commutes with colimits in each variable.
Homotopy coherent theorems of Dold–Kan type 111/118 between the ∞-categories of finitary k-excisive functors P?(Q) → D and (arbitrary) functors
P?≤k(Q)→D.
Proof. Left Kan extension alongP?<∞(Q),→ P?(Q)is fully faithful and induces equivalences (3.7.4) by Corollary 3.7.2.6. We need to show that its essential image agrees with Excdf(P?(Q),D) as full sucategories of Fun(P?(Q),D). The essential image is closed under colimits (because Fun(P?<∞(Q),D) has all colimits and left Kan extension preserves them), and contains all fini-tary k-excisive functors (for all k); hence the essential image contains Excdf. Conversely, every functor F in the essential image can be written as the colimit F ' colimk∈NLKE(F
P?≤k(Q)) because the category P?<∞(Q) is the ascending union of the full subcategoriesP?≤k(Q).
Let us now focus on the special case Q= {1}+ described in Example 3.7.2.3. We have the following commutative diagram where the left half is described by Corollary 3.7.2.7 and the right half is induced from the Dold–Kan type equivalence (3.3.2).
Excdf(S?,D) Fun(Fin?,D) Fun(Surj,D)
Exckf(S?,D) Fun(Fin≤k? ,D) Fun(Surj≤k,D)
Exck−1f (S?,D) Fun(Fin≤k−1? ,D) Fun(Surj≤k−1,D)
'
Pk a P k
'
'
Pk−1 a P k−1
LKE a Res
'
LKE a Res
' '
LKE a Res LKE a Res
(3.7.5)
We denote by P k:Excdf(S?,D)→Exckf(S?,D)the functor corresponding to restriction along Fin≤k? ,→ Fin?; it is right adjoint to the inclusion Exckf(S?,D) → Excdf(S?,D) hence deserves the name k-coTaylor approximation. It follows from Corollary 3.7.2.7 that each coanalytic functor F:S? →Dis the colimit of its coTaylor filtration
P 0(F)−→ P 1(F)−→ · · · −→ P k(F)−→ · · · −→F.
We say that a finitary functor S → D is k-cohomogeneous if it is k-excisive and has vanish-ing k-coTaylor approximation. Under the equivalence of (3.7.5), thek-cohomogeneous functors correspond precisely to those diagrams Surj→Dwhich are non-zero only in degreek.
Now we can describe the adjunctions appearing in the rightmost columns of (3.7.5) more explicitly. Fix k∈N. Consider the full subcategories Surj≤k−1 and Surj=k 'BSk of Surj≤k spanned by the objects hni ∈Surj≤k withn≤kand n=k respectively.
We have the following ladders of adjunctions given by Kan extension (left adjoints always on top).
Fun(Surj≤k−1,D) Fun(Surj≤k,D) Fun(BSk,D)
LKE
⊥
Res
RKE
Res
⊥
RKE Pk−1
LKE
(3.7.6)
Observe that in Surj≤k there are no arrows going from Surj≤k−1 to Surj=k; it follows that
• the essential image of left Kan extension alongSurj≤k−1 ,→Surj≤kis precisely the kernel of the restriction along BSk,→Surj≤k and
• the essential image of right Kan extension alongBSk ,→Surj≤k is precisely the kernel of the restriction alongSurj≤k−1,→Surj≤k.
This implies that the ladder (3.7.6) can be completed with the dashed adjoints to a ladder of recollements in the sense of [BBD82; BGS88; AKLY17]. Note that under the correspon-dence (3.7.5) the left dashed functor corresponds precisely to the Taylor approximation functor Pk−1; its kernel—which corresponds to the ∞-category of finitary k-homogeneous functors—
is precisely the essential image of left Kan extension along BSk ,→ Surj≤k. In other words, restriction and left Kan extension give rise to an equivalence
Homogkf(S?,D)←−→' Fun(BSk,D) =Sk−repD
of ∞-categories between finitaryk-homogeneous functors and coherent representations in D of the symmetric group Sk. Similarly, restriction and right Kan extension alongBSk ,→ Surj≤k give rise to an equivalence
coHomogkf(S?,D)←−→' Fun(BSk,D) =Sk−repD
of ∞-categories between finitary k-cohomogeneous functors and coherent representations in D of the symmetric group Sk.
Warning 3.7.2.8. The ∞-categories of finitary k-homogeneous and k-cohomogeneous functors S? → D are both abstractly equivalent to Sk−repD, hence to each other. However, they do not form the same subcategory of Excdf(S?,D) but are, in the language of semiorthogonal
decompositions [BK89],mutations of each other. ♦