Coalgebras for endofunctors

In this section we investigate the relation between the categories of coalgebras and fixed points for endofunctors. This will be applied in the proof of the equivalence between

“genuine” and “naive” cyclotomic spectra that will be given in the next section (Theo-remII.6.3 and TheoremII.6.9), where the key step is to upgrade the lax mapX!X^tCp to an equivalenceY'Φ^CpY of a related objectY.

Definition II.5.1. LetCbe an∞-category andF:C!Cbe an endofunctor. Then an F-coalgebra is given by an object X∈C together with a morphismX!F X. A fixpoint ofF is an objectX∈Ctogether with an equivalenceX−^∼!F X.

We define the∞-category CoAlg_F(C) as the lax equalizer CoAlg_F(C) = LEq

C ^id //

F //C

of∞-categories; cf. Definition II.1.4 for the notion of lax equalizers. If the ∞-category C is clear from the context, we will only write CoAlg_F. The∞-category FixF=FixF(C) is the full subcategory FixF⊆CoAlg_F spanned by the fixed points. The example that is relevant for us is thatC is the ∞-categoryCp^∞Sp of genuineCp^∞-spectra and F is the endofunctor Φ^Cpas discussed in DefinitionII.2.15. The reader should have this example in mind, but we prefer to give the discussion abstractly.

IfC is presentable andF is accessible, then it follows from Proposition II.1.5 that CoAlg_F is presentable and that the forgetful functor CoAlg_F!C preserves all colimits.

Now, assume that the functor F preserves all colimits, i.e. admits a right adjoint G.

Then, Fix_Fis an equalizer in the∞-categoryPr^Lof presentable∞-categories and colimit preserving functors (see [69, Proposition 5.5.3.13] for a discussion of limits inPr^L). As such, it is itself presentable and the forgetful functor Fix_F!C preserves all colimits.

As a consequence, the inclusion ι: Fix_F⊆CoAlg_F preserves all colimits, and thus, by the adjoint functor theorem [69, Corollary 5.5.2.9], the functorι admits a right adjoint R_ι: CoAlg_F!Fix_F. We do not know of an explicit way of describing the colocalization ιRι in general, but we will give a formula under some extra hypothesis now.

Construction II.5.2. There is an endofunctorFgiven on objects by F: CoAlg_F−!CoAlg_F,

(X−^ϕ!F X)7−!(F X−−^{F ϕ}!F²X),

which comes with a natural transformation µ:id! F. As a functor, Fis induced from the map of diagrams of∞-categories

using the functoriality of the lax equalizer. Similarly, the transformation µ can be de-scribed as a functor

which is given in components as the projection LEq C ^id//

F //C

!C^∆1, and the map C×_C×CC^∆1−!C^∆1×∆1∼=C^∆2×_C∆1C^∆2

which sendsX!F X to the 2-simplices

The last map, which we described on vertices refines to a map of simplicial sets since it is a composition of degeneracy maps and the mapF:C!C.

Recall the standing assumption that the underlying ∞-category C is presentable, and thatF preserves all colimits. By the adjoint functor theorem, the functorFfrom ConstructionII.5.2 admits a right adjointR_F: CoAlg_F!CoAlg_F. We will describeR_F more explicitly below (LemmaII.5.4). There is a transformationν:R_F!id obtained as the adjoint of the transformationµfrom ConstructionII.5.2.

Proposition II.5.3. The endofunctor ιRι: CoAlg_F!CoAlg_F is given by the limit of the following diagram of endofunctors

...−!R³_F ^R

2 Fν

−−−−!R²_F−−−−^RFν!R_F−−^ν!id.

Proof. As a first step, we use that the inclusionι: FixF⊆CoAlg_F not only admits a right adjoint, but also a left adjointLι: CoAlg_F!FixF which can be described concretely as follows: the composition of Lι with the inclusion ι: FixF!CoAlg_F is given by the colimit of functors precisely, the transformation id! F∞coming from the colimit structure maps), and that the local objects for this localization are precisely given by the fixed-point coalgebras.

We first note that, by definition ofF, the subcategory FixF⊆CoAlg_F can be described as those coalgebras X for which the transformation µ_X:X! F(X) is an equivalence.

Since Fpreserves colimits, it follows immediately that, for a given coalgebra X, the coalgebra F_∞X is a fixed point. We also make the observation that, for a given fixed pointX∈FixF, the canonical mapX! F_∞X is an equivalence. If follows now thatF_∞ applied twice is equivalent to F_∞, which shows that F_∞ is a localization (using [69, Proposition 5.2.7.4]), and also that the local objects are precisely the fixed points.

Now, the functorιL_ι: CoAlg_F!CoAlg_F is left adjoint to the functorιR_ι: CoAlg_F! CoAlg_F. Thus, the formula forF_∞ as a colimit implies immediately the claim, as the right adjoint to a colimit is the limit of the respective right adjoints.

Note that the limit in Proposition II.5.3 is taken in the ∞-category of endofunc-tors ofF-coalgebras which makes it potentially complicated to understand, since limits of coalgebras can be hard to analyze. However, if for a given coalgebra X!F X the endofunctorF:C!C preserves the limit of underlying objects of the diagram

...−!R³_FX−!R²_FX−!R_FX−!X,

then, by PropositionII.1.5(v), the limit of coalgebras is given by the underlying limit.

This will be the case in our application, and the reader should keep this case in mind.

Now, we give a formula for the functorR_F: CoAlg_F!CoAlg_F, which was by defini-tion the right adjoint of the functorF: CoAlg_F!CoAlg_F from ConstructionII.5.2. To simplify the treatment, we make some extra assumptions onF. Recall that, by assump-tion,F:C!Cadmits a right adjoint. Denote this right adjoint byR_F:C!C. We assume that the counit of the adjunctionF R_F!id_C be a natural equivalence. This is equivalent to R_F being fully faithful. We will denote byη: id_C!R_FF the unit of the adjunction.

We moreover assume thatF preserves pullbacks.

Lemma II.5.4. The functor R_F: CoAlg_F!CoAlg_F takes a coalgebra X−^ϕ!F X to the coalgebra R_F(X−^ϕ!F X)=Y−^ϕ!F Y whose underlying object Y is the pullback

Y //

RFX

R_Fϕ

X ^ηX //RFF X

in C,and where the coalgebra structure ϕ:Y!F Y is given by the left vertical map under the identification F Y−^∼!F RFX−^∼!X induced by applyingF to the upper horizontal map in the diagram and the counit of the adjunction between F and RF.

Moreover, the counit F R _F!id is an equivalence.

Proof. We will check the universal mapping property using the explicit description of mapping spaces in∞-categories of coalgebras given in PropositionII.1.5(ii). We get that maps from an arbitrary coalgebraZ!F Zinto the coalgebraY!F Y'X are given by the equalizer

Eq(Map_C(Z, Y) ////Map_C(Z, X))

We use the definition ofR_FX as a pullback to identify the first term with Map_C(Z, R_FX)×_{MapC(Z,RFF X)}Map_C(Z, X).

Under this identification, the two maps to Map_C(Z, X) are given as follows: the first map is given by the projection to the second factor. The second map is given by the projection to the first factor followed by the map

Map_C(Z, R_FX)−−^F!Map_C(F Z, F R_FX)−−^∼!Map_C(F Z, X)−!Map_C(Z, X), where the middle equivalence is induced by the counit of the adjunction, and the final map is precomposition withZ!F Z. As a result we find that

Eq(Map_C(Z, Y) ////Map_C(Z, X)) ' Eq(Map_C(Z, RFX) ////Map_C(Z, RFF X)) ' Eq(Map_C(F Z, X) ////Map_C(F Z, F X)) ' Map_CoAlg

F(F(Z), X).

This shows the universal property, finishing the identification of R_F(X). The explicit description implies that the counit F R _F(X)!X is an equivalence, as on underlying objects, it is given by the equivalenceF RFX'X.

Corollary II.5.5. Under the same assumptions as for Lemma II.5.4,the underly-ing object of the k-fold iteration R^k_FX is equivalent to

R^k_FX×_Rk

FF XR^k−1_F X×_Rk−1

F F X...×_{RFF X}X,

where the maps to the right are induced by the coalgebra map X!F X and the maps to the left are induced by the unit η. In this description, the map R^k_FX!R^k−1_F X can be described as forgetting the first factor.

Proof. This follows by induction on n using thatR_F, as a right adjoint, preserves pullbacks andF does so by assumption.

Now, let us specialize the general discussion given here to the case of interest. We letCbe the∞-categoryCp^∞Sp of genuineCp^∞-spectra andF be the endofunctor Φ^Cpas discussed in DefinitionII.2.15. Then the∞-category of genuinep-cyclotomic spectra is given by the fixed points of Φ^Cp (see DefinitionII.3.1). In this case, all the assumptions made above are satisfied: Cp^∞Sp is presentable as a limit of presentable ∞-categories along adjoint functors, the functor Φ^Cp preserves all colimits, and its right adjoint is fully faithful, as follows from PropositionII.2.14and the discussion after DefinitionII.2.15.

Theorem II.5.6. The inclusion ι: Cyc Sp^gen_p ⊆CoAlg_ΦCp(C_p^∞Sp) admits a right ad-joint R_ι such that the counit ιR_ι!id of the adjunction induces an equivalence of under-lying non-equivariant spectra.

Proof. Let us follow the notation of PropositionII.2.14and denote the right adjoint of Φ^Cp by RC_p:Cp^∞Sp!Cp^∞Sp. Then, the right adjoint of ¯Φ^Cp will be denoted by R_Cp: CoAlg_ΦCp!CoAlg_ΦCp.

We have that (RC_pX)^Cpm'X^Cpm−1 form>1 and (RC_pX)^{e}'0, as shown in Corol-lary II.2.16. If we use the formula given in Corollary II.5.5, then we obtain that, for (X, ϕ:X!F X)∈CoAlg_ΦCp, the underlying object of R^k_C

We see from this formula that, as soon askgets bigger thanm, this becomes independent ofk. Thus, the limit of the fixed points

...−!(R³_C

which is also eventually constant on all fixed points. In particular, ¯Φ^Cp commutes with the limit of the tower. Thus, by PropositionII.1.5(v), the limit of coalgebras is computed as the limit of underlying objects. Then, by PropositionII.5.3, we get that the limit of the sequence (2) are the C_pm-fixed points of the spectrum R_ιX. Specializing to m=0, we obtain that the canonical map (ιRιX)^{e}!X^{e} is an equivalence, as desired.

Now we study a slightly more complicated situation that will be relevant for global cyclotomic spectra. Therefore, letF1andF2be two commuting endomorphisms of an ∞-categoryC. By this, we mean that they come with a chosen equivalenceF1F2'F2F1, or equivalently they define an action of the monoid (N×N,+) on the ∞-category C, where (1,0) acts by F1 and (0,1) acts by F2. Since actions of monoids on∞-categories can always be strictified (e.g. using [69, Proposition 4.2.4.4]), we may assume without loss of generality thatF1 andF2 commute strictly as endomorphisms of simplicial sets.

We will make this assumption to simplify the discussion.

Then,F₂induces an endofunctor CoAlg_F

1(C)!CoAlg_F

1(C) which sends the coalge-braX!F₁Xto the coalgebraF₂X!F₂F₁X=F₁F₂X (defined as in ConstructionII.5.2).

This endofunctor restricts to an endofunctor of Fix_F1(C)⊆CoAlg_F1(C). Our goal is to study the∞-category

CoAlg_F

1,F₂(C) := CoAlg_F

2(CoAlg_F

1(C)).

In fact, we will directly consider the situation of countably many commuting endofunctors (F1, F2, ...). This can again be described as an action of the monoid (N>0,·)∼=(⊕^∞_i=1N,+) which we assume to be strict for simplicity. Then, we inductively define

CoAlg_F

Remark II.5.7. Informally, an object in the∞-category CoAlg_(F

i)i∈N(C) consists of

• an objectX∈C;

• morphismsX!F_i(X) for all i∈N;

• a homotopy between the two resulting mapsX!F_iF_j(X) for all pairs of distinct i, j∈N;

• a 2-simplex between the three 1-simplices in the mapping space X−!F_iF_jF_k(X)

for all triples of distincti, j, k∈N;

• ...

We will not need a description of this type, and therefore will not give a precise formulation and proof of this fact.

If we assume that C is presentable and all F_i are accessible, then the∞-category CoAlg_(F

i)i∈N(C) is presentable. To see this, we first inductively show that all∞-categories CoAlg_F

1,...,F_n(C) are presentable, by Proposition II.1.5(iv). Then, we conclude that CoAlg_(Fi)i∈N(C) is presentable as a limit of presentable ∞-categories along colimit-preserving functors. Also, this shows that the forgetful functor to CoAlg_(Fi)i∈N(C)!C preserves all colimits.

LemmaII.5.8. Let C be a presentable∞-category with commuting endofunctors F_i. Moreover,assume that all the functors Fi are accessible and preserve the terminal object of C.

(i) Assume that, for an object X∈C, the objects Fi(Fj(X)) are terminal for all distinct i, j∈N. Then, isomorphism classes of (Fi)_i∈N-coalgebra structures on X are in bijection with isomorphism classes of families of morphismsϕi:X!Fi(X)for all i,with no further dependencies between different ϕi’s.

(ii) In the situation of (i), assume that X has a coalgebra structure giving rise to morphisms ϕi:X!Fi(X) for all i. Then, for any coalgebra Y∈CoAlg_(Fi)i∈N(C), the

(iii) Assume that, for all distinct i, j∈N and all X∈C, the object Fi(Fj(X)) is terminal. Then CoAlg_(Fi)i∈N(C)is equivalent to the lax equalizer

LEq

Proof. We prove parts (i) and (ii) by induction on the number of commuting endo-morphisms (or rather the obvious analogue of the first two statements for a finite number of endofunctors). Then, part (iii) follows from parts (i) and (ii) and PropositionII.1.5.

Thus, fix X∈C such that Fi(Fj(X)) is terminal for all distinct i, j∈N. For a sin-gle endomorphism, the result follows from Proposition II.1.5(ii). Thus, assume that parts (i) and (ii) have been shown for endomorphisms F1, ..., Fn and we are given an additional endomorphism Fn+1. We know, by induction, that a refinement of X to an object of CoAlg_(F1,...,Fn)(C) is given by choosing maps X!Fi(X) for i6n. Now, a refinement to an object of CoAlg_{(F1,...,Fn,Fn+1)}(C)'CoAlg_Fn+1(CoAlg_(F1,...,Fn)(C)) con-sists of a mapX!Fn+1(X) of objects in CoAlg_(F1,...,Fn)(C). We observe thatFn+1(X) also satisfies the hypothesis of (i). Invoking part (ii) of the inductive hypothesis and the fact thatFi(Fn+1(X)) is terminal fori6n, we deduce that a map X!Fn+1(X) in CoAlg_(F1,...,Fn)(C) is equivalent to a mapX!Fn+1(X) inC. This shows part (i).

To prove part (ii), we use the formula for mapping spaces in a lax equalizer given in PropositionII.1.5: the space of maps for any coalgebra

Y∈CoAlg_Fn+1(CoAlg_(F1,...,Fn)(C)) toX is given by the equalizer

Eq Map_CoAlg

(F1,...,Fn)(C)(Y, X) ////Map_CoAlg

(F1,...,Fn)(C)(Y, Fn+1(X)) . By part (ii) of the inductive hypothesis, the first term is given by the equalizer

Eq Map_C(Y, X) ////Qn the mapping space in question is given by an iterated equalizer which is equivalent to

Eq Map_C(Y, X) ////Qn+1

i=1 Map_C(Y, Fi(X)) , which finishes part (ii) of the induction.

This induction shows parts (i) and (ii) for a finite number of endomorphisms. Now, the claim for a countable number follows by passing to the limit and noting that, in a limit of∞-categories, the mapping spaces are given by limits as well.

Now, we define

Fix_(Fi)i∈N(C)⊆CoAlg_(Fi)i∈N(C)

to be the full subcategory consisting of the objects for which all the morphismsX!Fi(X) are equivalences.

Lemma II.5.9. Let C be an ∞-category with commuting endofunctors (F₁, F₂, ...).

Then, we have an equivalence

Fix_(Fi)i∈N(C)' C^hN>0 Proof. We have that

C^hN>0' C^hN∞'lim

←−(C^hNk) and also thatC^hNk+1' C^hNkhN

. Using this decomposition and induction, we can reduce the statement to showing that, for a single endomorphism, FixF(C)'C^hN. But this follows, since the homotopy fixed points for N are the same as the equalizer of the morphismC!C given by acting with 1∈Nand the identity. This can, for example, be seen using thatNis the free A∞-space on one generator.

Now, we assume that all the commuting endofunctorsFi:C!Care colimit-preserving and thatC is presentable. Then, the∞-category

Fix_(Fi)i∈N(C)⊆CoAlg_(Fi)i∈N(C)

is presentable, since it is a limit of presentable∞-categories along left adjoint funtors, by Proposition II.5.9. Moreover, the inclusion into CoAlg_(Fi)i∈N preserves all colimits.

Thus, it admits a right adjoint. Our goal is to study this right adjoint. The idea is to factor it into inclusions

ιn: FixF₁,...,F_n(CoAlg_{Fn+1,Fn+2,...})⊆FixF₁,...,Fn−1(CoAlg_Fn,Fn+1,...)

and understand the individual right adjointsRι_n. To this end, we note that we have an endofunctor

Fn: FixF1,...,Fn−1(CoAlg_Fn,Fn+1,...)−!FixF1,...,Fn−1(CoAlg_Fn,Fn+1,...)

which is on underlying objects given byX7!F_nX. This follows using ConstructionII.5.2 and the isomorphism of simplicial sets

Fix_{F1,...,Fn−1}(CoAlg_F

n,F_n+1,...)'CoAlg_F

n(Fix_{F1,...,Fn−1}(CoAlg_F

n+1,F_n+2,...)).

Invoking the results given at the begining of the section, we also see thatF_n admits a right adjointR_Fn which comes with a natural transformationν:R_Fn!id, and that the following result is true.

LemmaII.5.10. The endofunctorιnRι_nis given by the limit of the following diagram

Now, as a last step, it remains to give a formula for the right adjointR_Fn. To get such a formula, we will make very strong assumptions (with the geometric fixed point functor Φ^Cp in mind). So, we will assume that, for all n, the functor Fn:C!C admits a fully faithful right adjointRF_nsuch that, for alli6=n, the canonical morphismFiRF_n!RF_nFi

adjoint toFnFiRF_n=FiFnRF_n'Fi is an equivalence. Moreover, we assume that all the functorsFi commute with pullbacks.

Lemma II.5.11. Under these assumptions, the underlying object of R_F^k

nX for a

where the maps to the right are induced by the coalgebra map X!FnX and the maps to the left are induced by the unit η of the adjunction between Fn and RF_n. In this description, the map R^k_F

nX!R^k−1_F

n X can be described as forgetting the first factor.

Proof. We deduce this from CorollaryII.5.5. To this end, consider the∞-category D:= FixF₁,...,Fn−1(CoAlg_{Fn+1,Fn+2,...}).

Then, as discussed above, the∞-category in question is CoAlg_FD

n(D), whereF_n^D:D!Dis the canonical extension of the functorF_ntoD. We claim that there is a similar canonical extension R_n^D:D!D of R_n; this follows from the assumption that R_n commutes with all the F_i for i6=n. With a similar argument, one can extend the unit and counit of the adjunction. In this way, we deduce that the functor R^D_n is right adjoint to F_n^D as endofunctors ofD. MoreoverR^D_n is also fully faithful, since this is equivalent to the fact that the counit of the adjunction is an equivalence.

Now, we can apply Corollary II.5.5 to compute the structure of the right adjoint R_Fn to F_n: CoAlg_F

n(D)!CoAlg_F

n(D). This then gives the formula for the underlying functor as well, since the forgetful functorD!Cpreserves all pullbacks as a consequence of the fact that allFi do.

For the application, we letC be the∞-category TSp_F and consider the commuting endofunctorsF_n:=Φ^Cpn, wherep₁, p₂, ... is the list of primes. These functors commute up to coherent equivalence, by the discussion before Definition II.3.3. Without loss of generality, we may assume that they strictly commute to be able to apply the preceding discussion. We first have to verify that the assumptions of LemmaII.5.11are satisfied.

Lemma II.5.12. The canonical morphism

Φ^CpRC_q−!RC_qΦ^Cp

of endofunctors TSpF!TSpF is an equivalence for all primes p6=q.

Proof. We show that the morphism

(Φ^CpRC_qX)^H−!(RC_qΦ^CpX)^H

is an equivalence for all objectsX∈TSpF and all finite subgroupsH⊆T.

We distinguish two cases. Assume first thatCq6⊆H. Then, we immediately get that (RCqΦ^CpX)^H'0. Next, we have to understand (Φ^CpRCqX)^H. There is a transformation of lax symmetric monoidal functors−^Cp!Φ^Cp.

Thus, (Φ^CpR_CqX)^His a module over (R_CqS)^He, whereHe is the preimage ofH under thepth power morphismT!T. Since pandq are different andH does not containC_q, it follows that alsoHe does not contain C_q. Thus, we get that (R_CqS)^Hf'0. Therefore, also (Φ^CpR_CqX)^H'0 which finishes the argument.

Note that, in particular, it follows that Φ^CpR_CqX lies in the essential image ofR_Cq. By PropositionII.2.14, it suffices to show that the natural map Φ^CpR_CqX!R_CqΦ^CpX is an equivalence after applying Φ^Cq. But

Φ^CqR_CqΦ^Cp'Φ^Cp'Φ^CpΦ^CqR_Cq'Φ^CqΦ^CpR_Cq, as desired.

Theorem II.5.13. The inclusion

ι: Cyc Sp^gen'Fix_(ΦCp)p∈P(TSp_F)⊆CoAlg_(ΦCp)p∈P(TSp_F)

admits a right adjoint Rι such that the counit ιRι!id of the adjunction induces an equivalence on underlying spectra.

Proof. We letC be the∞-categoryTSp_F and consider the commuting endofunctors Fn:=Φ^Cpn, where p1, p2, ... is the list of primes. We first prove that the inclusion

ιn:Cn+1:= FixF₁,...,F_n(CoAlg_{Fn+1,Fn+2,...})⊆FixF₁,...,Fn−1(CoAlg_Fn,Fn+1,...) :=Cn

admits a right adjointRι_n such that the counit of the adjunction induces an equivalence on underlying spectra. We use the notation of LemmaII.5.11and invoke CorollaryII.2.16 to deduce (similarly to the proof of TheoremII.5.6) that we have

(R^k_C

pX)^Cpm×Cr'X^Cpm−k×Cr×

(Φ^CpX)^Cpm−k×Cr×...×

(Φ^CpX)^Cpm−1×CrX^Cpm×Cr

forrcoprime top. For fixed value ofmandr, this term stabilizes ink. The same is true if we apply geometric fixed points to this sequence by an isotropy separation argument (or by a direct calulation that the resulting tower is equivalent to the initial tower as in the proof of TheoremII.5.6). Therefore, we deduce that the limit of the diagram

...−!R³_F

of objects inTSp_Fcommutes with all endofunctors Φ^Cp. Thus, it is the underlying object of the limit in the∞-category FixF₁,...,F_n−1(CoAlg_Fn,Fn+1,...). Therefore, LemmaII.5.10 implies that this is the underlying object of the right adjointRι_n which we try to under-stand. Then, specializing the formula above tor=0 andm=0, we obtain that the counit of the adjunction between Fn and R_Fn induces an equivalence (ιnRι_nX)^{e}'X^{e}as desired.

Now, the right adjoint of the full inclusion

ι: lim←−(Cn) = Fix_(ΦCp)p∈P(TSpF)⊆CoAlg_(ΦCp)p∈P(TSpF) =C1

is the colimit of the right adjoints C1

For each finite subgroup C_m⊆T, the genuine C_m-fixed points in this limit stabilizer;

namely, they are constant after passing all primes dividingm. This again implies that the limit is preserved by all Φ^Cp, and so the limit can be calculated on the underlying objects inTSp_F, which gives the claim of the theorem.

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