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PART IV.1. LIE ALGEBRAS AND CO-COMMUTATIVE CO-ALGEBRAS

Contents

Introduction 21. Lie algebras: recollections 31.1. The basics 31.2. Scaling the structure 31.3. Filtrations 41.4. The Chevalley complex 41.5. The functor of primitives 61.6. The enhanced adjunction 61.7. The symmetric Hopf algebra 82. Looping Lie algebras 92.1. Group-Lie algebras 102.2. Forgetting to group structure 102.3. Chevalley complex of group-Lie algebras 112.4. Chevalley complex and the loop functor 122.5. The tensor Hopf algebra 132.6. The co-symmetric algebra 152.7. Proof of Theorem 2.4.5 163. The universal enveloping algebra 173.1. Universal enveloping algebra: definition 173.2. Map from the tensor Hopf algebra 183.3. The PBW theorem 193.4. The Bar complex of the universal envelope 213.5. Modules for the Lie algebra 214. The universal envelope via loops 224.1. The main result 224.2. Proof of Theorem 4.1.2 234.3. Proof of Proposition 4.2.4 244.4. The map from the tensor Hopf algebra, revisited 24Appendix A. Commutative co-algebras and bialgebras 27A.1. Two incarnations of co-commutative bialgebras 27A.2. Modules over co-commutative Hopf algebras 29Appendix B. Actions of monoids and filtrations 31B.1. Equivariance with respect to a monoid 31B.2. Equivariance in algebraic geometry 31B.3. The category of filtered objects 33B.4. The associated graded 33Appendix C. Proof of the PBW theorem 34C.1. The monoidal category of symmetric sequences 34C.2. The PBW theorem at the level of operads 35

Date: October 22, 2013.

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2 LIE ALGEBRAS AND CO-COMMUTATIVE CO-ALGEBRAS

C.3. Proof of Theorem C.2.5 36Appendix D. Co-algebras over co-operads 37D.1. Co-operads 37D.2. Ind-nilpotent co-algebras over a co-operad 37D.3. Koszul duality functors 38D.4. (Usual) co-algebras over a co-operad 39D.5. Relation between two types of co-algebras 39D.6. The case of the co-commutative co-operad and the Lie operad 41Appendix E. A fully faithfulness conjecture 42E.1. Statement of the conjecture 42E.2. Some implications 43E.3. Calculation of co-primitives 44E.4. Computation of primitives 45E.5. Proof of Theorem E.1.5 46References 48

Introduction

LIE ALGEBRAS AND CO-COMMUTATIVE CO-ALGEBRAS 3

1. Lie algebras: recollections

In this section we review some basic (and well-known) fact about Lie algebras in symmetricmonoidal DG categories.

1.1. The basics.

1.1.1. In this section we let O be a symmetric monoidal DG category. Let LieAlg(O) denotethe category of Lie algebras in O (see Sect. C.1 for what we mean by this).

We have a pair of mutually adjoint functors

freeLie : O LieAlg(O) : oblvLie.

In particular, the forgetful functor oblvLie commutes with limits.

In addition, it is known that the functor oblvLie commutes with sifted colimits.

1.1.2. We also consider the functor

trivLie : O→ LieAlg(O)

that associates to an object of O the Lie algebra structure on this object with the zero operation.

The functor trivLie commutes with limits: its composition with oblvLie is the identity functor(and, hence, commutes with limits), while oblvLie commutes with limits and is conservative.

For the same reason, the functor trivLie commutes with sifted colimits.

1.2. Scaling the structure. In this subsection we formulate the procedure of deforming a Liealgebra by“scaling” the Lie bracket. This is a technically important tool as it allows to reducemany isomorphism claims to the case of abelian Lie algebras.

1.2.1. The category LieAlg(O) carries a canonical action 1 of the monoid A1. By definitionthis means that we have a functorial assignment for any S ∈ Sch of an action of the monoidMaps(S,A1) on the category LieAlg(O⊗QCoh(S)).

1.2.2. The action of 0 ∈ A1 is the endo-functor trivLie oblvLie, and the action of Gm ⊂ A1

is equipped with the canonical trivialization.

1.2.3. The above A1-action commutes with the functors

oblvLie : LieAlg(O)→ O and trivLie : O→ LieAlg(O).

The action of Gm on the functors oblvLie and trivLie, resulting from the trivialization ofthe Gm-action on LieAlg(O), is given by the standard action of Gm on the identity functor onO.

1This is not a feature of Lie algebras, but is true for any augmented operad.


1.2.4. For h ∈ LieAlg(O) consider h ⊗ OA1 ∈ LieAlg(O ⊗ QCoh(A1)). Let hscaled denote theobject of LieAlg(O ⊗ QCoh(A1)) obtained from h ⊗ OA1 by the action of the tautologicalA1-point of A1.

The object hscaled has the following properties:

• h0 ' trivLie oblvLie(h);• The image of hscaled under the restriction functor

LieAlg(O⊗QCoh(A1))→ LieAlg(O⊗QCoh(Gm))

is the Lie algebra, obtained from h by scaling the Lie operation by the underlying pointof Gm.

1.3. Filtrations. The main construction in this section expresses the following idea: any Liealgebra can be automatically viewed as a filtered Lie algebra. Hence, many functors fromthe category of Lie algebras in O (here O is a symmetric monoidal DG category) to O itselfautomatically lift to functors with values in the category of filtered objects in O.

1.3.1. Consider A1 as a scheme equipped with the tautological action of the monoid A1.

By construction, for h ∈ LieAlg(O), the object

hscaled ∈ LieAlg(O⊗QCoh(A1))

admits a structure of covariance with respect to A1 (see Sect. B.2.3(ii) for what this means).

1.3.2. Let nowO′ ΦO′ : LieAlg(O′)→ O′

be a system of functors that depends functorially on O′ ∈ DGCatSymMoncont .

We claim that for a given O ∈ DGCatSymMoncont , the functor

ΦO : LieAlg(O)→ O

can be canonically lifted to a functor

ΦFil : LieAlg(O)→ OFil,≥0,

where OFil,≥0 is as in Sect. B.3.1.

1.3.3. Recall the identification

OFil,≥0 '(O⊗QCoh(A1)

)A1 -contr,

see Lemma B.3.3.

Given h ∈ LieAlg(O) we define the corresponding object in(O⊗QCoh(A1)

)A1 -contrto be

ΦO⊗QCoh(A1)(hscaled),

where hscaled is as in Sect. 1.3.1.

1.4. The Chevalley complex. The focus of this chapter is the pair of mutually adjointoperations–the Chevalley complex and taking the primitives that connect Lie algebras withco-commutative co-Algebras. In this subsection we introduce the former.

1.4.1. Let us consider LieAlg(O) as a symmetric monoidal category with respect to Carte-sian product. Consider the category O as endowed with the Cartesian symmetric monoidalstructure.

The functor trivLie is tautologically left-lax symmetric monoidal. However, it is easy to seethat it is actually symmetric monoidal.


1.4.2. Consider the symmetric monoidal functor

trivLie [−1] : Vect→ LieAlg(O).

We letChev+ : LieAlg(O)→ Vect

denote its left adjoint.

By construction, the functor Chev+ acquires a left-lax symmetric monoidal structure, wherewe remind that O is considered as equipped with the Cartesian symmetric monoidal structure.

Applying the functor oblvLie to the unit of the adjunction, we obtain a natural transforma-tion

(1.1) [1] oblvLie → Chev+ .

1.4.3. Consider now the category O1O/. We consider it as endowed with the symmetricmonoidal structure compatible with the tautological forgetful functor O1O/ → O:

(1O →W1)⊗ (1O →W2) := (1O →W1 ⊗W2).

Consider the pair of adjoint functors

(1.2) O O1O/ : V 7→ V ⊕ 1O, (1O →W ) 7→W.

We regard O as endowed with the Cartesian symmetric monoidal structure, which is thesame as the co-Cartesian symmetric monoidal structure since O is stable. Note that the unitobject in O1O/ is the initial object. Hence, the right adjoint functor in (1.2) tautologicallyacquires a right-lax symmetric monoidal structure.

Hence, the left adjoint in (1.2) is naturally left-lax symmetric monoidal.

1.4.4. Composing the functor Chev+ with the left adjoint in (1.2), we obtain a functor that wedenote

Chev : LieAlg(O)→ O1O/.

By the above, the functor Chev is naturally left-lax symmetric monoidal.

We have the following fact, proved in Sect. 1.7.8 below:

Lemma 1.4.5. The left-lax symmetric monoidal structure on Chev is symmetric monoidal.

In other words, Lemma 1.4.5 says that for h1, h2 ∈ LieAlg(O), the natural map

Chev(h1)⊗ Chev(h2)→ Chev(h1 × h2)

is an isomorphism.

1.4.6. Being (left-lax) symmetric monoidal, the functor Chev upgrades to a functor

CocomCoalg(LieAlg(O))→ CocomCoalg(O1O/) =: CocomCoalgaug(O).

Since the symmetric monoidal structure on LieAlg(O) is Cartesian, the identity functor onLieAlg(O) canonically lifts to a functor

LieAlg(O)→ CocomCoalg(LieAlg(O)).

Composing, we obtain a functor

LieAlg(O)→ CocomCoalgaug(O),

that we shall denote by Chevenh.

This functor will be the main in this and the subsequent sections.


1.4.7. Consider again the functor


We note that it falls into the paradigm of Sect. 1.3. Hence, we can naturally upgrade it toa functor

ChevFil : LieAlg(O)→ (OFil,≥0)1O/.

Similarly, the functor Chevenh upgrades to a functor

Chevenh,Fil : LieAlg(O)→ CocomCoalgaug(OFil,≥0).

We note that CocomCoalgaug(OFil,≥0) is the category of augmented non-negatively filteredco-commutative co-alebras in O.

1.5. The functor of primitives.

1.5.1. Consider the category CocomCoalgaug(O).

In what follows we shall denote by oblvCocomaug the forgetful functor

CocomCoalgaug(O)→ O1O/,

and by oblvCocom, its composition with the forgetful functor

O1O/ → O.

In addition, we shall denote by oblvCocom+ the functor CocomCoalgaug(O) → O equal tothe composition of oblvCocomaug with the functor

O1O/ → O, (1O →W ) 7→ Cone(1O →W ).

1.5.2. Consider the functor

trivCocomaug : O→ CocomCoalgaug(O), V 7→ 1O ⊕ V.

We let

Prim : CocomCoalgaug(O)→ O

denote the functor right adjoint to trivCocomaug .

1.5.3. Let us note that by construction, the natural transformation (1.1) upgrades to

(1.3) [1] oblvLie → Prim Chevenh .

1.6. The enhanced adjunction. In this subsection we show that the functor Prim can beenhanced to a functor that takes values in the category of Lie algebras.



Chevenh : LieAlg(O)→ CocomCoalgaug(O),

and let ([−1] Prim)enh denote its right adjoint.

Proposition 1.6.2. There exists a canonical isomorphism of functors

oblvLie ([−1] Prim)enh ' [−1] Prim : CocomCoalgaug(O)→ O.

Proof. By adjunction, we have a canonical isomorphism of functors of O→ O1O/

Chev freeLie(V ) ' k ⊕ V [1].

By construction, this functor upgrades to an isomorphism of functors

(1.4) O→ CocomCoalg(O1O/) = CocomCoalgaug(O),

Chevenh freeLie(V ) ' trivCocomaug(V [1]).

The isomorphism stated in the proposition is obtained from one in (1.4) by passing to theright adjoint functors.

1.6.3. Let us apply the functor oblvLie to the unit of the adjunction

Id→ ([−1] Prim)enh Chevenh .

We obtain a natural transformation

oblvLie → oblvLie ([−1] Prim)enh Chevenh ' [−1] Prim Chevenh .

It is easy to see that the resulting natural transformation identifies with (1.3).

1.6.4. Consider the adjoint pair

Chevenh : LieAlg(O) CocomCoalgaug(O) : ([−1] Prim)enh.

It is easy to see that, in general, neither of the functors Chevenh and ([−1] Prim)enh is fullyfaithful (or even conservative).

However, we make the following conjectures:

Conjecture 1.6.5.

(a) The unit of the adjunction

Id→ ([−1] Prim)enh Chevenh

is an isomorphism, when evaluated on objects lying in the essential image of ([−1] Prim)enh.

(b) The co-unit of the adjunction

Chevenh ([−1] Prim)enh → Id

is an isomorphism, when evaluated on objects lying in the essential image of Chevenh.

We can rephrase point (a) of the conjecture as saying that the functor Chevenh is fullyfaithful on the essential image of the functor ([−1] Prim)enh. We can rephrase point (b) of

the conjecture as saying that the functor Chevenh is fully faithful on the essential image of thefunctor ([−1] Prim)enh.

See Sects. D.3.3 and E.2.7 for a refined version of the above conjecture.

Together, points (a) and (b) of Conjecture 1.6.5 imply:


Conjecture 1.6.6. The functors Chevenh and ([−1] Prim)enh define mutually inverse equiv-

alences between the essential image of ([−1] Prim)enh and the essential image of Chevenh.

1.7. The symmetric Hopf algebra. The symmetric (co)-algebra construction has multipleincarnations: we can view it as associating to an object of O the free commutative algebra, orthe corresponnding co-commutative co-algebra. Combining the two pieces of structure togetherwe will consider the functor

O→ CocomCoalg(ComAlg(O)).


freeCom : O→ ComAlg(O).

Since the operation of tensor product in ComAlg(O) equals the co-product, the functorfreeCom carries a natural right-lax symmetric monoidal structure (where we consider O asequipped with the Cartesian structure).

However, it is easy to see that this right-lax symmetric monoidal structure is in fact symmetricmonoidal.

In particular, we obtain that the functor freeCom naturally upgrades to a functor

SymHopf : O→ CocomCoalg(ComAlg(O)) ' CocomCoalgaug(ComAlg(O)).

1.7.2. Let

Sym : O→ CocomCoalgaug(O)

denote the composition

CocomCoalgaug(oblvCom) SymHopf .

Let

Sym : O→ O


oblvCom freeCom.

By the above, the functor Sym is naturally symmetric monoidal, where on the source copyof O we consider the Cartesian symmetric monoidal structure, and on the target, the one givenby tensor product.

We can regard Sym as the upgrading of Sym corresponding to the above symmetric monoidalstructure.

1.7.3. Note that by construction, we have a canonically defined natural transformation

(1.5) V → Prim Sym(V ), V ∈ O.

The map (1.5) admits a canonical retraction:

(1.6) Prim Sym(V ) ' oblvCocom+ trivCocomaug Prim Sym(V )→→ oblvCocom+ Sym(V ) ' Sym+(V )→ V,

where Sym+(V ) is the augmentation ideal of Sym(V ).

We have:

Theorem 1.7.4. The maps (1.5) and (1.6) are mutually inverse isomorphisms.

This theorem will be proved in Sect. E.


1.7.5. The following is another general fact about augmented operads, see Sect. D.6.5:

Proposition 1.7.6. There exists a canonical isomorphism of functors

(1.7) Chevenh trivLie(V [−1]) ' Sym(V ), O→ CocomCoalgaug(O).

The corresponding map

trivLie(V [−1])→ ([−1] Prim)enh(Sym(V ))

has the property that the underlying map

V [−1] ' oblvLie trivLie(V [−1])→ oblvLie ([−1] Prim)enh Sym(V ) = Prim Sym(V )[−1]

equals the map (1.5).

Combining Proposition 1.7.6 and Theorem 1.7.4, we obtain:

Corollary 1.7.7. The isomorphism

V → Prim Sym(V )

canonically lifts to an isomorphism of functors

trivLie(V [−1])→ ([−1] Prim)enh Sym(V ), O→ LieAlg(O).

1.7.8. We will now use Proposition 1.7.6 to prove Lemma 1.4.5.

Proof. We need to show that for h1, h2 ∈ LieAlg(O), the canonical map

Chev(h1 × h2)→ Chev(h1)⊗ Chev(h2)

is an isomorphism.

It suffices to show that the corresponding map

Chevenh,Fil(h1 × h2)→ Chevenh,Fil(h1)⊗ Chevenh,Fil(h2)

is an isomorphism. Further, by Lemma B.4.6, it suffices to show that the induced map

ass-gr Chevenh,Fil(h1 × h2)→(

ass-gr Chevenh,Fil(h1))⊗(

ass-gr Chevenh,Fil(h2))

is an isomorphism.

However, by Sect. 1.2.2, the latter map identifies with

Chevenh trivLie oblvLie(h1 × h2)→

→(

Chevenh trivLie oblvLie(h1))⊗(

Chevenh trivLie oblvLie(h2)),

which by Proposition 1.7.6 identifies with

Sym oblv(h1 × h2)→ (Sym oblv(h1))⊗ (Sym oblv(h2)) ,

and the isomorphism is mainfest.

2. Looping Lie algebras

In this section we study group-objects in the category of Lie algebras and the Chevalleyfunctor applied to such group-objects.


2.1. Group-Lie algebras. In this subsection we show that the category of Lie algebras hasthe feature that the functors of taking the loop space and the classifying space of a group-objectare mutually inverse equivalences of categories.

This discussion is not specific to Lie, but applies to any operad.

2.1.1. Note that LieAlg(O) is a pointed category. Consider the categories

Grp(LieAlg(O)) ⊂ Monoid(LieAlg(O)).

We claim:

Lemma 2.1.2. The inclusion Grp(LieAlg(O)) ⊂ Monoid(LieAlg(O)) is an equivalence.

Proof. The inclusion Grp(C) ⊂ Monoid(C) is an equivalence for any pointed category C (re-garded as a symmetric monoidal category with respect to the Cartesian product), for which amap c1 → c2 is an isomorphism whenever c1 ×

c2

∗ → ∗ is.

2.1.3. Consider now the pair of adjoint functors:

(2.1) BLie : Grp(LieAlg(O)) LieAlg(O) : ΩLie.

We claim:

Proposition 2.1.4. The functors (2.1) are mutually inverse equivalences.

Proof. We have to show that the natural transformations

Id→ ΩLie BLie and BLie ΩLie → Id .

It is enough to show that the resulting natural transformations

oblvLie oblvAssoc → oblvLie oblvAssoc ΩLie BLie and oblvLie BLie ΩLie → oblvLie

are isomorphisms.

The following diagram commutes tautologically

Grp(LieAlg(O))ΩLie←−−−− LieAlg(O)

oblvLieoblvGrp

y yoblvLie

O[−1]←−−−− O,

because the functor oblvLie commutes with limits.

The next diagram, obtained from one above by passing to left adjoints along the horizontalarrows,

Grp(LieAlg(O))BLie−−−−→ LieAlg(O)

oblvLieoblvGrp

y yoblvLie

O[1]−−−−→ O

also commutes, because oblvLie commutes with sifted colimits.

This implies the required assertion.

2.2. Forgetting to group structure. The main assertion of this subsection is the following:if we consider a group-object of the category of Lie algebras, and forget the group structure,then the resulting Lie algebra is canonically abelian.


2.2.1. In Sect. 2.4.10 we will prove:

Theorem 2.2.2. The composed functor

oblvGrp ΩLie : LieAlg(O)→ LieAlg(O)

is canonically isomorphic totrivLie [−1] oblvLie.

2.2.3. Combining with Proposition 2.1.4, we obtain:

Corollary 2.2.4. The functor

oblvGrp : Grp(LieAlg(O))→ LieAlg(O)

is canonically isomorphic totrivLie oblvLie oblvGrp.

2.3. Chevalley complex of group-Lie algebras. In this subsection we study what happensto the Chevalley functor when we apply it to group-objects in the category of Lie algebras. Weregard the resulting functor as taking values in the category of co-commutative Hopf algebras.

2.3.1. The functorChevenh : LieAlg(O)→ CocomCoalgaug(O)

has a natural left-lax symmetric monoidal structure, when we consider both LieAlg(O) andCocomCoalgaug(O) as symmetric monoidal categories with respect to the Cartesian product.

Recall that Cartesian product in CocomCoalgaug(O) is given by tensor product. Therefore,

by Lemma 1.4.5, the functor Chevenh is actually symmetric monoidal.

In particular, Chevenh gives rise to a functor

Grp(Chevenh) : Grp(LieAlg(O))→ AssocAlg(CocomCoalgaug(O)) =: CocomBialg(O).

Moreover, its essential image automatically lies in

CocomHopf(O) ⊂ CocomBialg(O).

2.3.2. We can view the functor Grp(Chevenh) slightly differently as follows:

Let us consider the symmetric monoidal functor


According to Lemma 1.4.5, this functor is symmetric monoidal, and hence gives rise to a functor

Grp(Chevenh) : Grp(LieAlg(O))→ AssocAlg(O1O/) ' AssocAlg(O).

The functor Grp(Chevenh) acquires a symmetric monoidal structure, induced by that onChev, where the corresponding symmetric monoidal structure on Grp(LieAlg(O)) is Cartesian.

In partcular, Grp(Chevenh) upgrades to a functor

CocomCoalg (Grp(LieAlg(O)))→ CocomCoalg (AssocAlg(O)) .

Pre-composing with the diagonal functor

Grp(LieAlg(O))→ CocomCoalg (Grp(LieAlg(O))) ,

we obtain a functor′Grp(Chevenh) : Grp(LieAlg(O))→ CocomCoalg (AssocAlg(O)) .

We claim:


Lemma 2.3.3. The functors ′Grp(Chevenh) and Grp(Chevenh) are canonically identified viathe equivalence

CocomCoalg (AssocAlg(O)) ' CocomBialg(O)

of Proposition A.1.2.

Proof. Follows from the fact that if C is a category endowed with the Cartesian symmetricmonoidal structure, then the diagonal functor on Monoid(C)

Monoid(C)→ CocomCoalg (Monoid(C))

and the functorMonoid(C)→ Monoid(CocomCoalg(C)),

obtained by applying Monoid(−) to the diagonal functor on C, correspond to each other underthe equivalence of Proposition A.1.2.

2.4. Chevalley complex and the loop functor. The principal actor in this section will bethe functor

Grp(Chevenh) ΩLie : LieAlg(O)→ CocomBialg(O).

We show that, unlike the functor Chevenh, the above functor is fully faithful (i.e., loopinghelps to preserve structure).

2.4.1. As in the case of Chevenh, the functor Grp(Chevenh) ΩLie naturally lifts to a functor

(Grp(Chevenh) ΩLie)Fil : LieAlg(O)→ CocomBialg(OFil,≥0).

Remark 2.4.2. As we will see in the next section, for h ∈ LieAlg(O), the object

Grp(Chevenh) ΩLie(h) ∈ CocomBialg(O)

(and the corresponding filtered version) identifies with another familiar object, namely theuniversal enveloping algebra UHopf(h) of h.

2.4.3. Recall the functor

([−1] Prim)enh : CocomCoalgaug(O)→ LieAlg(O).

Being the right adjoint of a symmetric monoidal functor, ([−1] Prim)enh acquires a naturalright-lax symmetric monoidal structure. In particular, it gives rise to a functor

Monoid([−1] Prim)enh : CocomBialg(O)→ Monoid(LieAlg(O)) = Grp(LieAlg(O)).

Since BLie and ΩLie are mutually inverse equivalences, the functor Grp(Chevenh) ΩLie isthe left adjoint of the functor

BLie Monoid([−1] Prim)enh, CocomBialg(O)→ LieAlg(O).

Note that

oblvLie BLie Monoid([−1] Prim)enh ' Prim oblvAssoc, CocomBialg(O)→ O.

2.4.4. Consider now the composed functor

oblvAssoc Grp(Chevenh) ΩLie : LieAlg(O)→ CocomCoalgaug(O).

In Sect. 2.7 we will prove:

Theorem 2.4.5. There exists a canonical isomorphism of functors

oblvAssoc Grp(Chevenh) ΩLie ' Sym oblvLie.

We will now derive some corollaries of Theorem 2.4.5.


2.4.6. First, we claim:

Corollary 2.4.7. The natural transformation

Id→ ([−1] Prim)enh Chevenh

is an isomorphism when evaluated on objects of the form oblvGrp ΩLie ∈ LieAlg(O).

Proof. Follows by combining Theorems 2.4.5 and 1.7.4.

Corollary 2.4.8. The unit of the adjunction

Id→(BLie Monoid([−1] Prim)enh

)(

Grp(Chevenh) ΩLie

)is an isomorphism.

Proof. It suffices to show that the natural transformation becomes an isomorphism after ap-plying the functor oblvGrp ΩLie, which reduces the assertion to that of Corollary 2.4.7.

Note that Corollary 2.4.8 can be restated as follows:

Corollary 2.4.9. The functor

Grp(Chevenh) ΩLie : LieAlg(O)→ CocomBialg(O)

is fully faithful.

2.4.10. We claim that Theorem 2.4.5 implies Theorem 2.2.2:

Proof. By Corollary 2.4.7, it suffuces to show that the functor

([−1] Prim)enh Chevenh oblvGrp ΩLie : LieAlg(O)→ LieAlg(O)

is canonically isomorphic totrivLie [−1] oblvLie.

We rewrite

([−1] Prim)enh Chevenh oblvGrp ΩLie ' ([−1] Prim)enh oblvAssoc Grp(Chevenh)ΩLie,

and further, using Theorem 2.4.5, as

([−1] Prim)enh Sym oblvLie.

Now, the desired assertion follows from Corollary 1.7.7.

2.5. The tensor Hopf algebra. In order to prove Theorem 2.4.5, we introduce another actor:the tensor Hopf algebra. The underlying associative algebra is the free object of the categoryof associative algebras. For our purposes, however, we will mostly need the underlying co-commutative co-algebra.

2.5.1. Consider the adjoint pair

freeAssoc : CocomCoalgaug(O) AssocAlg(CocomCoalgaug(O)) : oblvAssoc.

Consider the functor

trivCocomaug : O→ CocomCoalgaug(O),

and consider the composed functor

TenHopf := freeAssoc trivCocomaug , O→ CocomBialg(O).


2.5.2. We let

Ten : O→ CocomCoalgaug(O)


OTenHopf

−→ CocomBialg(O)oblvAssoc−→ CocomCoalgaug(O).

We let

Ten : O→ O

denote the functor oblvCocom Ten. I.e.,

Ten(V ) = ⊕n≥0

V ⊗n.

2.5.3. Note that since the forgetful functor

oblvCocomaug : CocomCoalgaug(O)→ O1O/

has a natural symmetric monoidal structure, the composition

AssocAlg(oblvCocomaug) TenHopf : O→ AssocAlg(O1O/) ' AssocAlg(O)

identifies with the functor

freeAssoc : O→ AssocAlg(O)

of the free (=tensor) associative algebra.

2.5.4. By construction, the functor TenHopf is the left adjoint of the functor

CocomBialg(O)oblvAssoc−→ CocomCoalgaug(O)

Prim−→ O,

the latter being canonically isomorphic to oblvLie BLie Monoid([−1] Prim)enh.

2.5.5. Note that we have a canonically defined natural transformations of functors

(2.2) LieAlg(O)→ CocomBialg(O), TenHopf oblvLie → Grp(Chevenh) ΩLie.

Indeed, by adjunction, the datum of a map (2.2) is equivalent to that of a map from oblvLie

to

oblvLie BLie Monoid([−1] Prim)enh Grp(Chevenh) ΩLie

and the required natural transformation is obtained by applying oblvLie to the unit of theadjunction

Id→(BLie Monoid([−1] Prim)enh

)(

Grp(Chevenh) ΩLie

).

2.5.6. Consider the map (2.2) evaluated on h = trivLie(V ). After applying the forgeful functoroblvAssoc, we obtain a map

(2.3) Ten(V ) = oblvAssoc TenHopf(V ) ' oblvAssoc TenHopf oblvLie trivLie(V )→

→ oblvAssoc Grp(Chevenh) ΩLie trivLie(V ) ' Chevenh oblvGrp ΩLie trivLie(V ) '

' Chevenh trivLie(V [−1])Proposition 1.7.6

' Sym(V ).

The following results from the construction:

Lemma 2.5.7. The map (2.3) is the usual map Ten(V ) → Sym(V ) from the tensor to thesymmetric co-algebra.


2.5.8. As in Sect. 2.4.1, the functor

TenHopf oblvLie : LieAlg(O)→ CocomBialg(O)

and the natural transformation (2.2) lift to a functor(TenHopf oblvLie

)Fil

: LieAlg(O)→ CocomBialg(OFil,≥0)

and a natural transformation

(2.4)(

TenHopf oblvLie

)Fil

→(

Grp(Chevenh) ΩLie

)Fil

.

Note, however, that by Sect. 1.2.3, the functor(

TenHopf oblvLie

)Fil

canonically factors as

LieAlg(O)→ CocomBialg(Ogr,≥0)→ CocomBialg(OFil,≥0),

where

Ogr → OFil

is the symmetric monoidal functor

(IdO⊗pA1)∗ : OA1 -contr → (O⊗QCoh(A1))A1 -contr.

Precomposing with trivLie : O→ LieAlg(O), we obtain that the functor

TenHopf : O→ LieAlg(O)

itself can be canonically lifted to a functor

(TenHopf)Fil : O→ CocomBialg(OFil,≥0),

and further, to a functor

(TenHopf)gr : O→ CocomBialg(Ogr,≥0).

Informally, the above is saying that the co-commutative Hopf algebra TenHopf(V ) for V ∈ Ois canonically filtered, and in fact graded.

2.6. The co-symmetric algebra.

2.6.1. Consider the functors Ten and coSym from O to O→ O1O/, defined by

V 7→ ⊕n≥0

V ⊗n and V 7→ ⊕n≥0

(V ⊗n)Σn ,

respectively.

There is a canonical natural transformation coSym→ Ten.


2.6.2. The above functors and the natural transformation between them have a natural sym-metric monoidal structure, where we regard O in the Cartesian symmetric monoidal structure,and O1O/ in the one given by the tensor product.

Hence, the above functors (and the natural transformation between them) upgrade to func-tors

Ten and coSym, O→ CocomCoalgaug(O).

It follows from the definitions that the above functor Ten is canonically isomorphic to thefunctor oblvAssoc TenHopf .

Lemma 2.6.3. The composed natural transformation of functors O→ CocomCoalgaug(O)

coSym→ Ten→ Sym

is an isomorphism.

Proof. It is enough to show that the map in question becomes an isomorphism after applyingthe forgetful functor oblvCocomaug . The needed assertion follows now from the fact that, sincewe work over a field of characteristic zero, for each n, the composed map

(V ⊗n)Σn → V ⊗n → (V ⊗n)Σn

is an isomorphism.

2.6.4. As before, we can cannically lift the functors Ten and coSym (and the natural transfor-mation between them) to functors (and a natural transformation)

TenFil and coSymFil, O→ CocomCoalgaug(OFil,≥0),

and further to

Tengr and coSymgr, O→ CocomCoalgaug(Ogr,≥0).

2.7. Proof of Theorem 2.4.5. We are now ready to prove Theorem 2.4.5.

2.7.1. By Lemma 2.6.3, it suffices to construct an isomorphism between the functors

coSym oblvLie and oblvAssoc Grp(Chevenh) ΩLie, LieAlg(O)→ CocomCoalgaug(O).

We construct the natural transformation

(2.5) coSym oblvLie → oblvAssoc Grp(Chevenh) ΩLie

to be the composition

coSym oblvLie → Ten oblvLie → oblvAssoc Grp(Chevenh) ΩLie,

where the second arrow is obtained by applying the functor oblvAssoc to the natural transfor-mation (2.2).

2.7.2. We note that the natural transfrmation (2.5) lifts to a natural transformation of functors

LieAlg(O)→ CocomCoalg(OFil,≥01O/ ),

namely,

(2.6) coSymFil oblvLie → oblvAssoc (

Grp(Chevenh) ΩLie

)Fil

.


2.7.3. Consider the functor of the “associated graded”

ass-gr : OFil → Ogr

(see Sect. B.4.1), and the corresponding functor

ass-gr : CocomCoalgaug(OFil,≥0)→ CocomCoalgaug(Ogr,≥0).

Since the latter functor is conservative, it is enough to show that the natural transformation(2.6) becomes an isomorphism after applying the functor ass-gr.

2.7.4. Consider the natural transformation

(2.7) ass-gr coSymFil oblvLie → ass-gr oblvAssoc (

Grp(Chevenh) ΩLie

)Fil

,

obtained by applying ass-gr to (2.6).

By Sect. 1.2.2, after we apply to (2.7) the (conservative) forgetful functor

CocomCoalgaug(Ogr)→ CocomCoalgaug(O),

the resulting natural transformation identifies with

coSym oblvLie → oblvAssoc Grp(Chevenh) ΩLie trivLie oblvLie,

which comes via pre-composing with oblvLie from the natural transformation of functors

coSym→ Ten→ oblvAssoc Grp(Chevenh) ΩLie trivLie ' Sym(V ),

where the composition of the second and third arrows is the map (2.3).

Hence, by Lemma 2.5.7, the above map identifies with the composition

coSym→ Ten→ Sym,

and hence is an isomorphism by Lemma 2.6.3.

3. The universal enveloping algebra

In this section we recall some basic facts about the functor of universal enveloping algebrain the setting of higher categories.

3.1. Universal enveloping algebra: definition.

3.1.1. We have a naturally defined restriction functor

resAssoc→Lie : AssocAlg(O)→ LieAlg(O),

coming from the corresponding map of operads.

The functor

U : LieAlg(O)→ AssocAlg(O)

is defined to be the left adjoint of resAssoc→Lie.


3.1.2. The functor resAssoc→Lie has a natural right-lax symmetric monoidal structure, whereAssocAlg(O) is a symmetric monoidal category via the tensor product, and LieAlg(O) a sym-metric monoidal category via the Cartesian product.

Hence, the functor U acquires a natural left-lax symmetric monoidal structure (as we shallsee shortly, this left-lax symmetric monoidal structure is actually symmetric monoidal).

In particular, the functor U gives rise to a functor

CocomCoalg(LieAlg(O))→ CocomCoalg(AssocAlg(O)).

Pre-composing with the diagonal functor

LieAlg(O)→ CocomCoalg(LieAlg(O)),

and composing with the equivalence

CocomCoalg(AssocAlg(O)) ' CocomBialg(O)

of Proposition A.1.2, we obtain a functor

UHopf : LieAlg(O)→ CocomBialg(O).

3.1.3. By Sect. 1.3, we can upgrade the functor UHopf to a functor

(UHopf)Fil : LieAlg(O)→ CocomBialg(OFil,≥0).

We will also consider the functor

UFil : LieAlg(O)→ AssocAlg(OFil,≥0).

3.2. Map from the tensor Hopf algebra.

3.2.1. By adjunction, we have an isomorphism of functors

(3.1) U freeLie ' freeAssoc, O→ AssocAlg(O).

Furthermore, by construction, this isomorphism upgrades to

(3.2) UHopf freeLie ' TenHopf , O→ CocomBialg(O).

3.2.2. By adjunction, from (3.2) we obtain a natural transformation

(3.3) TenHopf oblvLie → UHopf , LieAlg(O)→ CocomBialg(O).

We will also consider the natural transformations

freeAssoc oblvLie → U, LieAlg(O)→ AssocAlg(O)

and

Ten oblvLie → oblvAssoc U, LieAlg(O)→ CocomCoalgaug(O),

obtained from (3.3) by applying the corresponding forgetful functors.


3.2.3. The map (3.3) can also be interpreted as follows. We start with the unit of the adjunction

Id→ resAssoc→Lie U,

and apply to it the functor oblvLie:

(3.4) oblvLie → oblvAssoc U, LieAlg(O)→ O.

By construction, the natural transformation (3.4) upgrades to

(3.5) oblvLie → Prim oblvAssoc U, LieAlg(O)→ O.

The natural transformation (3.3) is obtained from (3.5) via the pair of adjoint functors

(TenHopf ,Prim oblvAssoc).

3.2.4. Let (UHopf)R be the functor

CocomBialg(O)→ LieAlg(O),

right adjoint to UHopf .

From (3.1) we obtain:

oblvLie (UHopf)R ' Prim oblvAssoc, CocomBialg(O)→ O.

I.e., like the functor BLie Grp([−1] Prim)enh, the functor (UHopf)R upgrades the functor

Prim oblvAssoc : CocomBialg(O)→ O

to a functor CocomBialg(O)→ LieAlg(O).

Remark 3.2.5. In Sect. 4.4 we will see that there is canonical isomorphism

BLie Grp([−1] Prim)enh ' (UHopf)R,

compatible with the identifications

oblvLie (UHopf)R ' Prim oblvAssoc ' oblvLie BLie Grp([−1] Prim)enh.

3.3. The PBW theorem.

3.3.1. Consider the composed functor

U trivLie : O→ AssocAlg(O).

We claim that it admits a canonically defined natural transformation

(3.6) U trivLie → resCom→Assoc freeCom,

where resCom→Assoc is the forgetful functor ComAlg(O)→ AssocAlg(O).

Indeed, the map (3.6) is given by adjunction by the natural transformation

trivLie(V )→ trivLie(Sym(V )) ' resAssoc→Lie resCom→Assoc freeCom(V ).

Here we are using the fact that the composed functor

ComAlg(O)resCom→Assoc

−→ AssocAlg(O)resAssoc→Lie

−→ LieAlg(O)

is canonically isomorphic to trivLie oblvCom.


Remark 3.3.2. It is easy to see that the map (3.6) can be upgraded to a natural transformationof functors

UHopf trivLie → CocomCoalg(resCom→Assoc) SymHopf , O→ CocomBialg(O),

where we regard CocomCoalg(resCom→Assoc) as a functor

CocomCoalg(ComAlg(O))→ CocomCoalg(AssocAlg(O))Proposition A.1.2

'' AssocAlg(CocomCoalg(O)) =: CocomBialg(O).

3.3.3. The PBW theorem says:

Theorem 3.3.4. The natural transformation (3.6) is an isomorhism.

We will prove Theorem 3.3.4 in Sect. C. See Corollary 3.3.6 below for the relation with themore usual version of the PBW theorem.

3.3.5. Recall the symmetric monoidal functor

ass-gr : OFil → Ogr,

and the corresponding functor

Assoc(ass-gr) : AssocAlg(OFil)→ AssocAlg(Ogr).

Consider the functor

Ugr := Assoc(ass-gr) UFil, LieAlg(O)→ AssocAlg(Ogr).

By Sect. 1.2.2, the above functor is canonically isomorphic to

LieAlg(O)oblvLie−→ O

deg=1−→ Ogr trivLie−→ LieAlg(Ogr)U−→ AssocAlg(Ogr).

From Theorem 3.3.4 we obtain:

Corollary 3.3.6. There exists a canonical isomorphism of functors

LieAlg(O)→ AssocAlg(Ogr)

from Ugr to the composition

LieAlg(O)oblvLie−→ O

deg=1−→ Ogr −→ AssocAlg(Ogr),

where the last arrow is resCom→Assoc freeCom.

Corollary 3.3.6 is the more usual formulation of the PBW theorem.

3.3.7. From Corollary 3.3.6 we shall now deduce:

Lemma 3.3.8. The left-lax symmetric monoidal structure on the functors

U : LieAlg(O)→ AssocAlg(O) and UHopf : LieAlg(O)→ CocomBialg(O)

is symmetric monoidal.

Proof. We have to show that for h1, h2 ∈ LieAlg(O), the morphism

U(h1 × h2)→ U(h1)⊗ U(h2)

is an isomorphism.

It is enough to prove the corresponding fact for the functor UFil, and hence also for thefunctor Ugr. Now the assertion follows via Corollary 3.3.6 from the fact that the functorfreeCom is symmetric monoidal.


3.3.9. As another corollary of Theorem 3.3.4, we obtain:

Corollary 3.3.10. For h ∈ LieAlg(O), the composed map in CocomCoalg(O)

coSym oblvLie(h)→ Ten oblvLie → oblvAssoc UHopf(h)

is an isomorphism.

Proof. It is enough to show that the map in question is an isomorphism at the associated gradedlevel, in which case the assertion follows from Lemma 2.6.3.

3.4. The Bar complex of the universal envelope.

3.4.1. Note that we have a canonically defined natural transformation from the functor U tothe constant functor with value 1O.

By a slight abuse of notation, let us denote by the same symbol U , the functor

LieAlg(O)→ (AssocAlg(O))/1O' AssocAlgaug(O), .

3.4.2. The following assertion follows by adjunction:

Lemma 3.4.3. There exists a canonical isomorphism of functors LieAlg(O)→ O

Bar U ' Chev .

3.4.4. Consider now the functor

Bar : CocomBialg(O) ' AssocAlgaug(CocomCoalgaug(O))→ CocomCoalgaug(O).

By the construction of the functors U and Chevenh, we have:

Lemma 3.4.5. The isomorphism of Lemma 3.4.3 canonically upgrades to an isomorphism offunctors

Bar UHopf ' Chevenh, LieAlg(O)→ CocomCoalgaug(O).

3.4.6. Finally, we note that the isomorphism of Lemma 3.4.5 can be further upgraded to anisomorphism of functors with values in CocomCoalgaug(OFil,≥0).

3.5. Modules for the Lie algebra.

3.5.1. Let O be a symmetric monoidal DG category, and let h ∈ LieAlg(O). In this book wedefine the category

h-mod(O)

to U(h)-mod(O).

Remark 3.5.2. One can show that the category h-mod(O) as defined above, identifies withh-modules in the operadic sense.

3.5.3. The upgrading U(h) UHopf(h) defines on h-mod(O) a structure of symmetric monoidalcategory, such that the tautological forgetful functor

oblvh : h-mod(O)→ O

is symmetric monoidal.


3.5.4. We lettrivh : O→ h-mod(O)

denote the functor corresponding to the augmentation on U(h). The functor trivh has a naturalsymmetric monoidal structure.

We letcoinvh : h-mod(O)→ trivh

the left adjoint of trivh.

Tautologicallycoinvh(−) ' Bar(U(h),−).

3.5.5. By adjunction, the functor coinvh has a natural left-lax symmetric monoidal structure.

In particular, coinvh(1O) acquires a structure of object of CocomCoalg(O). Moreover, thefunctor coinvh upgrades to a functor

h-mod(O)→ coinvh(1O)-comod(O).

However, by construction and Lemma 3.4.5, we have a canonical identification betweencoinvh(1O) and Chevenh(h) as objects of CocomCoalg(O).

Thus, we obtain that the functor coinvh upgrades to a functor

(3.7) coinvenhh : h-mod(O)→ Chevenh(h)-comod(O).

4. The universal envelope via loops

The point this section is that we can express the universal enveloping algebra of a Lie algebravia the Chevalley functor.

4.1. The main result.

4.1.1. The main result of this section is the following:

Theorem 4.1.2. There exists a canonical isomorphism of functors

UHopf ' Grp(Chevenh) ΩLie, LieAlg(O)→ CocomBialg(O).

Several remarks are in order:

Remark 4.1.3. The isomorphism of Theorem 4.1.2 automatically upgrades to an isomorphismat the filtered level:

(UHopf)Fil '(

Grp(Chevenh) ΩLie

)Fil

.

Remark 4.1.4. One can generalize the proof of Theorem 4.1.2 to estabish the isomorphims offunctors

(4.1) UEn' Chev Ω×nLie ,

where UEn is the left adjoint to the forgetful functor

resEn→Lie : En -Alg(O)→ LieAlg(O),

arising from the corresponding map of operads.

Moreover, the isomorphism (4.1) automatically upgrades to an isomorphism of the corre-sponding functors

LieAlg(O)→ CocomCoalg(En -Alg(O)) ' En -Alg(CocomCoalg(O)),


(4.2) UHopfEn

' Chevenh Ω×nLie .

Furthermore, the isomorphism (4.2) can be upgraded to an isomorphism of functors withvalues in CocomCoalg(En -Alg(OFil,≥0)).

Remark 4.1.5. A very natural proof of the isomorphism (4.1) can be given using the languageof factorization algebras. When n = 2, it is essentially one in [FG, Proposition 6.1.2].

4.2. Proof of Theorem 4.1.2.

4.2.1. For h ∈ LieAlg(O), consider the object ΩLie(h) ∈ Monoid(LieAlg(O)). Since the functorUHopf is symmetric monoidal (see Lemma 3.3.8), it gives rise to a functor

Assoc(UHopf) : Monoid(LieAlg(O))→ AssocAlg (AssocAlg(CocomCoalg(O))) =:

= E2 -Alg(CocomCoalg(O)).

We consider the resulting object

(4.3) Assoc(UHopf) ΩLie(h) ∈ AssocAlg (AssocAlg(CocomCoalg(O))) .

4.2.2. We consider two functors

Assoc(Bar) : AssocAlg (AssocAlg(CocomCoalg(O)))→ AssocAlg(CocomCoalg(O)),

and

Bar : AssocAlg (AssocAlg(CocomCoalg(O)))→ AssocAlg(CocomCoalg(O)),

corresponding to taking the Bar-complex with respect to the “inner” and “outer” associativealgebra structure, respectively.

We claim:

(4.4) Bar Assoc(UHopf) ΩLie(h) ' UHopf(h)

and

(4.5) Assoc(Bar) Assoc(UHopf) ΩLie(h) ' Grp(Chevenh) ΩLie(h).

Indeed, since the functor UHopf is symmetric monoidal, we have

(4.6) Bar Assoc(UHopf) ΩLie(h) ' UHopf BLie ΩLie(h) ' UHopf(h),

which gives the isomorphism in (4.4).

To establish the isomorphism in (4.5), we note that the isomorphism of Lemma 3.4.5 iscompatible with the (right-lax) symmetric monoidal structures, and, hence, upgrades to anisomorphism

Assoc(Bar) Assoc(UHopf) ' Grp(Chevenh).

This gives rise to the isomorphism in (4.5) by precomposing with ΩLie.


4.2.3. Recall that the symmetric monoidal structure on CocomCoalg(O) is Cartesian. In par-ticular, we can consider the full subcategories

CocomHopf(O) := Grp(CocomCoalg(O)) ⊂ AssocAlg(CocomCoalg(O)),

andGrp(Grp(CocomCoalg(O))) ⊂ AssocAlg(AssocAlg(CocomCoalg(O))).

Note, however, that the inclusions

Grp(Grp(CocomCoalg(O))) ⊂ Grp(AssocAlg(CocomCoalg(O)))

andGrp(Grp(CocomCoalg(O))) ⊂ Grp(AssocAlg(CocomCoalg(O)))

are actually equalities.

We have the following basic fact proved below:

Proposition 4.2.4. For an ∞-category C endowed with the Cartesian symmetric monoidalstructure, there exists a canonical isomorphism of functors

Assoc(B) ' B, Grp(Grp(C))→ Grp(C).

We apply the isomorphism of Proposition 4.2.4 to the object (4.3), and obtain an isomorphism

(4.7) Assoc(Bar) Assoc(UHopf) ΩLie(h) ' Bar Assoc(UHopf) ΩLie(h).

Combining with the isomorphism (4.7), (4.4) and (4.4), we arrive at the conclusion of thetheorem.

4.3. Proof of Proposition 4.2.4.

4.3.1. By adjunction, the assertion of the proposition amounts to a canonical isomorphism offunctors

(4.8) Ω ' Assoc(Ω) : Grp(C)→ Grp(Grp(C)).

The latter reduces the assertion to the proposition when C = Spc is the category of spaces.

4.3.2. We start with the isomorphism of functors

(4.9) Assoc(Ω) Ω ' Ω Ω, Spc∗/ → Grp(Grp(Spc)).

By adjunction, we obtain a natural transformation

(4.10) B Assoc(Ω)→ Ω B ' Id, Grp(Spc)→ Grp(Spc).

Applying Ω : Grp(Spc)→ Grp(Grp(Spc)) to (4.10), we obtain the desired map

Assoc(Ω) ' Ω B Assoc(Ω)→ Ω.

4.3.3. To show that the resulting map Assoc(Ω)→ Ω is an isomorphism, it is enough to do soafter precomposing with Ω : Spc∗/ → Grp(Spc). However, the resulting map

Assoc(Ω) Ω→ Ω Ω

equals that of (4.9), and hence is an isomorphism.

4.4. The map from the tensor Hopf algebra, revisited. Having identified UHopf withGrp(Chevenh) ΩLie, we will now show that this identification is compatible with the map ofthe tensor Hopf algebra to both.


4.4.1. Recall now that natural transformations

TenHopf oblvLie → UHopf

(see (3.3)) and

TenHopf oblvLie → Grp(Chevenh) ΩLie,

of (2.2). We claim:

Proposition 4.4.2. The natural transformations (3.3) and (2.2) and match under the isomor-phism of Theorem 4.1.2.

4.4.3. From Proposition 4.4.2, by passing to right adjoints, we obtain:

Corollary 4.4.4. There exists a canonical of functors

(UHopf)R ' BLie Assoc([−1] Prim)enh, CocomBialg(O)→ LieAlg(O),

compatible with the identifications

oblvLie (UHopf)R ' Prim oblvAssoc ' oblvLie BLie Assoc([−1] Prim)enh.

4.4.5. We can rephrase Corollary 4.4.4 as follows. For h ∈ LieAlg(O) we have the map

(4.11) oblvLie(h)→ Prim oblvAssoc UHopf(h)

of (3.5).

We also have the map

(4.12) oblvLie(h) ' [1] oblvLie ΩLie(h)→

→ Prim Chevenh oblvAssoc ΩLie(h) ' Prim oblvAssoc Grp(Chevenh) ΩLie(h),

where the middle arrow is (1.3).

We claim:

Corollary 4.4.6. The natural transformations (4.11) and (4.12) match under the isomorphismof Theorem 4.1.2.

4.4.7. Proof of Proposition 4.4.2. By adjunction, it suffices to match the natural transforma-tions of of Corollary 4.4.6.

Consider the isomorphism of Proposition 4.2.4 for the category C equal to O. We identifythe category of En-algebras in O (with respect to the Cartesian symmetric monoidal structure)with O via the forgetful functor oblvEn

. Under this identification, the functor B correspondsto [1], and the isomorphism of Proposition 4.2.4 is the identity automorphism of the functor[1].

By the functoriality of the isomorphism of Proposition 4.2.4 with respect to the functor

Prim : CocomCoalgaug(O)→ O,

the following diagram diagram commutes:


(4.13)

[1] Prim oblvE2 Assoc(UHopf) ΩLie(h)

Assoc(B) E2(Prim) Assoc(UHopf) ΩLie(h)

Assoc(Prim) Assoc(Bar) Assoc(UHopf) ΩLie(h)

Prim oblvAssoc Assoc(Bar) Assoc(UHopf) ΩLie(h)

[1] Prim oblvE2Assoc(UHopf) ΩLie(h)

B E2(Prim) Assoc(UHopf) ΩLie(h)

Assoc(Prim) Bar Assoc(UHopf) ΩLie(h)

Prim oblvAssoc Bar Assoc(UHopf) ΩLie(h)

++

++

++

++

Consider now the map

(4.14) oblvLie oblvAssoc ΩLie(h)→ Prim oblvE2Assoc(UHopf) ΩLie(h),

obtained by applying (4.11) to ΩLie(h).

By the construction of the isomorphism of Theorem 4.1.2, it suffices to show that the com-posed map

oblvLie(h) ' [1] oblvLie oblvAssoc ΩLie(h)(4.14)−→

→ [1] Prim oblvE2Assoc(UHopf) ΩLie(h)

(4.13)−→

→ Prim oblvAssoc Assoc(Bar) Assoc(UHopf) ΩLie(h) '

' Prim oblvAssoc Grp(Chevenh) ΩLie(h)

is canonically isomorphic to (4.12), and that the composed map

oblvLie(h) ' [1] oblvLie oblvAssoc ΩLie(h)(4.14)−→

→ [1] Prim oblvE2Assoc(UHopf) ΩLie(h)

(4.13)−→

→ Prim oblvAssoc Bar Assoc(UHopf) ΩLie(h) ' Prim oblvAssoc UHopf(h)


is canonically isomorphic to (4.11).

Both isomorphisms follow by unwinding the definitions.

Appendix A. Commutative co-algebras and bialgebras

In this section we establish two basic facts about co-commutative bialgebras in the settingof symmetric monoidal ∞-categories. Both these facts are obvious in ordinary categories, butnot altogether obvious in the ∞-categorical context.

A.1. Two incarnations of co-commutative bialgebras. Co-commutative bialgebras can bethought of in two different ways: as co-commutative co-algebras in the category of associativealgebras, or as associative algebras in the category of co-commutative co-algebras. In thissubsection we show that the two are equivalent.

A.1.1. In this subsection we let O be a symmetric monoidal category, which contains colimits,and for which the functor of tensor product distributes over colimits.

The category CocomBialg(O) is defined as

AssocAlg(CocomCoalg(O)),

where the (symmetric) monoidal structure on CocomCoalg(O) is given by tensor product, whichcoincides with the Cartesian product in CocomCoalg(O).

Consider now the category AssocAlg(O), endowed with a symmetric monoidal structuregiven by tensor product.

In this subsection we are going to prove:

Proposition-Construction A.1.2. There exists a canonical equivalence of categories

CocomCoalg(AssocAlg(O)) ' AssocAlg(CocomCoalg(O)) =: CocomBialg(O).

The rest of this subsection is devoted to the proof of the proposition.

A.1.3. Step 1. For any monoidal category O′, we have a canonical equivalence

AssocAlg(O′) ' AssocAlg(O′1′O/) and ComAlg(O′) ' ComAlg(O′1′O/).

In particular, we have a canonical equivalence

CocomCoalg(AssocAlg(O))→ CocomCoalg(AssocAlgaug(O)),

where

AssocAlgaug(O) := AssocAlg(O)/1O.

A.1.4. Step 2. Recall now that we have a canonically defined symmetric monoidal functor

Bar• : AssocAlgaug(O)→ (O)∆op

.

In particular, we obtain a functor

Cocom(Bar•) : CocomCoalg(AssocAlgaug(O))→ CocomCoalg((O)∆op

) '

' (CocomCoalg(O))∆op

.

Combining with Step 1, we obtain a functor

(A.1) CocomCoalg(AssocAlg(O))→ (CocomCoalg(O))∆op

.


A.1.5. Step 3. Since the symmetric monoidal structure on CocomCoalg(O) is Cartesian, thefunctor

(A.2) Bar• : AssocAlg(CocomCoalg(O))→ (CocomCoalg(O))∆op

is fully faithful.

Now, it is easy to see that the essential image of the functor (A.1) lies in that of (A.2).

This defines the sought-for functor

(A.3) CocomCoalg(AssocAlg(O))→ AssocAlg(CocomCoalg(O)).

A.1.6. Step 4. Let us now prove that the functor (A.3) is an equivalence. By construction, thecomposed functor

CocomCoalg(AssocAlg(O))→ AssocAlg(CocomCoalg(O))oblvAssoc−→ CocomCoalg(O)

is the tautological functor

(A.4) Cocom(oblvAssoc) : CocomCoalg(AssocAlg(O))→ CocomCoalg(O).

It suffices to show that the functor (A.4) and

(A.5) AssocAlg(CocomCoalg(O))oblvAssoc−→ CocomCoalg(O)

are both monadic, and that the map of monads, induced by (A.3), is an isomorphism as plainendo-functors of CocomCoalg(O).

A.1.7. Step 5. The functor

AssocAlg(O′)oblvAssoc−→ O′

is monadic for any monoidal category O′ (satisfying the same assumtion as O); its left adjointis given by

V 7→ freeAssoc(V ).

In particular, the functor (A.5) is monadic.

A.1.8. Step 6. We have a pair of adjoint functors

freeAssoc : O AssocAlg(O) : oblvAssoc,

with the right adjoint being symmetric monoidal.

Hence, the above pair induces an adjoint pair

CocomCoalg(O) CocomCoalg(AssocAlg(O)).

Hence, we obtain that the functor (A.4) is also monadic.

A.1.9. Step 7. To show that the map of monads on CocomCoalg(O), induced by (A.3) is anisomorphism as plain endo-functors, it is enough to do so after composing with the (conservative)forgetful functor oblvCocom : CocomCoalg(O)→ O.

Now, it follows from the construction that the map in question is the identy map on thefunctor

CocomCoalg(O)→ O, A 7→ freeAssoc(oblvCocom(A)).


A.2. Modules over co-commutative Hopf algebras. The goal of this subsection is toestablish the following basic fact: given a co-commutative Hopf algebra A, the category ofmodules over A as an associative algebra is equivalent to the totalization of the co-simplicialcategory of co-modules over Bar•(A), where Bar•(A) is considered as a simplicial co-algebra.

A.2.1. Since the symmetric monoidal structure on CocomCoalg(O) is Cartesian, it makes senseto talk about the subcategory

Grp(CocomCoalg(O)) ⊂ Monoid(CocomCoalg(O)) =

= AssocAlg(CocomCoalg(O)) =: CocomBialg(O)

of group-like objects. We denote this category by CocomHopf(O).

A.2.2. Let A be a co-commutative bi-algebra in O. Consider the corresponding object

Bar•(A) ∈ (CocomCoalg(O))∆op

.

Consider the resulting simplicial category

Bar•(A)-comod(O),

i.e., the simplicial category formed by co-modules in O over the terms of Bar•(A), viewed as asimplicial co-algebra.

Passing to right adjoints, we obtain a co-simpicial category

Bar•(A)-comod(O)R.

The goal of this subsection is to establish the following:

Proposition-Construction A.2.3. Assume that A ∈ CocomHopf(O), and let

A := Assoc(oblvCocom(A))

be the underlying associative algebra. Then there is a canonical equivalence of categories:

(A.6) A-mod ' Tot(Bar•(A)-comod(O)R

).

The rest of this subsection is devoted to the proof of Proposition A.2.3.

Remark A.2.4. Note that, by definition, a structure on A ∈ CocomCoalg(O) of co-commutativeHopf algebra is equivalent to lifting the Yoneda functor

MapsCocomCoalg(O)(−, A) : (CocomCoalg(O))op → Spc

to a functor

(CocomCoalg(O))op → Grp(Spc).

Similarly, for M ∈ O, a datum of lifting it to an object of Tot(Bar•(A)-comod(O)R

)is

equivalent to a functorial assignment to A′ ∈ CocomCoalg(O) of an action of the group

MapsCocomCoalg(O)(A′, A)

on the A′-comodule A′ ⊗M .


Remark A.2.5. The upgrading of A to an object of CocomCoalg(AssocAlg(O)) defines on the

category A-mod a symmetric monoidal structure.

Similarly, the category Tot(Bar•(A)-comod(O)R

)is naturally symmetric monoidal (the lat-

ter is especially evident from the description in Remark A.2.4 above).

Now, it follows from the construction, given below, that the equivalence (A.6) is naturallycompatible with the above symmetric monoidal structures.

A.2.6. Using Proposition A.1.2, to A we can canonically attach an object

A ∈ CocomCoalg(AssocAlgaug(O)).

Moreover, by construction, under the equivalence

(CocomCoalg(O))∆op

' CocomCoalg((O)∆op

)

the objectBar•(A) ∈ (CocomCoalg(O))∆op

identifies with the corresponding object

Cocom(Bar•)(A)-comod(O) ∈ CocomCoalg((O)∆op

).

A.2.7. Consider the category P that consists of pairs (A,M), where A ∈ AssocAlgaug(O) and

M ∈ A-mod. This is a symmetric monoidal category under the operation of tensor product.

If A ∈ CocomCoalg(AssocAlgaug(O)), then the object (A,1O) ∈ P has a natural structureof object of CocomCoalg(P). Moreover, we have a naturally defined functor

(A.7) A-mod(O)→ (A,1O)-comod(P), M 7→ (A,M).

We have a naturally defined symmetric monoidal functor

(A.8) Bar•with module : P→ (O)∆op

, (A,M) 7→ Bar•(A,M),

so that for A ∈ AssocAlgaug(O), we have

Bar•with module(A,1O) ' Bar•(A),

and for A ∈ CocomCoalg(AssocAlgaug(O))

Bar•with module(A,1O) ' Cocom(Bar•)(A),

as objects of (CocomCoalg(O))∆op

.

A.2.8. Combining (A.7) and (A.8) we obtain a functor

(A.9) A-mod(O)→ Sect (∆op,Bar•(A)-comod(O)) ,

where Sect(∆op,−) denotes the category of sections of a given simplicial category.

Lemma A.2.9. If the bi-algebra A is a Hopf algebra, then for M ∈ A-mod(O), the section(A.9) defines, by passing to right adjoints, an object of

Tot(Bar•(A)-comod(O)R

)(i.e., the corresponding morphisms are isomorphisms for every arrow in ∆).

A.2.10. Thus, by Lemma A.2.9, we obtain the desired functor

(A.10) A-mod(O)→ Tot(Bar•(A)-comod(O)R

).

Let us now show that the functor (A.10) is an equivalence.


A.2.11. Let

ev0 : Tot(Bar•(A)-comod(O)R

)→ O

denote the functor of evaluation on 0-simplices.

It is easy to see that the semi co-simplicial category underlying Bar•(A)-comod(O)R satisfiesthe monadic Beck-Chevalley condition. Hence, the functor ev0 is monadic, and the resultingmonad on O, regarded as a plain endo-functor, is given by tensor product with

oblvCocom oblvAssoc(A) ' oblvAssoc(A).

By construction, the composite functor

A-mod(O)→ Tot(Bar•(A)-comod(O)R

) ev0

−→ O

is the tautological forgetful functor oblvA : A-mod(O) → O. Hence, it is also monadic, andthe resulting monad on O, regarded as a plain endo-functor, is given by tensor product with

oblvAssoc(A).

Hence, it remains to see that the homomorphism of monads on O, induced by (A.10), is anisomorphism as plain endo-functors of O. However, it follows from the construction that the

map in question is the identity map on the functor oblvAssoc(A)⊗−.

Appendix B. Actions of monoids and filtrations

In the bulk of this chapter we made heavy technical use of filtrations on objects attached toLie algebras. In this section we provide the foundations of the theory.

B.1. Equivariance with respect to a monoid. The notion of equivariance with respect toa group is completely standard. The situation with monoids is less familiar: in fact, there aretwo different notions of equivariance.

B.1.1. Let X be an object of Spc and let G be a monoid in Spc, equipped with an action onX. Let C be an ∞-category, and let

Φ : X → C

be a functor.

Informally, a structure of covariance (resp., contravariance) on Φ with respect to G is ahomotopy-coherent system of assignments for every g ∈ G and x ∈ X of a 1-morphism Φ(x)→Φ(g · x) (resp., Φ(g · x)→ Φ(x)), in a way compatible with the monoid structure.

A structure of invariance is when the above maps are isomorphisms.

B.1.2. This definition can be formalized as follows. Consider the corresponding simplicial object(X/G)• in Spc.

LetX/G denote the∞-category |(X/G)•|, where the geometric realization is taken in∞ -Cat.

We have a canonical map π : X → X/G. For x, y ∈ X the space MapsX/G(π(x), π(y)) is bydefinition

g ∈ G, g(x) ' y.

A structure of covariance (resp., contravariance) on Φ with respect to G is the datum offactorization of Φ through a functor X/G→ C (resp., (X/G)op → C).

B.2. Equivariance in algebraic geometry.


B.2.1. Let X be a prestack, equipped with an action of a monoidal prestack G. Let F be anobject of QCoh(X).

A datum of covariance (resp., contravariance) on F with respect to G is a functorial assign-

ment for every S ∈ Schaff of a datum of covariance (resp., contravariance) with respect to themonoid Maps(S,G) on the functor

Maps(S,X)→ (QCoh(S))op, x 7→ x∗(F).

In other words, the datum of covariance (resp., contravariance) equivariance assigns to everyx : Maps(S,X) and g ∈ Maps(S,G) a map

(g · x)∗(F)→ x∗(F) (resp., x∗(F)→ (g · x)∗(F)),

in a way compatible with the product of g’s, and compatible with pullbacks S1 → S2.

We let QCoh(X)G -cov (resp., QCoh(X)G -contr) denote the category of objects in QCoh(X),equipped with a structure of covariance (resp., contravariance) with respect to G.

We let QCoh(X)G -inv be the category of objects equipped with a structure of G-invariance.By definition, we have the fully faithful embeddings

QCoh(X)G -cov ← QCoh(X)G -inv → QCoh(X)G -contr.

B.2.2. A similar definition applies when instead of QCoh∗Schaff we consider any presheaf ofcategories

P : (Schaff)op →∞ -Cat .

We will denote the resulting categories by

P(X)G -inv, P(X)G -cov and P(X)G -contr,

respectively.

Similar definitions apply when instead of the category Schaff we consider <∞Schaffft , i.e.,

when we deal with prestacks and presheaves locally almost of finite type. An example of thisis IndCoh(−).

B.2.3. Here are several examples that we will use:

(i) The case in Sect. B.2.1 corresponds to P(S) := QCoh(S).

(ii) Another example considered in this chapter is

P(S) := LieAlg(O⊗QCoh(S)) or LieAlg(O⊗ IndCoh(S))

for a fixed symmetric monoidal DG category O.

(iii) The example that we will consider in [Book-IV.4] is

P(S) := QCoh(S)- mod,

the category of module categories over QCoh(S).

(iv) Take P(S) := PreStk/S . Note that in this case, for Y → X, the datum of contravarianceon Y is equivalent to the lift of the given G-action on X to a G-action on Y. This structure is aninvariance if and only if the square

G× Yaction−−−−→ Yy y

G× Xaction−−−−→ X


is Cartesian.

B.2.4. Suppose now that both X and P are both smooth classical schemes. In this case in thecategory

(Schaff)X×P

cofinal is the full subcategory spanned by those S, which are themselves classical and smooth.

Hence, the structure of covariance or contravariance with respect to G of a section

p ∈ P(X)

is determined by taking S being classical and smooth.

An instance of this that we will in practice consider most often is when X = G = A1, equippedwith the natural action on itself.

B.3. The category of filtered objects. The formalism of equivariance with respect tomonoids allows to give an algebro-geometric interpretation to the notion of non-positive (resp.,non-negative) filtrations.

B.3.1. Let C be a DG category. We define

CFil := Funct(Z,C).

We let

CFil,≥0 ⊂ CFil

be the full subcategory of functors that take value 0 on Z<0. We can identify

CFil,≥0 := Funct(Z≥0,C).

We let

CFil,≤0 ⊂ CFil

be the full subcategory of functors that take a constant value on Z≥0. We can identify

CFil,≤0 := Funct(Z≤0,C).

B.3.2. It is well-known that the category CFil is canonically equivalent to(C⊗QCoh(A1)

)Gm.

Consider Gm as the subgroup of the monoid A1. We have:

Lemma B.3.3.

(a) The forgetful functor(C⊗QCoh(A1)

)A1 -cov →(C⊗QCoh(A1)

)Gm

is fully faithful and its essential image identifies with CFil,≥0 ⊂ CFil.

(b) The forgetful functor(C⊗QCoh(A1)

)A1 -contr →(C⊗QCoh(A1)

)Gm

is fully faithful and its essential image identifies with CFil,≤0 ⊂ CFil.

B.4. The associated graded.


B.4.1. Consider the category

Cgr := CZ.

We can rewrite

CZ ' C⊗VectZ ' C⊗ Rep(Gm).

Now, by [1-aff, Theorem 2.2.2], the naturally defined functor

C⊗ Rep(Gm)→ CGm

is an equivalence.

B.4.2. We have the following graded analog of Lemma B.3.3:

Lemma B.4.3.

(a) The forgetful functor

CA1 -cov → CGm

is fully faithful and its essential image identifies with Cgr,≥0 ⊂ Cgr.

(b) The forgetful functor

CA1 -contr → CGm

is fully faithful and its essential image identifies with Cgr,≤0 ⊂ Cgr.

B.4.4. We can consider the functor of the “associated graded”

ass-gr : OFil → Ogr.

In terms of the identification

OFil '(C⊗QCoh(A1)

)Gm,

the functor ass-gr corresponds to

M 7→ (IdO⊗i)∗(M),

where i : 0 → A1.

B.4.5. The following well-known assertion plays an important technical role:

Lemma B.4.6. The restriction of the functor ass-gr to OFil,≥0 ⊂ OFil is conservative.

Appendix C. Proof of the PBW theorem

In this section we recall the formalism of operads, from which the proof of the PBW theoremwill follow easily.

C.1. The monoidal category of symmetric sequences. A convenient ways to define op-erads (resp., algebras over them) as algebras in a certain monoidal category (modules for thesealgebras in a particular module category).


C.1.1. Let VectΣ denote the category of symmetric sequences. As a DG category, we have:

VectΣ := Πn≥1

Rep(Σn),

i.e., consists of objects

P := P(n) ∈ Rep(Σn), n ≥ 1.

The category VectΣ is endowed with a natural (non-symmetric) monoidal structure and anaction on O, given by the formula

P ? V := ⊕n

(P(n)⊗ V ⊗n

)Σn.

The unit object

1VectΣ ∈ VectΣ

is one satisfying

1VectΣ(1) = k, 1VectΣ(n) = 0 for n > 1.

C.1.2. A (unital) operad is by definition an associative algebra in VectΣ with respect to theabove monoidal structure.

Convention: We will only consider operads P, for which the unit map defines an isomorphismk → P(1). In particular, such operads are automatically augmented.

C.1.3. For an operad P ∈ AssocAlg(VectΣ), the category of P -Algaug(O) of augmented P-algebras on O is by definition the category P-mod(O).

We shall denote by

freeP : O P -Algaug(O) : oblvP+

the corresponding adjoint pair of functors, where

oblvP+(A) = ker (oblvP(A)→ k) .

C.1.4. We will consider the following operads: Assoc, Com and Lie. By definition

Assoc(n) = kΣn , Com(k) = k,

and Lie is the usual (i.e., classical) Lie operad, where we set by definition Lie(1) = k.

C.2. The PBW theorem at the level of operads.

C.2.1. We have the canonical maps

φ : Lie→ Assoc and ψ : Assoc→ Com,

such that the composition ψ φ factors through the augmentation/unit

Lie→ 1VectΣ → Com .


C.2.2. The map φ gives rise to the forgetful functor

resAssoc→Lie : AssocAlgaug(O)→ LieAlg(O),

and the map ψ gives rise to the forgetful functor

resCom→Assoc : Comaug(O)→ AssocAlgaug(O).

The functor

U : LieAlg(O)→ AssocAlgaug(O)

is given by

h 7→ Assoc ?Lie

h.

C.2.3. The functor

U trivLie : O→ AssocAlgaug(O)

is given by

V 7→ (Assoc ?Lie

1VectΣ) ? V.

The canonical map

U trivLie(V )→ Sym(V )

comes from the map in VectΣ:

(C.1) Assoc ?Lie

1VectΣ → Com,

which arises via the description of the map ψ φ in Sect. C.2.1.

C.2.4. The operadic PBW theorem says:

Theorem C.2.5. The map (C.1) is an isomorphism in VectΣ.

It is clear that Theorem C.2.5 implies Theorem 3.3.4.

C.3. Proof of Theorem C.2.5.

C.3.1. First, it is known, that when viewed as a right Lie-module in VectΣ, the object Assoc isnon-canonically isomorphic to

Com ?Lie .

Hence, there exists some isomorphism between Assoc ?Lie

1VectΣ and Com. In particular, for

every n, we have: (Assoc ?

Lie1VectΣ

)(n) ∈ Vect♥ .

C.3.2. Hence, it remains to show that for any V ∈ Vect♥, the map

H0((

Assoc ?Lie

1VectΣ

)? V)→ freeCom(V )

is an isomorphism.

Note, however, that the object H0((

Assoc ?Lie

1VectΣ

)? V)

identifies with

H0 (U trivLie(V )) ,

i.e., the universal enveloping algebra of the trivial Lie algebra, taken in the world of classicalassociative algebras.

However, the latter is easily seen to map isomorphically to freeCom(V ).


Appendix D. Co-algebras over co-operads

In this section we will put the (Chev,Prim)-adjunction into the framework of Koszul dualityfor algebras over an operad. In addition, we introduce another version of co-algebras (calledind-nilpotent co-algebras), in which the Koszul duality is closer to being an equivalence.

D.1. Co-operads.

D.1.1. By a co-operad we shall mean a co-associative co-algebra object in VectΣ.

As in the case of operads (see Sect. C.1.2), we will only consider co-operads Q for which theco-unit defines an isomorphism Q(1)→ k. (In particular, all our co-operads are augmented.)

D.1.2. Let VectΣf.d. ⊂ VectΣ be the full subcategory spanned by those objects P, for which

P(n) ∈ Vect is finite-dimensional in each cohomological degree for every n.

Term-wise dualization P 7→ P∗ defines a monoidal equivalence

(VectΣf.d.)

op → VectΣf.d. .

In particular, it defines an anti-equivalence between the subcategories of operads and co-operads that belong to VectΣ

f.d..

We set

Cocom := Com∗ .

This is the co-operad responsible for co-commutative co-algebras.

D.1.3. Let P be an operad. We can form a co-operad

P∨ := Bar .

Vice versa, for a co-operad Q we can consider the operad

Q∨ := coBar(Q).

It is known that the above functors

P 7→ P∨ and Q 7→ Q∨

define mutually inverse equivalences of categories, referred to as Koszul duality.

Moreover, if P ∈ VectΣf.d., then P∨ has the same property, and vice versa.

For example, it is known that

Cocom∨ ' Lie[−1],

and hence

Lie∨ ' Cocom[1].

D.2. Ind-nilpotent co-algebras over a co-operad.

D.2.1. Recall the action of VectΣ on O, considered in Sect. C.1.1.

Let Q be a co-operad. By definition, the category

Q -Coalgaug,ind-nilp(O)

is that of Q-comodules in O with respect to the ?-action.

Remark D.2.2. Modules for the above monad should be more properly called “ind-nilpotentco-algebras with divided powers”, see [FG, Sect. 3.5]. However, we shall omit the reference todivided powers from the notation because we are working over a field of characteristic zero.


D.2.3. We have the pair of adjoint functors

oblvQ+,ind-nilp : Q -Coalgaug,ind-nilp(O) O : cofreeQind-nilp ,

with oblvQ+,ind-nilp being co-monadic. In particular, oblvQ+,ind-nilp is conservative, commuteswith all colimits and with totalizations of oblvQ+,ind-nilp -split co-simplicial objects.

D.2.4. The augmentation on Q gives rise to a functor

trivQind-nilp : O→ Q -Coalgaug,ind-nilp(O).

The co-Bar construction defines a functor

Primind-nilpQ := coBar(Q,−) : Q -Coalgaug,ind-nilp(O)→ O,

right adjoint of trivQind-nilp .

By adjunction, Primind-nilpQ cofreeQind-nilp ' Id.

D.3. Koszul duality functors.

D.3.1. Let P be an operad. The augmentation on P defines the functor

trivP : O→ P -Algaug(O).

The Bar-construction defines a functor

coPrimP := Bar(P,−) : P -Algaug(O)→ O,

left adjoint to trivP.

D.3.2. Let Q := P∨ be the Koszul dual co-operad.

According to [FG, Corollary 3.3.5], the functor coPrimP can be naturally upgraded to afunctor

coPrimenh,ind-nilpP : P -Algaug(O)→ Q -Coalgaug,ind-nilp(O),

and the functor Primind-nilpQ can be upgraded to a functor

Primenh,ind-nilpQ : Q -Coalgaug,ind-nilp(O)→ P -Algaug(O),

so that

coPrimP ' oblvQ+,ind-nilp coPrimenh,ind-nilpP

and

Primind-nilpQ ' oblvP+ Primenh,ind-nilp

Q .

Furthermore, the functors

coPrimenh,ind-nilpP : P -Algaug(O) Q -Coalgaug,ind-nilp(O) : Primenh,ind-nilp

Q

form an adjoint pair.

We also note a functorial isomorphism

(D.1) coPrimenh,ind-nilpP freeP ' trivQind-nilp .


D.3.3. The following is part of [FG, Conjecture 3.4.5]:

Conjecture D.3.4. The functor

Primenh,ind-nilpQ : Q -Coalgaug,ind-nilp(O)→ P -Algaug(O)

is fully faithful.

In the sequel, we will relate Conjecture D.3.4 to several other plausible conjectures, seeSect. E.2.

D.4. (Usual) co-algebras over a co-operad.

D.4.1. We have another right-lax action of VectΣ on O, given by

P ∗ V = Πn≥1

(P(n)⊗ V ⊗n)Σn ,

and a natural transformation between the (lax) actions

(D.2) P ? V → P ∗ V.

Remark D.4.2. Note that the natural transformation (D.2) involves the operation of averagingwith respect to symmetric groups, see [FG, Sect. 3.5.5].

D.4.3. For a co-operad Q, the category Q -Coalgaug(O) of augmented Q-coalgebras is that ofQ-modules in O with respect to the ∗-action.

For example, for Q = Cocom, we obtain the usual category CocomCoalg(O).

D.4.4. We let

oblvQ+ : Q -Coalgaug(O)→ O

denote the corresponding forgetful functor.

The functor oblvQ+ is conservative and commutes with all colimits (in fact, one can showthat oblvQ+ admits a right adjoint, but it is not easy to describe this right adjoint explicitly).

In addition, it is known that the functor oblvQ+ commutes with totalizations of oblvQ+ -splitco-simplicial objects.

D.4.5. The augmentation on Q defines the functor

trivQ : O→ Q -Coalgaug(O).

The functor trivQ+ commutes with colimits and hence admits a right adjoint, denoted PrimQ.In Sect. E.4 we will describe the functor PrimQ more explicitly.

D.5. Relation between two types of co-algebras.


D.5.1. The natural transformation (D.2) gives rise to the forgetful functor

(D.3) res?→∗ : Q -Coalgaug,ind-nilp(O)→ Q -Coalgaug(O).

The following assertion will be useful in the sequel:

Lemma D.5.2. The functor res?→∗ commutes with totalizations of oblvQ+,ind-nilp-split co-simplicial objects.

Proof. Follows from the combintaion of the following three facts:

(1) the functor oblvQ+,ind-nilp commutes with totalizations of oblvQ+,ind-nilp -split co-implicialobjects;

(2) the functor res?→∗ sends oblvQ+,ind-nilp -split co-simplicial objects to co-simplicial objects;that are oblvQ+ -split;

(3) the functor oblvQ+ commutes with totalizations of oblvQ+ -split co-implicial objects.

D.5.3. We have:

(D.4) oblvQ+ res?→∗ ' oblvQ+,ind-nilp , Q -Coalgaug,ind-nilp(O)→ O

and

trivQ ' res?→∗ trivQind-nilp , O→ Q -Coalgaug(O).

We shall denote

cofreefakeQ := res?→∗ cofreeQind-nilp : O→ Q -Coalgaug(O).

D.5.4. Let P := Q∨ be the Koszul dual operad. We denote

coPrimenhP := res?→∗ coPrimenh,ind-nilp

P , P -Algaug(O)→ Q -Coalgaug(O).

It follows from (D.4) that the functor res?→∗ commutes with colimits. Hence, it admits aright adjoint, denoted (res?→∗)R.

We define

PrimenhQ := Primenh,ind-nilp

Q (res?→∗)R : Q -Coalgaug(O)→ P -Algaug(O).

By adjunction, the functors

coPrimenhP : P -Algaug(O) Q -Coalgaug(O) : Primenh

Q

form an adjoint pair.

D.5.5. We have the following tautological isomorphism:

(D.5) PrimQ ' Primenh,ind-nilpQ (res?→∗)R,

and hence

oblvQ+ PrimenhQ ' PrimQ .


D.5.6. Consider the unit of the adjunction

Id→ (res?→∗)R res?→∗.

Composing with Primenh,ind-nilpQ and pre-composing with cofreefake

Q , we obtain a naturaltransformation

Primenh,ind-nilpQ cofreeQind-nilp →

→ Primenh,ind-nilpQ (res?→∗)R res?→∗ cofreeQind-nilp ' PrimQ cofreefake

Q ,

and, combining with the unit of the (cofreeQind-nilp ,Primenh,ind-nilpQ )-adjunction, a natural trans-

formation:

(D.6) Id→ PrimQ cofreefakeQ .

D.6. The case of the co-commutative co-operad and the Lie operad. Note that thefunctor

Chevenh : LieAlg(O)→ CocomCoalgaug(O),

introduced in Sect. 1.4.4 was introduced not via Koszul duality, but by using the Cartesiansymmetric monoidal structure on LieAlg(O). In this subsection we will prove that the twodefinitions in fact agree.

D.6.1. Applying the definition of the category Q -Coalgaug(O) to Q = Cocom, we recover thecategory CocomCoalgaug(O). The functors oblvQ+ and trivQ correspond to the functorsoblvCocom+ and trivCocom, respectively, introduced earlier.

D.6.2. The category LieAlg(O) identifies with P -Algaug(O) for P = Lie. Recall that Lie∨ 'Cocom[1].

Note that the functor

Chev+ : LieAlg(O)→ O,

defined in Sect. 1.4.2 identifies with

h 7→ coPrimLie(h)[1].

D.6.3. Set

Chevenh,ind-nilp(h) := k ⊕(

[1] coPrimenh,ind-nilpLie (h)

).

This defines a functor

Chevenh,ind-nilp : LieAlg(O)→ CocomCoalgaug,ind-nilp(O).

We claim:

Lemma D.6.4. The composition

res?→∗ Chevenh,ind-nilp : LieAlg(O)→ CocomCoalgaug(O)

identifies canonically with the functor Chevenh of Sect. 1.4.6.


Proof. Let us denote by

Chevenh′ : LieAlg(O)→ CocomCoalgaug(O) = CocomCoalg(O1O/)

the functor res?→∗ Chevenh,ind-nilp.

This functor has a tautological left-lax symmetric monoidal structure, when we considerboth categories as symmetric monoidal with respect to the Cartesian structure. Recall that theCartesian symmetric monoidal structure on Cocom(O1O/) is given by tensor product.

Hence, using the diagonal functor

LieAlg(O)→ CocomCoalg(LieAlg(O)),

we can upgrade the functor Chevenh′ to a functor

Chevenh′′ : LieAlg(O)→ CocomCoalg(CocomCoalg(O1O/)).

The functors Chevenh′ and Chevenh are obtained from the functor Chevenh′′ by applying theforgetful functors

(D.7) CocomCoalg(CocomCoalg(O1O/))⇒ CocomCoalg(O1O/),

corresponding to forgetting the “outer” and “inner” operations, respectively.

Now, the required assertion follows from the fact that the functors (D.7) are canonicallyisomorphic, by Eckmann-Hilton.

D.6.5. In the same way as in Lemma D.6.4 we prove:

Lemma D.6.6. There exists a canonical isomorphism of functors

cofreefakeCocom ' Sym, O→ CocomCoalgaug(O).

Combining Lemma D.6.6 with the isomorphism (D.1), we obtain the statement of Proposi-tion 1.7.6.

Note also that the natural transformation (D.6) for Q = Cocom identifies, under the isomor-phism of Lemma D.6.6, with the natural transformation (1.5).

Appendix E. A fully faithfulness conjecture

In this subsection we study more closely the forgetful functor from in-nilpotent co-algebrasto usual co-algebras. A particular case of this conjecture is Theorem 1.7.4.

E.1. Statement of the conjecture.

E.1.1. We propose2:

Conjecture E.1.2. The functor

res?→∗ : Q -Coalgaug,ind-nilp(O)→ Q -Coalgaug(O)

of (D.3) is fully faithful.

2In [FG, Remark 3.5.3] it was erroneously stated that the authors knew how to prove this statement. Un-fortunately, this turned out not be the case.


E.1.3. Tautologically, Conjecture E.1.2 says that the unit of the adjunction

Id→ (res?→∗)R res?→∗

is an isomorphism.

Hence, from Conjecture E.1.2 we obtain:

Conjecture E.1.4. Then the natural transformation (D.6) is an isomorphism.

The main result of this section reads:

Theorem E.1.5. Conjecture E.1.4 holds if the operad Q is such that Q and Q∨[1] are bothclassical and finite-dimensional.

Note that in view of Sect. D.6.5, Theorem E.1.5 implies Theorem 1.7.4.

E.2. Some implications. In this subsection we will assume that Conjecture E.1.4 holds for agiven operad Q. In particular, it applies to Q := Cocom, since Cocom∨ ' Lie[−1].

E.2.1. Note that the fact that the natural transformation (D.6) is an isomorphism can bereformulated as saying that the functor res?→∗ induces an isomorphism

(E.1) MapsQ -Coalgaug,ind-nilp(O) (trivQaug,ind-nilp(V ), cofreeQind-nilp(W ))→

→ MapsQ -Coalgaug(O)

(trivQaug(V ), cofreefake

Q (W ))

is an isomorphism for any V,W ∈ O.

Yet another way of reformulating the fact that (D.6) is an isomorphism is the following:

Corollary E.2.2.

(a) The natural transformation

coBarQ → PrimQ res?→∗

is an isomorphism.

(b)

Primenh,ind-nilpQ → Primenh

Q res?→∗

is an isomorphism.

E.2.3. We now claim:

Proposition E.2.4. The functor res?→∗ defines an isomorphism

MapsQ -Coalgaug,ind-nilp(O) (trivQaug,ind-nilp(V ), A)→ MapsQ -Coalgaug(O) (trivQaug(V ), res?→∗(A))

for any V ∈ O and A ∈ Q -Coalgaug,ind-nilp(O).

Proof. Follows from the isomorphism (E.1) using the fact that the functor res?→∗ commuteswith with totalizations of oblvQ+,ind-nilp -split co-simplicial objects, while every object

A ∈ Q -Coalgaug,ind-nilp(O)

can be written as such a totalization, whose terms are of the form cofreeQind-nilp(W ) for W ∈O.

From here, we obtain:


Corollary E.2.5. The functor res?→∗ defines an isomorphism defines an isomorphism

MapsQ -Coalgaug,ind-nilp(O) (A′, A)→ MapsQ -Coalgaug(O) (res?→∗(A′), res?→∗(A))

for any A′ lying in the essential image of the functor coPrimenh,ind-nilpP , where P := Q∨.

Proof. Follows from the fact that any object of P -Algaug(O) can be written as a colimit of onesof the form freeP(V ), while

coPrimenh,ind-nilpP freeP(V ) ' trivQaug,ind-nilp(V ).

E.2.6. Relation to the Koszul duality conjecture. We now claim that Conjecture D.3.4 (for ourco-operad Q) implies Conjecture E.1.2:

Proof. Taking into account Corollary E.2.5, it suffices to know that the functor

coPrimenh,ind-nilpP : P -Algaug(O)→ Q -Coalgaug,ind-nilp(O)

is essentially surjective. However, the latter follows from Conjecture D.3.4.

E.2.7. Recall now Conjecture E.2.8. Generalizing for our co-operad Q, we propose:

Conjecture E.2.8.

(a) The unit of the adjunction

Id→ PrimenhQ coPrimenh

P

is an isomorphism, when evaluated on objects lying in the essential image of the functor PrimenhQ .

(b) The co-unit of the adjunction

coPrimenhP Primenh

Q → Id

is an isomorphism, when evaluated on objects lying in the essential image of the functor

coPrimenh,fakeP .

We claim that Conjecture D.3.4 implies Conjecture E.2.8:

Proof. Point (b) of the conjecture follows by combining Corollary E.2.2(b) with the assumptionthat the co-unit of the adjunction

coPrimenh,ind-nilpP Primenh,ind-nilp

Q → Id

is an isomorphism.

By Corollary E.2.2(a), the assertion of point (a) of the conjecture is equivalent to the unitof the adjunction

Id→ Primenh,ind-nilpQ coPrimenh,ind-nilp

P

being an isomorphism on the essential image of Primenh,ind-nilpQ . However, Conjecture D.3.4

implies that the functor coPrimenhP is a localization, and the assertion follows.

E.3. Calculation of co-primitives. The rest of this section is devoted to the proof of Con-jecture E.1.4.


E.3.1. For n ≥ 1, let

ιn : Vect→ VectΣ

be the tautological functor that produces symmetric sequences with only the n-th non-zerocomponent.

We have the following basic fact:

Lemma E.3.2. For an operad P, the object 1VectΣ ∈ VectΣ, regarded as a right P-module canbe canonically written as a colimit

colimn≥0

Mn,

with

Cone(Mn−1 →Mn) ' ιn(P∨(n)) ? P.

E.3.3. The assertion of Lemma E.3.2 gives rise to the following more explicit way to expressthe functor coPrimP:

Corollary E.3.4. The functor

A 7→ coPrimP(A), P -Algaug(O)→ O

admits a canonical filtration by functors of the form

A 7→Mn ?PA,

where Mn are right P-modules. The associated graded of this filtration is canonically identifiedwith

n 7→ P∨(n) ? oblvPaug(A).

E.4. Computation of primitives. Our current goal is to formulate and prove an analog ofCorollary E.3.4 for co-algebras over a co-operad, namely Proposition E.4.3 below.

E.4.1. Let Q be a co-oparad, N a right Q-comodule in VectΣ, and A ∈ Q -Coalgaug(O).

We can form a co-simpicial object of O with the n-th term

coBar•(N,Q, A) :=

N ? Q ? ... ? Q︸︷︷︸n

∗A.We define

NQ∗ A := Tot (coBar•(N,Q, A)) .

E.4.2. We are going to prove:

Proposition E.4.3. The functor

A 7→ PrimQ(A), Q -Coalgaug(O)→ O

can be canonically written as a limit of functors of the form

NnQ∗ A, n ≥ 1,

where Nn are right Q-comodules in VectΣ with

ker(Nn → Nn−1) ' ιn(Q∨(n)) ? Q.

The rest of this section is devoted to the proof of this proposition.


E.4.4. By definition, the functor PrimQ is calculated as

A 7→ coBar•(1VectΣ ,Q, A).

Now, we have the following assertion, which is an analog of Lemma E.3.2 for co-operads:

Lemma E.4.5. For a co-operad Q, the object 1VectΣ ∈ VectΣ, regarded as a right Q-comodulecan be canonically written as a limit

limn≥0

Nn,

with

ker(Nn → Nn−1) ' ιn(Q∨(n)) ? Q.

Since functor of totalization commutes with the formation of limits of terms, in order toprove Proposition E.4.3, it suffices to show that for every m ≥ 0, the natural map

coBarm(1VectΣ ,Q, A)→ limn∈(Z≥1)op

coBarm(Nn,Q, A)

is an isomorphism.

E.4.6. For the latter, by the definition of the ∗-action, it suffices to show that for any i ≥ 0,the map 1VectΣ ? Q ? ... ? Q︸︷︷︸

m

(i)⊗A⊗i → limn∈(Z≥1)op

Nn ? Q ? ... ? Q︸︷︷︸m

(i)⊗A⊗i

is an isomorphism.

However, the required isomorphism follows from the fact that for every given i, the family

n 7→

Nn ? Q ? ... ? Q︸︷︷︸m

(i)

stabilizes to 1VectΣ ? Q ? ... ? Q︸︷︷︸m

(i)

for n ≥ k.

E.5. Proof of Theorem E.1.5.

E.5.1. Step 1. We take A := cofreefakeQ (V ) for V ∈ O. We need to show that the natural map

V → PrimQ cofreefakeQ (V )

is an isomorphism.

We calculate the right-hand side via Proposition E.4.3. We will prove that for every n ≥ 1,the map

Cone

(V → Nn

Q∗ cofreefakeQ (V )

)→ Cone

(V → Nn−1

Q∗ cofreefakeQ (V )

)is zero. This will prove the required assertion.


E.5.2. Step 2. We note that for any n ≥ 1, the object Nn has a finite filtration by objects ofthe form ιm(Q∨(m)) ? Q, m ≤ n.

We obtain that for any A ∈ Q -Coalgaug,ind-nilp(O) the canonical map

NnQ? A→ Nn

Q∗ res?→∗(A)

is an isomorphism.

Hence, we obtain that it suffices to show that the map

Cone

(V → Nn

Q? cofreeQind-nilp(V )

)→ Cone

(V → Nn−1


)is zero.

E.5.3. Step 3. Note that each NnQ? cofreeQind-nilp(V ) is naturally graded by integers d ≥ 1,

such that the map

V → NnQ? cofreeQind-nilp(V )

is an isomorphism on the degree 1 part for all n.

Hence, it remains to show that for all d > 1, the map

(E.2)

(Nn


)d →

(Nn−1


)d

is zero.

E.5.4. Step 4. Note now that the functor

V 7→(Nn


)d

(resp., the natutal transformation (E.2)) is given by

V 7→ (Kdn ⊗ V ⊗d)Σd

for some Kdn ∈ Rep(Σd) (resp., a map Kd

n → Kdn−1).

Hence, it remains to show that for every d > 1 and every n, the map

(E.3) Kdn → Kd

n−1

is zero.

E.5.5. Step 5. Since the category Rep(Σd) is semi-simple, the fact that (E.3) is equivalent tothe map in question inducing the zero map on cohomology.

The latter reduces the assertion of the theorem to the case of O = Vect. Namely, it sufficesto show that for some/any V ∈ Vect♥f.d. with dim(V ) ≥ d, the map (E.2) induces the zero mapon cohomology.


E.5.6. Step 6. Note that the assumption on Q∨ implies that for any m,

ker(Nm → Nm−1)Q? cofreeQind-nilp(V )

lives in the cohomological degree m− 1.

Hence, the fact that the map (E.2) is zero on the cohomology is equivalent to the fact thatthe object

(E.4)

(Nn


)d ∈ Vect

is acyclic in degrees 0, 1, ..., n− 1.

E.5.7. Step 7. Consider the operad Q∗, and set Mn := N∗n. We obtain that the object (E.4) isthe linear dual of the object

(E.5)

(Mn ?

Q∗freeQ∗(V

∗)

)d.

Hence, is is enough to show that the object (E.5) is acyclic in degrees −(n− 1), ...,−1, 0.

E.5.8. Step 8. We note that (Q∗)∨ ' (Q∨)∗. So, by the assumption on Q∨,

Cone

((Mm−1 ?

Q∗freeQ∗(V

∗)

)d →

(Mm ?

Q∗freeQ∗(V

∗)

))d

is concentrated in degree −m+ 1.

Hence, the acyclicity of (E.5) in the specified degrees is equivalent to the acyclicity in alldegrees of

(E.6) colimm

(Mm ?

Q∗freeQ∗(V

∗)

)d.

E.5.9. Step 9. By Lemma E.3.4, the colimit (E.6) identifies with the degree d part of

1VectΣ ?Q∗

freeQ∗(V∗).

However,1VectΣ ?

Q∗freeQ∗(V

∗) ' V ∗

and hence its degree d part for d 6= 1 vanishes.

References

[Fra] J. Francis.

[FG] J. Francis and D. Gaitsgory, Chiral Koszul duality.[1-aff] D. Gaitsgory, Sheaves of categories and the notion of 1-affineness.

[GL:DG] D. Gaitsgory, Notes on Geometric Langlands: Generalities on DG categories,available at http://www.math.harvard.edu/ gaitsgde/GL/.

[GL:Stacks] D. Gaitsgory, Notes on Geometric Langlands: Stacks,available at http://www.math.harvard.edu/ gaitsgde/GL/.

[GL:QCoh] D. Gaitsgory, Notes on Geometric Langlands: Quasi-coherent sheaves on stacks,available at http://www.math.harvard.edu/ gaitsgde/GL/.

[GL:IndCoh] D. Gaitsgory, Ind-coherent sheaves, arXiv:1105.4857.[GL:IndSch] D. Gaitsgory and N. Rozenblyum, DG indschemes, arXiv:1108.1738.[GL:Crystals] D. Gaitsgory and N. Rozenblyum, Crystals.[Book-II.1][Book-II.2]

[Book-III.1]


[Book-III.2]

[Book-III.3]

[Book-IV.4][Lu0] J. Lurie, Higher Topos Theory, Princeton Univ. Press (2009).

[Lu1] J. Lurie, DAG-VIII, available at http://www.math.harvard.edu/ lurie.

[Lu?] J. Lurie, 2-Categories, available at http://www.math.harvard.edu/ lurie.

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