Table Of ContentNATURAL OPERATIONS ON THE HIGHER HOCHSCHILD
HOMOLOGY OF COMMUTATIVE ALGEBRAS
ANGELA KLAMT
Abstract. Wegivethedefinitionofhigher(co)Hochschildhomologyofdg-functorsin
thesenseofPirashvilianddefinetheirformaloperationsinthesenseofWahl,whichgive
acomplexofoperationsonthehigherHochschildhomologyofcommutativealgebras. In
certaincasesweobtainsmallermodelsoftheoperationsandidentifythemwiththedual
of the chains on the mapping space of simplicial sets. Moreover, we show that some of
theresultsaretrueinamoregeneralsetupworkingoverchaincomplexesortopological
spaces. WeusethistoinvestigatetheformaloperationsoftopologicalChiralhomology.
Introduction
Given a simplicial set X and a commutative algebra A one can associate to this data
•
a chain complex CH (A), the higher Hochschild complex of A with respect to X defined
X• •
in [Pir00], where the classical Hochschild complex is the one associated to the standard
simplicialdecompositionofthecircle. Inthispaper,weareinterestedinthechaincomplex
of natural operations on the higher Hochschild complex of given types of algebras such
as commutative algebras, Poisson algebras or commutative Frobenius algebras. Following
the approach of [Wah12], we approximate this chain complex by a complex of formal
operations which we identify in certain cases. Our methods differ from [Wah12] in that
weonlyworkwithstrictlyassociativealgebras. Thisallowsustousesimplicialtechniques
to give easier proofs of many results in [Wah12] in the case of strictly associative algebras.
LetE beacommutative PROP,i.e. asymmetricmonoidaldg-categorywithobjectsthe
natural numbers equipped with a symmetric monoidal dg-functor i : Com → E which is
theidentityonobjects. AnE-algebraisastrongsymmetricmonoidalfunctorΦ : E → Ch.
Let X be a simplicial finite set. The higher Hochschild complex of Φ with respect to
•
X (in the sense of [Pir00]) denoted by CH (Φ(1)) is the total complex of a simplicial
• X•
chain complex which in simplicial degree k is given by Φ(1)⊗|Xk|. The boundary maps are
induced by the boundary maps of the simplicial set. Similarly, one can define the higher
Hochschild homology for any dg-functor Φ : E → Ch (not necessary strong symmetric
monoidal) by taking the total complex of the simplicial chain complex with simplicial
degree k equal to Φ(|X |). Again, the boundaries are induced by the boundary maps
k
of X which act on Φ via the functor i : Com → E (see Definition 2.3). This defines a
•
functor
C (−) : Fun(E,Ch) → Ch
X•
and the construction can be extended to arbitrary simplicial sets using homotopy colim-
its. When restricted to strong symmetric monoidal functors, C (Φ) is isomorphic to the
X•
higher Hochschild complex CH (Φ(1)). On the other hand the higher Hochschild con-
X•
struction can be defined via an enriched tensor product which then, working in the model
category of topological spaces instead of chain complexes looks similar to the definition
of topological Chiral homology.
In the first part of this paper we work in the category of chain complexes and are
interested in the natural transformations of the (iterated) higher Hochschild homology
of E-algebras with respect to simplicial sets X and Y denoted by Nat⊗(X ,Y ) =
• • E • •
Hom(CH (−),CH (−)). We define the complex of formal operations as the com-
X• Y•
plex Nat (X ,Y ) = Hom(C (−),C (−)), i.e. we test on all functors and not only on
E • • X• Y•
1
2 ANGELAKLAMT
the strong symmetric monoidal ones. Hence, there is a restriction from Nat (X ,Y ) to
E • •
Nat⊗(X ,Y ). Analogously to [Wah12, Theorem 2.9] we give conditions on the PROP
E • •
implying that the restriction is injective/surjective/a quasi-isomorphism (see Theorem
3.4).
The higher Hochschild complex is invariant under quasi-isomorphisms of functors and
quasi-isomorphic to its reduced version (see Section 2.2). It actually can be defined as a
functor C (Φ)(−) : Ch → Ch and so we can consider the iterated Hochschild complex.
X•
We show that for two simplicial sets X and X(cid:48) there is a quasi-isomorphism between
• •
C (C (Φ)) and C (Φ) (see Theorem 2.13) and so the general case is covered (up
X• X•(cid:48) X•(cid:113)X•(cid:48)
to quasi-isomorphism) by taking the higher Hochschild complex once.
We similarly define the higher coHochschild complex D (Φ) of a coalgebra (see Def-
X•
inition 2.4) and in general of dg-functors Ψ : Eop → Ch. The formal operations between
these are defined as NatD(X ,Y ) := Hom(D (−),D (−)). Our first technical theorem
E • • X• Y•
connects the two complexes of formal operations to a third more computable complex:
Theorem A (Theorem 3.2 and Theorem 3.6). For any commutative PROP E and sim-
plicial sets X and Y there are isomorphisms of chain complexes
• •
Nat (X ,Y ) ∼= D (C (E(−,−))) ∼= NatD(Y ,X ).
E • • X• Y• E • •
In the second part of the paper we consider the case E = Com. Under two types
of conditions on X• and Y• we identify the complex NatCom(X•,Y•) with other, better
known complexes.
First, working over a field F, for X arbitrary and Y a simplicial set that is weakly
• •
equivalent to a simplicial finite set, a quasi-isomorphism of functors C∗(Y×−) (cid:39) A⊗− :
•
Com → Ch induces a quasi-isomorphism
NatCom(X•,Y•) (cid:39) CHX•(A)∗
(see Proposition 4.2). In particular if Q ⊂ F, the deRham algebra Ω•(Y ;F) fulfills this
•
property (see Appendix A.1) and therefore
NatCom(X•,Y•) (cid:39) CHX•(Ω•(Y•;F))∗.
Another example are strongly F-formal simplicial sets, simplicial sets weakly equivalent
to simplicial finite sets such that there is a multiplicative quasi-isomorphism C∗(Y ) →
•
H∗(Y ). In this case, we get
•
NatCom(X•,Y•) (cid:39) CHX•(H∗(Y•))∗.
(cid:96)This∗is,(cid:96)usedSin1(cid:113)[K(cid:96)la13]∗t)o),ctohmepfuortme tahleopopereartaitoinosnsonintthheeHhoocmhosclohgilydHh∗o(mNoaltoCgoymo(f(cid:96)cno1mSm•1u(cid:113)-
m1 n2 • m2
tative algebras.
Our second computation of NatCom(X•,Y•) is when the dimension of the simplicial
set X is smaller than the connectivity of Y . Using Bousfield’s spectral sequence (see
• •
[Bou87]), we get a quasi-isomorphism between NatCom(X•,Y•) and the simplicial chains
on the topological mapping space hom (|X |,|Y |). We show moreover, that this quasi-
Top • •
isomorphism preserves some extra structure close to a coproduct:
Theorem B (SeeTheorem4.12). ForanarbitrarysimplicialsetY andafinitesimplicial
•
set X such that dim(X ) ≤ Conn(Y ), there is weak equivalence
• • •
C∗(HomTop(|X•|,|Y•|)) (cid:39) NatCom(X•,Y•).
If we take homology with coefficients in a field F, the comultiplication on the homology
H∗(NatCom(X•,Y•);F) induced by the one on H∗(HomTop(|X•|,|Y•|);F) commutes with
OPERATIONS ON HIGHER HOCHSCHILD HOMOLOGY 3
restriction to the filtration of Nat, i.e.
H∗(NatCom(X•,Y•);F) (cid:47)(cid:47) H∗(NatCom(X•,Y•);F)⊗H∗(NatCom(X•,Y•);F)
(cid:15)(cid:15) (cid:15)(cid:15)
H (Nat2m(X ,Y );F) H∗(∆2m)(cid:47)(cid:47) H (Natm(X ,Y );F)⊗H (Natm(X ,Y );F)
∗ • • ∗ • • ∗ • •
and
H∗(NatCom(X•,Y•);F) (cid:47)(cid:47) H∗(NatCom(X•,Y•);F)⊗H∗(NatCom(X•,Y•);F)
(cid:15)(cid:15) (cid:15)(cid:15)
H (Nat2m+1(X ,Y );FH)∗(∆2m+(cid:47)(cid:47)1H) (Natm+1(X ,Y );F)⊗H (Natm(X ,Y );F)
∗ • • ∗ • • ∗ • •
commute.
Here, Natm(X•,Y•) is the filtration of NatCom(X•,Y•) by its cosimplicial degree. The
families of maps ∆ : Nat2m(X ,Y ) → Natm(X ,Y ) ⊗ Natm(X ,Y ) and ∆ :
2m • • • • • • 2m+1
Nat2m+1(X ,Y ) → Natm+1(X ,Y )⊗Natm(X ,Y ) come from a comultiplication on the
• • • • • •
cosimplicial simplicial abelian group underlying NatCom(X•,Y•).
The proof of Theorem B is similar to the proof of [PT03, Theorem 2] and [GTZ10a,
Proposition 2.4.2] but since we are in a kind of dual situation and we do not know a
reference for the theorem in this situation, we need to check the compatibility of the
maps again.
Thelastpartofthepaperisanattempttocarryoverthetechniquestoamuchbroader
generality. Hence this part of the paper can be read independently of the first part. We
workwithM themonoidalmodelcategoryofchaincomplexes(withtheprojectivemodel
structure) or topological spaces (with the mixed model structure) and E a small category
enrichedoverM withallmorphismspacesbeingcofibrantandtheinclusionoftheidentity
a cofibration. In these cases the enriched bar construction gives a model of the derived
enrichedtensorproduct. GivenalevelwisecofibrantfunctorA : Eop → M,forΦ : E → Ch
we define the Hochschild complex of Φ as
C (Φ) := Φ⊗B(E,E,A)
A
E
in the case of chain complexes and as
C (Φ) := |s Φ|⊗B(E,E,A)
A •
E
for topological spaces, where |s (−)| is the geometric realization of the singular chains
•
of a space, a choice of a cofibrant replacement in topological spaces. In both cases the
Hochschild complex is a model of the derived tensor product. Similarly for Ψ : Eop → Ch,
we define the coHochschild construction as
D (Ψ) = hom (B(E,E,A),Ψ)
A Eop
in both cases. Hence for any category E(cid:48) with a map E → E(cid:48),the Hochschild construction
defines a functor C (−) : Fun(E(cid:48),M) → M and given A,A(cid:48) : E(cid:48) → Ch we can define
A
the formal transformations Nat (A,A(cid:48)) as all transformations of these functors. We can
E(cid:48)
prove more general versions of Theorem A and in particular deduce:
Corollary C (see Cor. 5.15 and Cor. 5.17). Let E and E(cid:48) be small categories cofibrantly
enriched over Ch or Top together with a functor E → E(cid:48) and let A,A(cid:48) : Eop → M be two
functors. Then
Nat (A,A(cid:48)) (cid:39) D C (E(cid:48)(−,−)).
E(cid:48) A A(cid:48)
4 ANGELAKLAMT
The statement even holds if we start with any (topological) h–projective replacement
B → A (see Def. 5.2 and Def. 5.3), even if E was not cofibrantly enriched. By the
A
last theorem and the invariance of D and C under weak equivalences, it follows that
A A(cid:48)
the definition of Nat (A,A(cid:48)) (up to weak equivalence) is independent of the choice of
E(cid:48)
h–projective resolution.
Furthermore, we can rewrite the definition of the higher Hochschild and coHochschild
complex with respect to a simplicial set in this setup and use the results of this section to
deduce a prove of Theorem A. On the other hand, taking E to be the framed little disk
operad Diskfr and E = Embfr(−,M) the framed embeddings into an n–dimensional
n M
framed manifold M, the functor CE (Φ) equals the topological chiral homology of Φ (de-
(cid:82) M
noted by (Φ)) in the sense of [Lur12]. Moreover, in the topological case the Hochschild
M
complex is cofibrant and hence the formal operations can actually be defined via the
derived mapping space, i.e. we get
(cid:90) (cid:90)
Nat (M,M(cid:48)) := Nat (E ,E ) (cid:39) Rhom( (−), (−)).
E(cid:48) E(cid:48) M M(cid:48)
M M(cid:48)
Then from the above corollary we can deduce an equivalence
NatDisknfr(M,M(cid:48)) (cid:39) DEMCEM(cid:48)(Disknfr(−,−)).
We hope that the results generalize further and that one gets similar theorems for ex-
ampleworkingoverspectra,givingawaytocomputeoperationsontopologicalHochschild
homology. The topological proof uses that |s (−)| is a strong monoidal, comonadic re-
•
placement. Even though a more general theory for cofibrant replacements via comonadic
functors is known (see [Gar09, Theorem 3.3] and [BR12, Cor. 3.1]) we do not see a way
to get the monoidality simultaneously for example for spectra.
The paper is organized as follows: In Section 1 we fix notations and conventions on
(double) chain complexes and simplicial sets. More details are given in Appendix B,
which we will refer to if needed. In Section 2.1 we give the definitions of the higher
Hochschild and coHochschild complexes which are the main subject of the paper. In
Section 2.2 we establish basic properties of these using the simplicial structure given in
our situation. In Section 2.3 we show that the iterated higher (co)Hochschild complex
up to quasi-isomorphism is covered by the single one by applying it with respect to the
disjoint union of simplicial sets. In Section 3 we define the formal operations of the
(co)Hochschild construction and state Theorem A. We also explain the connection to
monoidal functors and define the ∆ and ∆ maps which are used in Theorem B. In
2m 2m+1
Section 4 we fix the commutative PROP and state examples where the formal operations
can be identified with the dual of the higher Hochschild complex of some algebra. The
proof of the examples is given in Appendix A. Finally, in the last part of this section we
establish the details from [Bou87] to give a proof of Theorem B.
Section 5 deals with the more general setup in monoidal model categories. This section
can be read independently of what was done before. We first recall the definitions of the
enriched tensor product and mapping spaces and their derived versions together with the
enriched bar and cobar construction. We recall [Shu06, Theorem 23.12] to give conditions
on when the bar construction can be used to compute the derived functors. In Section
5.2 we define the (co)Hochschild complex in chain complexes and topological spaces and
state the analogs of Theorem A. In Section 5.5.1 we explain how to see higher Hochschild
homologyinthismoregeneralsetupandhowtodeduceTheoremAfromthemoregeneral
theorems given before. Last, in Section 5.5.2 we show how to apply the theorem in the
topological situation to Chiral homology.
Acknowledgements. I would like to thank my advisor Nathalie Wahl for many helpful
discussions, questions and comments. I would also like to thank Gr´egory Ginot and
OPERATIONS ON HIGHER HOCHSCHILD HOMOLOGY 5
DustinClausenforfruitfuldiscussions. TheauthorwassupportedbytheDanishNational
Research Foundation through the Centre for Symmetry and Deformation (DNRF92).
Contents
Introduction 1
Acknowledgements 4
1. Homological algebra setup 5
1.1. Chain complexes and double complexes 5
1.2. Simplicial sets 7
2. Higher Hochschild homology 8
2.1. Definition 8
2.2. Basic properties of the higher Hochschild and coHochschild functors 11
2.3. Iterated Hochschild functors 13
3. Formal and natural operations of the higher Hochschild complex functors 16
3.1. Formal operations 16
3.2. Restriction to natural transformations of algebras 17
3.3. Formal operations of the coHochschild construction 18
3.4. Coalgebra structures 18
4. Formal operations for the commutative PROP 21
4.1. Small models for the formal operations 21
4.2. Relationship of the formal operations and the mapping space 23
5. The general setup in closed monoidal model categories 29
5.1. Derived tensor products and mapping spaces 30
5.2. Hochschild and coHochschild construction and formal operations 32
5.3. The proofs for chain complexes 35
5.4. The proof for topological spaces 36
5.5. Applications to higher Hochschild and Chiral homology 38
Appendix A. More details for Theorem 4.5 and Theorem 4.6 41
A.1. The DeRham coalgebra 41
A.2. Simplicial sets with a multiplicative quasi-isomorphism C∗(X ) → H∗(X ) 42
• •
Appendix B. Algebraic tools 43
B.1. Functorality of Tot and sTot 43
B.2. The Eilenberg-Zilber Theorem 44
B.3. Spectral sequences for double complexes 45
B.4. The model structures on topological spaces and chain complexes used in
Section 5 47
References 48
1. Homological algebra setup
1.1. Chain complexes and double complexes. Throughout this paper we will use
chain and double chain complexes as dg-categories. In this section we give the sign
conventions and notations used later on.
Notation 1.1 (Sign Conventions). In this paper Ch means the category of Z-graded
chain complexes over Z, unless otherwise specified. For two chain complexes A and B
∗ ∗
we fix the differential on A ⊗B to be d ⊗id+(−1)kid⊗d . A dg-category C is a
k l A B
category enriched over Ch, i.e. it has morphism spaces C(a,b) which are chain complexes
together with chain maps k → C(a,a) and C(a,b)⊗C(b,c) → C(a,c) which fulfill the
unit and associativity conditions. Note that by this convention postcomposition with
morphisms acts from the right. For an abelian category A the dg-category Ch(A) has
chain complexes C in A as objects. A morphism f of degree k in Ch(C ,D ) is a
∗ ∗ ∗
6 ANGELAKLAMT
family of maps (f ) : C → D in A. The differential on Ch(C ,D ) is defined as
p p p+k ∗ ∗
d(f) = (−1)i(d ◦f −f ◦d ) : C → D (note that by the convention of functions
i D i i−1 C i i+d−1
acting from the right the sign differs from the usual one). For A the category of abelian
groups, we define the dual (A )∗ := Ch(A ,Z) with Z the trivial complex concentrated
∗ ∗
in degree 0. For an element f ∈ A∗, i.e. f = 0 for i (cid:54)= k, one gets d(f) = 0 for i (cid:54)= k+1
k i i
and d(f) = (−1)|f|f ◦d .
A
A map f : C → D in Ch(A) is called a chain map if it is a degree zero cycle
∗ ∗
in Ch(C ,D ). A chain map is a quasi-isomorphism if it induces an isomorphism on
∗ ∗
homology. Two chain maps f,g : C → D are chain homotopic if there is a degree one
∗ ∗
map s ∈ Ch(C ,D ) such that d◦s+s◦d = f −g. A chain map f : C → D is a chain
∗ ∗ 1 ∗ ∗
homotopy equivalence if there exists a map g : D → C such that f ◦g and g ◦f are
∗ ∗
homotopic to the identity of D and C , respectively.
∗ ∗
For a dg-category C a dg-functor Φ : C → Ch is an enriched functor, i.e. the structure
maps
c : Φ(a)⊗C(a,b) → Φ(b)
Φ
are chain maps.
Notation 1.2. A double chain complex C is for each p,q an abelian group C with
∗,∗ p,q
maps d : C → C and d : C → C such that d ◦d = 0, d ◦d = 0 and
h p,q p−1,q v p,q p,q−1 h h v v
d ◦d = d ◦d . By this, a double chain complex can be viewed as a chain complex
h v v h
of chain complexes in two ways: The first one has in each degree p the chain complex
B = C (i.e. the differential d : B → B is the horizontal one). The second one
p p,∗ h p p−1
has in degree q the chain complex D = C and the differential is the vertical one.
q ∗,q
These two ways of seeing double chain complexes as objects in the abelian category of
chain complexes induce two structures of a dg-category on the category of double chain
complexes.
More precisely, we define the dg-category dChh to have as objects double chain com-
plexes C . A morphism of degree k in dChh(C ,D ) is a map f : C → D
∗,∗ ∗,∗ ∗,∗ ∗,∗ ∗+k,∗
which is a chain map with respect to d (i.e. d ◦f = f ◦d ). The differential of f is
v v v
given by dh(f) = (−1)p(dh ◦f −f ◦dh).
p,q D p,q p−1,q C
Similarly,thecategorydChv isthecategorywiththesameobjectsbutinheritingtheen-
riched structure with respectto the vertical differential. An elementf ∈ dChv(C ,D )
∗,∗ ∗,∗
is a map f : C → D which is a chain map with respect to d and has differential
∗,∗ ∗,∗+k h
dv(f) = (−1)q(dv ◦f −f ◦dv).
p,q D p,q p,q−1 C
Chain maps of double chain complexes, i.e. maps of degree zero commuting with both
differentials, are precisely the degree zero cycles of the morphism complexes of either
category.
Wewanttodefinethetotalcomplexofadoublecomplexsuchthatitgivesadg-functor
dChv → Ch. This is done as follows: For a double complex C define Tot(cid:81)(C) to be
p,q
the product double complex with
(cid:81) (cid:89)
Tot (C) = C
n p,q
p+q=n
and the direct sum double complex
(cid:77)
Tot⊕(C) = C
n p,q
p+q=n
both with differential d = dh+(−1)pdv. Note that for a first or third quadrant double
p,q p q
complex both complexes agree.
WedefinesTot(cid:81)(C)andsTot⊕(C)tobetheswitched double complexes withtheroleof
(cid:81) (cid:81)
the horizontal and vertical direction switched. As an abelian group sTot (C) = Tot (C)
and sTot⊕(C) = Tot⊕(C) but the differentials are d = (−1)qdh+dv. In both cases, the
p,q p q
OPERATIONS ON HIGHER HOCHSCHILD HOMOLOGY 7
switched and unswitched complex are isomorphic via the isomorphism
(1.1) x ∈ C , x (cid:55)→ (−1)pqx.
p,q
For f ∈ dChv(C ,D ) of degree |f| (i.e. f : C → D ) and x ∈ C we
∗,∗ ∗,∗ p,q p,q p,q+|f| p,q
define(Tot(f)(x)) = f (x ) and sTot(f)(x) = (−1)|f|pf(x) for both the product and
p,q p,q p,q
the direct sum total complexes. With these definitions both functors are dg-functors
dChv → Ch (see Proposition B.1).
For a double complex C we define its filtration by columns as
p,q
(cid:89)
F = C .
s p,q
p≤s
For a right half plane (first and fourth quadrant) double complex, the product becomes
a direct sum, whereas for a left half plane (second and third quadrant) double complex
it can be non-finite. This filtration yields the spectral sequence of double complexes (see
Appendix B.3). The spectral sequence of a right half plane double complex converges
to the direct sum total complex, the one of a left half plane double complex converges
conditionally to the product total complex. We show that this implies that for C and
p,q
D botheitherrightorlefthalfplanedoublecomplexesandachainmapf : C → D
p,q p,q p,q
which is a quasi-isomorphism in dChv (i.e. a quasi-isomorphism with homology taken in
the vertical direction), f induces a quasi-isomorphism of their direct sum or product
complexes, respectively (see Corollary B.12). If on the other hand f : C → D is
p,q p,q
a quasi-isomorphism in dChh (i.e. in the category with the horizontal differential) the
spectral sequence argument used in the previous case does not work. However, if f is
a chain homotopy equivalence in dChh it still induces a quasi-isomorphism of the direct
sum or product total complexes, respectively (see Corollary B.14).
1.2. Simplicial sets. In this section we recall the sign conventions and notation for
(co)simplicial sets.
Let A be an abelian category. A simplicial object A in A is a functor A : ∆op → A.
• •
We denote the boundary maps by d and the degeneracy maps s .
i i
The chain complex C (A ) ∈ Ch(A) is given by A in the k-th degree and differential
∗ • k
d = (cid:80)n (−1)id .
i=0 i
Definition 1.3 (cf. [Wei95, Chapter 8.3]). The normalized chain complex N (A ) ∈
∗ •
Ch(A) is defined to be
n−1
(cid:92)
N (A ) = ker(d : A → A ).
n • i n n−1
i=0
The degenerate subcomplex D (A) is given by
∗
(cid:91)
D (A ) = im(s ).
n • i
AcosimplicialobjectB• inanabeliancategoryA isafunctorB• : ∆ → A. Wedenote
the boundary maps by di and the degeneracy maps si. The cochain complex C∗(B•) ∈
coCh(A) is given by Bk in the k-th degree and it has differential d = (cid:80)n+1(−1)i+kdi.
i=0
By our sign conventions in Notation 1.1, given a simplicial abelian group A with dual
•
cosimplicial abelian (A )∗, we get C∗((A )∗) = (C (A ))∗.
• • −∗ •
For a simplicial set X we write C (X ) for the chain complex given by the chain
• ∗ •
complex associated to the linearization of X , i.e. C (X ) := C (Z[X ]). Since this
• ∗ • ∗ •
linearization is never applied to simplicial abelian groups, we use the same notation in
both cases.
8 ANGELAKLAMT
Definition 1.4. The normalized cochain complex N∗(B•) ∈ coCh(A) is given by
n−1
(cid:92)
Nn(B•) = ker(si : Bn → Bn−1).
i=0
The degenerate subcomplex D∗(B•) is defined to be
(cid:91)
Dn(B•) = im(d ).
i
Notation 1.5. We define the reduced Moore complex C of a simplicial object A as
∗ •
C (A) := C (A )/D (A )
∗ ∗ • ∗ •
and the reduced Moore cocomplex C∗ of a cosimplicial object B• as
C∗(B•) := N∗(B•).
Proposition 1.6 ([Wei95, Lemma 8.3.7 and Theorem 8.3.8],[Fre12, Lemma 4.2.5]). For
a simplicial object A in an abelian category A we have
•
∼
C (A ) = N (A )⊕D (A )
∗ • ∗ • ∗ •
and there is a natural chain homotopy equivalence
N (A ) (cid:39)h C (A ).
∗ • ∗ •
Together, we have
C (A ) (cid:39)h C (A ).
∗ • ∗ •
Dually, for a cosimplicial object B• we have
C∗(B•) ∼= N∗(B•)⊕D∗(B•)
and a natural chain homotopy equivalence
C∗(B•) = N∗(B•) (cid:39)h C∗(B•).
For a simplicial abelian group A and its dual cosimplicial abelian group A∗,
• •
(C (A ))∗ ∼= C∗(A∗) and (C (A ))∗ ∼= C∗(A∗).
∗ • • ∗ • •
2. Higher Hochschild homology
Our definitions of the higher Hochschild complex and coHochschild complex of com-
mutative algebras are analog to the definition of the Hochschild complex for A -algebras
∞
given by Wahl and Westerland in [WW11] and the coHochschild complex defined in
[Wah12]. Many of the statements proved in this article have been proven by the afore-
mentionedauthorsintheircase. Theproofsgeneralizebutsometimesalsosimplifybythe
toolsofsimplicialsetswehaveinoursetup. Furthermore, thedefinitionoftheHochschild
complex for functors in the ungraded setup already occurs in [Pir00].
2.1. Definition.
Notation 2.1. Let FinSet be the category of all finite sets with all maps between them
and FinOrd the category of sets n = {1,...,n} with all maps between those. For this
paper we fix an equivalence of categories S : FinSet → FinOrd.
Notation 2.2. Denote by Com(−,1) the unital commutative operad which has one
operation of degree zero in each degree m ≥ 0. Let Com(m,n) be the induced linearized
Prop. We note that as categories Com ∼= Z[FinOrd], i.e. the category with the same
objects but the linearized homomorphism sets and that we have the embedding functor
L : FinOrd → Com which is the identity on objects.
Let E be a symmetric monoidal dg-category equipped with a functor i : Com → E. We
assume that i is a bijection on objects.
OPERATIONS ON HIGHER HOCHSCHILD HOMOLOGY 9
Definition 2.3. Let Φ be a dg-functor from E to Ch. Let Y be a simplicial finite set
•
(i.e. Y is finite for each k). We define the higher Hochschild complex of Φ with respect
k
to Y as C (Φ) : E → Ch via
• Y•
(2.1) C (Φ) : E −F−Y−•(−Φ→) Ch∆op −C→∗ dCh −s−T−o−t→⊕ Ch
Y•
where F (Φ) sends a set n to the simplicial chain complex
Y•
F (Φ)(n) : ∆op −Y→• FinSet −→S FinOrd −L−(−−−(cid:113)−n→) Com →−i E −→Φ Ch.
Y•
The reduced Higher Hochschild complex of Φ is defined via
(2.2) C (Φ) : E −F−Y−•(−Φ→) Ch∆op −C−→∗ dCh −s−T−o−t→⊕ Ch
Y•
whereC isthereducedchaincomplexfunctordefinedinDefinition1.3. Theconstruction
∗
so far is functorial in Y so we can generalize to arbitrary simplicial sets as follows:
•
If Y is any simplicial set we define
•
C (Φ)(n) := colim C (Φ)(n)
Y• K•→Y•, K•
K• finite
and
C (Φ)(n) := colim C (Φ)(n)
Y• K•→Y•, K•
K• finite
as the colimit over all simplicial finite subsets of Y .
•
Following [Pir00] there is a different definition using enriched tensor products. Given
a simplicial finite set X we define
•
(cid:77)
L (n) = Com(n,X +m)[k]
X•,m k
k
withdifferentiald : Com(n,X +m) → Com(n,X +m)givenbypostcompositionwith
k k−1
d(cid:48) = (cid:80)k (−1)id where the d ∈ Com(X +m,X +m) are the maps induced by the
i=0 i i k k−1
simplicial boundary maps d : X → X . L is a covariant functor Com → Com.
i k k−1 X•,m
If X is an arbitrary simplicial set, we set
•
L = colim L .
X•,m K•→X•, K•,m
K• finite
Then we have an isomorphism
C (Φ)(m) ∼= L ⊗ Φ
X• X•,mCom
where the right hand side is the enriched tensor product as defined in Definition 5.1.
Throughout the first part of the paper we will not work with this definition.
Definition 2.4. Let Ψ be a functor from Eop to Ch and let Y be a simplicial finite set.
•
We define the higher coHochschild complex of Ψ with respect to Y as D (Ψ) : Eop → Ch
• Y•
via
(2.3) D (Ψ) : Eop −G−Y−•−(−Ψ→) Ch∆ −r−◦−C→∗ dCh −T−o−t(cid:81)→ Ch
Y•
where G (Ψ) sends a set n to the cosimplicial chain complex
Y•
G (Ψ)(n) : ∆ −Y→• FinSetop −→S FinOrdop −L−(−−−(cid:113)−n→) Comop →−i Eop −→Ψ Ch.
Y•
C∗ istheMoorefunctordefinedinDefinition1.4andr turnsacochainobjectintoachain
object with the opposite grading (i.e. sending a cochain complex Ai to a chain complex
A ).
−i
Similar to above we can define the reduced Higher coHochschild complex of Ψ to be
(2.4) D (Ψ) : Eop −G−Y−•−(−Ψ→) Ch∆ −r−◦−C→∗ dCh −T−o−t(cid:81)→ Ch.
Y•
10 ANGELAKLAMT
Again, if Y is any simplicial set we define
•
D (Ψ)(n) := lim D (Ψ)(n)
Y• (cid:16)K•→Y•,(cid:17)op K•
K• finite
and
D (Ψ)(n) := lim D (Ψ)(n)
Y• (cid:16)K•→Y•,(cid:17)op K•
K• finite
as the limit over all finite sets.
Again, we can define D (Ψ) in terms of L equivalently via
Y• Y•
DY•(Ψ)(m) ∼= homComop(LY•,m,Ψ)
where homComop(−,−) denotes the enriched hom as defined in Definition 5.1.
Remark 2.5. For a simplicial finite set the functor C (Φ) can be described more ex-
Y•
plicitly:
(cid:77)
C (Φ)(n) = Φ(Y (cid:113)n)
Y• j k l
k+l=j
with the differential of x ∈ Φ(Y (cid:113)n) given by
k
k
(cid:88)
d(x) = d (x)+(−1)|x| (−1)iΦ(d (cid:113)id )(x).
Φ i n
i=0
Here, d : Y → Y is the face map of the simplicial set.
i k k−1
The reduced functor is given as the quotient
(cid:77)
C (Φ)(n) = Φ(Y (cid:113)n) /U
Y• j k l k
k+l=j
with
(cid:88)
U = im(s (cid:113)id ).
k i n
Similarly, we have
(cid:89)
D (Ψ)(n) = Ψ(Y (cid:113)n) .
Y• j k l
l−k=j
For y ∈ D (Ψ) the differential is
Y•
k+1
(cid:88)
d(y) = (−1)i+k+1Ψ(d (cid:113)id )(y )+(−1)kd (y )
k i n k+1 Ψ k
i=0
k+1
(cid:88)
= (−1)k(d (y )− (−1)iΨ(d (cid:113)id )(y )).
Ψ k i n k−1
i=0
The reduced complex is the subcomplex
k−1
(cid:89) (cid:92)
D (Ψ)(n) = ker(Ψ(si(cid:113)n)) .
Y• j l
l−k=j i=0
Remark 2.6. Taking E = Com and Φ : Com → Ch strong symmetric monoidal (i.e.
∼
Φ(n) ⊗ Φ(m) = Φ(n + m) in a natural and symmetric way), Φ(1) is a commutative
differential graded algebra. Φ(n) is isomorphic to A⊗n and in this case our definition of
the higher Hochschild complex for a simplicial finite set agrees (up to sign twist) with the
higher Hochschild complex CH (A) defined by Pirashvili in [Pir00] (see also [GTZ10b]).
X•
Foranelementx ∈ A⊗|Xk|, theisomorphismcorrectingthesignisgivenbyx (cid:55)→ (−1)|x|kx.
For E arbitrary, a strong monoidal functor Φ : E → Ch induces a strong monoidal functor
Φ ◦ i : Com → Ch by precomposing with the inclusion of Com into E and the higher
Hochschild complex of Φ agrees with the higher Hochschild complex of Φ◦i.
