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AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
Existencetheorems
Algebraic Relativism:Encoding the Higher Structure of
Morphisms
Kathryn Hess
Institute of Geometry, Algebra and TopologyEcole Polytechnique Fédérale de Lausanne
CRM-University of Ottawa Distinguished Lecture23 February 2007
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
Existencetheorems
Slogan
Operadsparametrize n-ary operations, andgovern the identities that they must satisfy.
Co-rings over operadsparametrize higher, “up to homotopy” structure onhomomorphisms, andgovern the relations among the “higher homotopies"and the n-ary operations.
Co-rings over operads should therefore be considered asrelative operads.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
Existencetheorems
Slogan
Operadsparametrize n-ary operations, andgovern the identities that they must satisfy.
Co-rings over operadsparametrize higher, “up to homotopy” structure onhomomorphisms, andgovern the relations among the “higher homotopies"and the n-ary operations.
Co-rings over operads should therefore be considered asrelative operads.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
Existencetheorems
Slogan
Operadsparametrize n-ary operations, andgovern the identities that they must satisfy.
Co-rings over operadsparametrize higher, “up to homotopy” structure onhomomorphisms, andgovern the relations among the “higher homotopies"and the n-ary operations.
Co-rings over operads should therefore be considered asrelative operads.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
Existencetheorems
Outline
1 Algebra in monoidal categoriesMonoids and modulesCo-rings
2 Operads and “relative operads”Operads(Co)algebras and their morphismsCo-rings as “relative operads”
3 Existence theoremsSetting the stageDiffraction and cobar dualityExistence of parametrized morphisms
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Monoidal categories: definition
Let M be a category.
A monoidal structure on M is given bya bifunctor −⊗− : M×M → M,a distinguished object I
such thatA⊗ I ∼= A ∼= I ⊗ A
and(A⊗ B)⊗ C ∼= A⊗ (B ⊗ C)
naturally.
(M,⊗, I) is symmetric if A⊗ B ∼= B ⊗ A naturally.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Monoidal categories: definition
Let M be a category.
A monoidal structure on M is given bya bifunctor −⊗− : M×M → M,a distinguished object I
such thatA⊗ I ∼= A ∼= I ⊗ A
and(A⊗ B)⊗ C ∼= A⊗ (B ⊗ C)
naturally.
(M,⊗, I) is symmetric if A⊗ B ∼= B ⊗ A naturally.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Monoidal categories: examples
Set or Top, together with cartesian product ×
Ab, together with ⊗Z
Vectk, together with tensor product ⊗k
Ch, together with graded tensor product ⊗k
All of these examples are symmetric.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Monoids: definition
Let (M,⊗, I) be a monoidal category.
A monoid in M is an object A in C, together with twomorphisms in C:
µ : A⊗ A → A and η : I → A
such that
A⊗ A⊗ A
µ⊗IdA
IdA⊗µ // A⊗ A
µ
A A⊗ Aµ //µoo A
A⊗ Aµ // A A⊗ I
IdA⊗η
::vvvvvvvvv∼=
OO
I ⊗ Aη⊗IdA
ddHHHHHHHHH∼=
OO
commute.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Monoids: examples
In (Set,×): monoids (of sets) (e.g., (N,+,0) )
In (Top,×): topological monoids (e.g., Lie groups,H-spaces)
In (Ab,⊗Z): rings!
In (Vectk,⊗k): k-algebras (e.g., group rings k[G])
In (Ch,⊗k): differential graded algebras (e.g., thecochains on topological space)
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Morphisms of monoids
Let (A, µ, η) and (A′, µ′, η′) be monoids in a monoidalcategory (M,⊗, I).
A morphism of monoids from (A, µ, η) to (A′, µ′, η′) is amorphism f : A → A′ such that the diagrams
A⊗ A
µ
f⊗f // A′ ⊗ A′
µ′
Iη′
===
====
=η
Af // A′ A
f // A′
commute.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Modules
Let (A, µ, η) be a monoid in (M,⊗, I).
A right A-module is an object M in M together with amorphism in M
ρ : M ⊗ A → M
such that
M ⊗ A⊗ A
ρ⊗IdA
IdA⊗µ // M ⊗ A
ρ
M M ⊗ Aρoo
M ⊗ Aρ // A M ⊗ I
IdM⊗η
::uuuuuuuuu∼=
OO
commute.
Dually, (M, λ), with λ : A⊗M → M is a left A-module ifthe analogous diagrams commute.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Modules
Let (A, µ, η) be a monoid in (M,⊗, I).
A right A-module is an object M in M together with amorphism in M
ρ : M ⊗ A → M
such that
M ⊗ A⊗ A
ρ⊗IdA
IdA⊗µ // M ⊗ A
ρ
M M ⊗ Aρoo
M ⊗ Aρ // A M ⊗ I
IdM⊗η
::uuuuuuuuu∼=
OO
commute.
Dually, (M, λ), with λ : A⊗M → M is a left A-module ifthe analogous diagrams commute.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Modules: homomorphisms
Let (M, ρ) and (M ′, ρ′) be right A-modules.
A morphism of right A-modules from (M, ρ) to (M ′, ρ′) is amorphism g : M → M ′ in M such that
M ⊗ A
ρ
g⊗IdA // M ′ ⊗ A
ρ′
M
g // M ′
commutes. The category of right A-modules and theirmorphisms is denoted ModA.
There is an analogous definition of AMod, the category ofleft A-modules and their morphisms.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Modules: homomorphisms
Let (M, ρ) and (M ′, ρ′) be right A-modules.
A morphism of right A-modules from (M, ρ) to (M ′, ρ′) is amorphism g : M → M ′ in M such that
M ⊗ A
ρ
g⊗IdA // M ′ ⊗ A
ρ′
M
g // M ′
commutes. The category of right A-modules and theirmorphisms is denoted ModA.
There is an analogous definition of AMod, the category ofleft A-modules and their morphisms.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Bimodules
Suppose that λ : A⊗M → M and ρ : M ⊗ A → M are leftand right actions of A.
(M, λ, ρ) is an A-bimodule if
A⊗M ⊗ A
λ⊗IdA
IdA⊗ρ // A⊗M
λ
M ⊗ Aρ // M
commutes.
A morphism g : M → M ′ that is a morphism of both leftA-modules and right A-modules is a morphism ofA-bimodules. The category of A-bimodules and theirmorphisms is denoted AModA.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Bimodules
Suppose that λ : A⊗M → M and ρ : M ⊗ A → M are leftand right actions of A.
(M, λ, ρ) is an A-bimodule if
A⊗M ⊗ A
λ⊗IdA
IdA⊗ρ // A⊗M
λ
M ⊗ Aρ // M
commutes.A morphism g : M → M ′ that is a morphism of both leftA-modules and right A-modules is a morphism ofA-bimodules.
The category of A-bimodules and theirmorphisms is denoted AModA.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Bimodules
Suppose that λ : A⊗M → M and ρ : M ⊗ A → M are leftand right actions of A.
(M, λ, ρ) is an A-bimodule if
A⊗M ⊗ A
λ⊗IdA
IdA⊗ρ // A⊗M
λ
M ⊗ Aρ // M
commutes.A morphism g : M → M ′ that is a morphism of both leftA-modules and right A-modules is a morphism ofA-bimodules. The category of A-bimodules and theirmorphisms is denoted AModA.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Monoidal products of bimodules
Let (M,⊗, I) be a bicomplete, closed monoidal category.Let (A, µ, η) be a monoid in M.
RemarkThe category of A-bimodules is also monoidal, withmonoidal product ⊗
Agiven by the coequalizer
M ⊗ A⊗ Nρ⊗N⇒
M⊗λM ⊗ N −→ M ⊗
AN.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Definition of co-rings
An A-co-ring is a comonoid (R, ψ) in the category ofA-bimodules, i.e.,
ψ : R −→ R ⊗A
R
is a coassociative morphism of A-bimodules.
CoRingA is the category of A-co-rings and theirmorphisms.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Example: the canonical co-ringIf ϕ : B → A is a monoid morphism, then
R = A⊗B
A
is an A-co-ring, where
ψ : R → R ⊗A
R
is the following composite of A-bimodule maps.
A⊗B
A ∼= //
ψ
A⊗B
B ⊗B
A
A⊗Bϕ⊗
BA
(A⊗
BA)⊗
A(A⊗
BA) A⊗
BA⊗
BA∼=oo
This example arose in Galois theory.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
(A,R)Mod
Ob (A,R)Mod = Ob AMod
(A,R)Mod(M,N) = AMod(R ⊗A
M,N)
Given
ϕ ∈ (A,R)Mod(M,M ′) and ϕ′ ∈ (A,R)Mod(M ′,M ′′)
their composite in (A,R)Mod,
ϕ′ϕ ∈ (A,R)Mod(M,M ′′),
is given by
R ⊗A
Mψ⊗
AM
−−−→ R ⊗A
R ⊗A
MR⊗
Aϕ
−−−→ R ⊗A
M ′ ϕ′−→ M ′′.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
(A,R)Mod
Ob (A,R)Mod = Ob AMod
(A,R)Mod(M,N) = AMod(R ⊗A
M,N)
Given
ϕ ∈ (A,R)Mod(M,M ′) and ϕ′ ∈ (A,R)Mod(M ′,M ′′)
their composite in (A,R)Mod,
ϕ′ϕ ∈ (A,R)Mod(M,M ′′),
is given by
R ⊗A
Mψ⊗
AM
−−−→ R ⊗A
R ⊗A
MR⊗
Aϕ
−−−→ R ⊗A
M ′ ϕ′−→ M ′′.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Mod(A,R)
Ob Mod(A,R) = Ob ModA
Mod(A,R)(M,N) = ModA(M ⊗A
R,N)
Given
ϕ ∈ Mod(A,R)(M,M ′) and ϕ′ ∈ Mod(A,R)(M ′,M ′′)
their composite in Mod(A,R),
ϕ′ϕ ∈ Mod(A,R)(M,M ′),
is given by
M ⊗A
RM⊗
Aψ
−−−→ M ⊗A
R ⊗A
Rϕ⊗
AR
−−−→ M ′ ⊗A
Rϕ′−→ M ′′.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategoriesMonoids and modules
Co-rings
Operads and“relativeoperads”
Existencetheorems
Mod(A,R)
Ob Mod(A,R) = Ob ModA
Mod(A,R)(M,N) = ModA(M ⊗A
R,N)
Given
ϕ ∈ Mod(A,R)(M,M ′) and ϕ′ ∈ Mod(A,R)(M ′,M ′′)
their composite in Mod(A,R),
ϕ′ϕ ∈ Mod(A,R)(M,M ′),
is given by
M ⊗A
RM⊗
Aψ
−−−→ M ⊗A
R ⊗A
Rϕ⊗
AR
−−−→ M ′ ⊗A
Rϕ′−→ M ′′.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Generating n-ary operations
A binary operation
µ : X × X → X : (x , y) 7→ x · y
on a set X gives rise to numerous n-ary operations, e.g.,
X × X → X : (x , y) 7→ y · x ,
X × X × X → X : (x , y , z) 7→ (x · y) · z,
X × X × X → X : (x , y , z) 7→ x · (y · z),
X × X × X × X → X : (w , x , y , z) 7→ w · ((z · x) · y),
X×X×X×X×X → X : (v ,w , x , y , z) 7→ ((x ·v)·z)·(y ·w),
Etc.!
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Generating n-ary operations
A binary operation
µ : X × X → X : (x , y) 7→ x · y
on a set X gives rise to numerous n-ary operations, e.g.,
X × X → X : (x , y) 7→ y · x ,
X × X × X → X : (x , y , z) 7→ (x · y) · z,
X × X × X → X : (x , y , z) 7→ x · (y · z),
X × X × X × X → X : (w , x , y , z) 7→ w · ((z · x) · y),
X×X×X×X×X → X : (v ,w , x , y , z) 7→ ((x ·v)·z)·(y ·w),
Etc.!
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Generating n-ary operations
A binary operation
µ : X × X → X : (x , y) 7→ x · y
on a set X gives rise to numerous n-ary operations, e.g.,
X × X → X : (x , y) 7→ y · x ,
X × X × X → X : (x , y , z) 7→ (x · y) · z,
X × X × X → X : (x , y , z) 7→ x · (y · z),
X × X × X × X → X : (w , x , y , z) 7→ w · ((z · x) · y),
X×X×X×X×X → X : (v ,w , x , y , z) 7→ ((x ·v)·z)·(y ·w),
Etc.!
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Generating n-ary operations
A binary operation
µ : X × X → X : (x , y) 7→ x · y
on a set X gives rise to numerous n-ary operations, e.g.,
X × X → X : (x , y) 7→ y · x ,
X × X × X → X : (x , y , z) 7→ (x · y) · z,
X × X × X → X : (x , y , z) 7→ x · (y · z),
X × X × X × X → X : (w , x , y , z) 7→ w · ((z · x) · y),
X×X×X×X×X → X : (v ,w , x , y , z) 7→ ((x ·v)·z)·(y ·w),
Etc.!
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Parametrizing n-ary operations
Need to organize and systematize all this information,i.e., to
enumerate in a reasonable way all possible n-aryoperations;encode relations among various n-ary operations.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
A helpful descriptive tool
Planar trees labeled with permutations are useful forencoding n-ary operations.
Operads formalize this parametrization via trees.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
A helpful descriptive tool
Planar trees labeled with permutations are useful forencoding n-ary operations.
Operads formalize this parametrization via trees.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Symmetric sequences
Let (M,⊗, I) be a bicomplete, closed, symmetric monoidalcategory.
MΣ is the category of symmetric sequences in M.
X ∈ Ob MΣ =⇒ X = X(n) ∈ Ob M | n ≥ 0,
where X(n) admits a right action of the symmetric groupΣn, for all n.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The level monoidal structure
Let−⊗− : MΣ ×MΣ −→ MΣ
be the functor given by(X⊗ Y
)(n) = X(n)⊗ Y(n), with
diagonal Σn-action.
Proposition(MΣ,⊗,C) is a closed, symmetric monoidal category,where C(n) = I with trivial Σn-action, for all n.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The level monoidal structure
Let−⊗− : MΣ ×MΣ −→ MΣ
be the functor given by(X⊗ Y
)(n) = X(n)⊗ Y(n), with
diagonal Σn-action.
Proposition(MΣ,⊗,C) is a closed, symmetric monoidal category,where C(n) = I with trivial Σn-action, for all n.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The graded monoidal structure
Let−− : MΣ ×MΣ −→ MΣ
be the functor given by(X Y
)(n) =
∐i+j=n
(X(i)⊗ Y(j)
)⊗
Σi×Σj
I[Σn],
where I[Σn] is the free Σn-object on I.
Proposition(MΣ,,U) is a closed, symmetric monoidal category,where U(0) = I and U(n) = O (the 0-object), for all n > 0.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The graded monoidal structure
Let−− : MΣ ×MΣ −→ MΣ
be the functor given by(X Y
)(n) =
∐i+j=n
(X(i)⊗ Y(j)
)⊗
Σi×Σj
I[Σn],
where I[Σn] is the free Σn-object on I.
Proposition(MΣ,,U) is a closed, symmetric monoidal category,where U(0) = I and U(n) = O (the 0-object), for all n > 0.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The composition monoidal structure
Let− − : MΣ ×MΣ −→ MΣ
be the functor given by(X Y
)(n) =
∐m≥0
X(m) ⊗Σm
(Ym)(n).
Proposition(MΣ, , J) is a right-closed, monoidal category, whereJ(1) = I and J(n) = O (the 0-object), for all n 6= 1.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The composition monoidal structure
Let− − : MΣ ×MΣ −→ MΣ
be the functor given by(X Y
)(n) =
∐m≥0
X(m) ⊗Σm
(Ym)(n).
Proposition(MΣ, , J) is a right-closed, monoidal category, whereJ(1) = I and J(n) = O (the 0-object), for all n 6= 1.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Operads as monoids
An operad in M is a monoid (P, γ, η) in (MΣ, , J).
More explicitly, there is a family of morphisms in M
γ~n : P(k)⊗(
P(n1)⊗ · · · ⊗ P(nk )
)→ P
( k∑i=1
ni),
for all k ≥ 0 and all ~n = (n1, ...,nk ) ∈ Nk , that areappropriately equivariant, associative and unital.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Operads as monoids
An operad in M is a monoid (P, γ, η) in (MΣ, , J).
More explicitly, there is a family of morphisms in M
γ~n : P(k)⊗(
P(n1)⊗ · · · ⊗ P(nk )
)→ P
( k∑i=1
ni),
for all k ≥ 0 and all ~n = (n1, ...,nk ) ∈ Nk , that areappropriately equivariant, associative and unital.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The associative operad A
For all n ∈ N,A(n) = I[Σn],
on which Σn acts by right multiplication, and
γ~n : A(k)⊗(
A(n1)⊗ · · · ⊗A(nk )
)→ A
( k∑i=1
ni)
is given by “block permutation.”
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The endomorphism operad EX
Let X be an object of M.
Let hom(Y ,−) denote the right adjoint to −⊗ Y .
For all n ∈ N,EX (n) = hom(X⊗n,X )
on which Σn acts by permuting inputs, and
γ~n : EX (k)⊗(
EX (n1)⊗ · · · ⊗ EX (nk )
)→ EX
( k∑i=1
ni)
is given by “composition of functions”.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
The coendomorphism operad EX
Let X be an object of M.
Let hom(Y ,−) denote the right adjoint to −⊗ Y .
For all n ∈ N,EX (n) = hom(X ,X⊗n)
on which Σn acts by permuting outputs, and
γ~n : EX (k)⊗(
EX (n1)⊗ · · · ⊗ EX (nk )
)→ EX
( k∑i=1
ni)
is given by “composition of functions”.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Algebras and coalgebras over operads
Let (P, γ, η) be an operad in M.A P-algebra consists of an object A of M, togetherwith a morphism of operads µ : P → EA.
⇐⇒ ∃ µn : P(n)⊗ A⊗n → An≥0
–appropriately equivariant, associative and unital.
A P-coalgebra consists of an object C of M, togetherwith a morphism of operads δ : P → EC .
⇐⇒ ∃ δn : C ⊗ P(n) → C⊗nn≥0
–appropriately equivariant, associative and unital.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Algebras and coalgebras over operads
Let (P, γ, η) be an operad in M.A P-algebra consists of an object A of M, togetherwith a morphism of operads µ : P → EA.
⇐⇒ ∃ µn : P(n)⊗ A⊗n → An≥0
–appropriately equivariant, associative and unital.
A P-coalgebra consists of an object C of M, togetherwith a morphism of operads δ : P → EC .
⇐⇒ ∃ δn : C ⊗ P(n) → C⊗nn≥0
–appropriately equivariant, associative and unital.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Algebras and coalgebras over operads
Let (P, γ, η) be an operad in M.A P-algebra consists of an object A of M, togetherwith a morphism of operads µ : P → EA.
⇐⇒ ∃ µn : P(n)⊗ A⊗n → An≥0
–appropriately equivariant, associative and unital.
A P-coalgebra consists of an object C of M, togetherwith a morphism of operads δ : P → EC .
⇐⇒ ∃ δn : C ⊗ P(n) → C⊗nn≥0
–appropriately equivariant, associative and unital.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Example
A-algebras are monoids in M.
A-coalgebras are comonoids in M.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Morphisms of P-(co)algebrasA P-algebra morphism from (A, µ) to (A′, µ′) is amorphism ϕ : A → A′ in M such that the followingdiagram commutes for all n.
P(n)⊗ A⊗n µn //
P(n)⊗ϕ⊗n
A
ϕ
P(n)⊗ (A′)⊗n µ′n // A′
A P-coalgebra morphism from (C, δ) to (C′, δ′) is amorphism ϕ : C → C′ in M such that the followingdiagram commutes for all n.
C ⊗ P(n)δn //
ϕ⊗P(n)
C⊗n
ϕ⊗n
C′ ⊗ P(n)
δ′n // (C′)⊗n
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Morphisms of P-(co)algebrasA P-algebra morphism from (A, µ) to (A′, µ′) is amorphism ϕ : A → A′ in M such that the followingdiagram commutes for all n.
P(n)⊗ A⊗n µn //
P(n)⊗ϕ⊗n
A
ϕ
P(n)⊗ (A′)⊗n µ′n // A′
A P-coalgebra morphism from (C, δ) to (C′, δ′) is amorphism ϕ : C → C′ in M such that the followingdiagram commutes for all n.
C ⊗ P(n)δn //
ϕ⊗P(n)
C⊗n
ϕ⊗n
C′ ⊗ P(n)
δ′n // (C′)⊗n
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
P-algebras as left P-modules
Let z : M → MΣ be the functor defined on objects byz(X )(0) = X and z(X )(n) = O for all n > 0.
Proposition (Kapranov-Manin,?)The functor z restricts and corestricts to a full and faithfulfunctor
z : P-Alg → PMod.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
P-coalgebras as right P-modules
Let T : M → MΣ be the functor defined on objects byT(X )(n) = X⊗n, where Σn acts by permuting factors.
Proposition (H.-Parent-Scott)The functor T restricts and corestricts to a full and faithfulfunctor
T : P-Coalg → AModP.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Co-rings over operads
Given a co-ring (R, ψ) over an operad P, i.e.,R is a P-bimodule,
ψ : R → R P
R is a coassociative, counital morphism
of P-bimodules,
get “fattened” categories of modules
(P,R)Mod and Mod(P,R)
giving rise to “fattened” categories of P-algebras and ofP-coalgebras
(P,R)-Alg and (P,R)-Coalg.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
Co-rings over operads
Given a co-ring (R, ψ) over an operad P, i.e.,R is a P-bimodule,
ψ : R → R P
R is a coassociative, counital morphism
of P-bimodules,get “fattened” categories of modules
(P,R)Mod and Mod(P,R)
giving rise to “fattened” categories of P-algebras and ofP-coalgebras
(P,R)-Alg and (P,R)-Coalg.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
How co-rings parametrize morphisms I
Let (A, µ) and (A′, µ′) be P-algebras.
A morphism in (P,R)-Alg from (A, µ) to (A′, µ′) is amorphism of left P-modules
R P
z(A) → z(A′),
i.e., a collection of morphisms in MR(n)⊗ A⊗n → A′ | n ≥ 1
that are “appropriately compatible” with the P-actions onR, A and A′.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”Operads
(Co)algebras and theirmorphisms
Co-rings as “relativeoperads”
Existencetheorems
How co-rings parametrize morphisms II
Let (C, δ) and (C′, δ′) be P-algebras.
A morphism in (P,R)-Coalg from (C, δ) to (C′, δ′) is amorphism of right P-modules
T(C) P
R → T(C′),
i.e., a collection of morphisms in MC ⊗ R(n) → (C′)⊗n | n ≥ 1
that are “appropriately compatible” with the P-actions onR, C and C′.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Chain complexes
Ch is the category of chain complexes over acommutative ring R that are bounded below.Ch is closed, symmetric monoidal with respect to thetensor product:
(C,d)⊗ (C′,d ′) := (C′′,d ′′)
whereC′′
n =⊕
i+j=n
Ci ⊗R C′j
andd ′′ = d ⊗R C′ + C ⊗R d ′.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The cobar constructionLet C denote the category of chain coalgebras, i.e., ofcomonoids in Ch. Let A denote the category of chainalgebras, i.e., of monoids in Ch.
The cobar construction is a functor
Ω : C −→ A : C 7−→ ΩC =(T (s−1C),dΩ
),
whereT is the free tensor algebra functor on gradedR-modules,(s−1C)n = Cn+1 for all n, anddΩ is the derivation specified by
dΩs−1 = −s−1d + (s−1 ⊗ s−1)∆,
where d and ∆ are the differential and coproduct onC.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The cobar constructionLet C denote the category of chain coalgebras, i.e., ofcomonoids in Ch. Let A denote the category of chainalgebras, i.e., of monoids in Ch.
The cobar construction is a functor
Ω : C −→ A : C 7−→ ΩC =(T (s−1C),dΩ
),
whereT is the free tensor algebra functor on gradedR-modules,(s−1C)n = Cn+1 for all n, anddΩ is the derivation specified by
dΩs−1 = −s−1d + (s−1 ⊗ s−1)∆,
where d and ∆ are the differential and coproduct onC.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The category DCSH
Ob DCSH = Ob C.DCSH(C,C′) := A(ΩC,ΩC′).
Morphisms in DCSH are called
strongly homotopy-comultiplicative maps.
ϕ ∈ DCSH(C,C′) ⇐⇒ ϕk : C → (C′)⊗kk≥1 + relations!
The chain map ϕ1 : C → C′ is called a DCSH-map.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The category DCSH
Ob DCSH = Ob C.DCSH(C,C′) := A(ΩC,ΩC′).
Morphisms in DCSH are called
strongly homotopy-comultiplicative maps.
ϕ ∈ DCSH(C,C′) ⇐⇒ ϕk : C → (C′)⊗kk≥1 + relations!
The chain map ϕ1 : C → C′ is called a DCSH-map.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The category DCSH
Ob DCSH = Ob C.DCSH(C,C′) := A(ΩC,ΩC′).
Morphisms in DCSH are called
strongly homotopy-comultiplicative maps.
ϕ ∈ DCSH(C,C′) ⇐⇒ ϕk : C → (C′)⊗kk≥1 + relations!
The chain map ϕ1 : C → C′ is called a DCSH-map.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Topological significance
Let K be a simplicial set.
Theorem (Gugenheim-Munkholm)The natural coproduct ∆K : C∗K → C∗K ⊗ C∗K isnaturally a DCSH-map
Thus, there exists a chain algebra map
ϕK : ΩC∗K → Ω(C∗K ⊗ C∗K
)such that (ϕK )1 = ∆K .
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The diffracting functorComon⊗ is the category of comonoids in (ChΣ,⊗,C).
Theorem (H.-P.-S.)There is a functor
Φ : Comon⊗ → CoRingA
such that the underlying A-bimodule of Φ(X) is free, forall objects X in Comon⊗.
Corollary
There are functors from Comonop⊗ to the category of
small categories, given on objects by:
X 7−→ (A,Φ(X)
)Mod and X 7−→ Mod(A,Φ(X)
).
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The diffracting functorComon⊗ is the category of comonoids in (ChΣ,⊗,C).
Theorem (H.-P.-S.)There is a functor
Φ : Comon⊗ → CoRingA
such that the underlying A-bimodule of Φ(X) is free, forall objects X in Comon⊗.
Corollary
There are functors from Comonop⊗ to the category of
small categories, given on objects by:
X 7−→ (A,Φ(X)
)Mod and X 7−→ Mod(A,Φ(X)
).
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Induction
Let C and C′ be chain coalgebras. Let X be an object inComon⊗.
A (transposed tensor) morphism of right A-modules
θ : T(C) A
Φ(X) −→ T(C′)
naturally induces a (multiplicative) morphism ofsymmetric sequences
Ind(θ) : T(ΩC) X −→ T(ΩC′).
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Linearization
Let C and C′ be chain coalgebras. Let X be an object inComon⊗.
A (multiplicative) morphism of symmetric sequences
θ : T(ΩC) X −→ T(ΩC′)
can be naturally linearized to a (transposed tensor)morphism of right A-modules
Lin(θ) : T(C) A
Φ(X) −→ T(C′).
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The Cobar Duality Theorem
Theorem (H.-P.-S.)Induction and linearization define mutually inverse,natural bijections
(A,Φ(X))-Coalg(C,C′)Ind−−→ ChΣ
mult(T(ΩC) X,T(ΩC′)
)and
ChΣmult
(T(ΩC) X,T(ΩC′)
) Lin−−→ (A,Φ(X))-Coalg(C,C′)
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The Alexander-Whitney co-ring
The Alexander-Whitney co-ring is F = Φ(J).
The level comultiplication J → J⊗ J is composed of theisomorphisms I
∼=−→ I ⊗ I and O∼=−→ O ⊗O.
Theorem (H.-P.-S.)F admits a counit ε : F → A inducing a homologyisomorphism in each level. (In fact, F is exactly thetwo-sided Koszul resolution of A.)F admits a coassociative, level coproduct, i.e., F isan object in Comon⊗.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The Alexander-Whitney co-ring
The Alexander-Whitney co-ring is F = Φ(J).
The level comultiplication J → J⊗ J is composed of theisomorphisms I
∼=−→ I ⊗ I and O∼=−→ O ⊗O.
Theorem (H.-P.-S.)F admits a counit ε : F → A inducing a homologyisomorphism in each level. (In fact, F is exactly thetwo-sided Koszul resolution of A.)F admits a coassociative, level coproduct, i.e., F isan object in Comon⊗.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
An operadic description of DCSH
Theorem (H.-P.-S.)DCSH is isomorphic to (A,F)-Coalg, where
Ob (A,F)-Coalg = Ob C, and
(A,F)-Coalg(C,C′) = Mod(A,F)
(T(C),T(C′)
).
Remark(A,F)-Coalg inherits a monoidal structure from the levelcomonoidal structure of F.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Acyclic models
Let D be a category, and let M be a set of objects in D.Let X : D → Ch be a functor.
X is free with respect to M if there is a setxm ∈ X (m) | m ∈ M such that
X (f )(xm) | f ∈ D(m,d),m ∈ M
is a Z-basis of X (d) for all objects d in D.X is acyclic with respect to M if X (m) is acyclic for allm ∈ M.
More generally, if C is a category with a forgetful functorU to Ch and X : D → C is a functor, we say that X is free,respectively acyclic, with respect to M if UX is.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Acyclic models
Let D be a category, and let M be a set of objects in D.Let X : D → Ch be a functor.
X is free with respect to M if there is a setxm ∈ X (m) | m ∈ M such that
X (f )(xm) | f ∈ D(m,d),m ∈ M
is a Z-basis of X (d) for all objects d in D.X is acyclic with respect to M if X (m) is acyclic for allm ∈ M.
More generally, if C is a category with a forgetful functorU to Ch and X : D → C is a functor, we say that X is free,respectively acyclic, with respect to M if UX is.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
The general existence theorem
Theorem (H.-P.-S.)Let D be a small category, and let F ,G : D → C befunctors. Let U : C → Ch be the forgetful functor.
If there is a set of models in D with respect to which F isfree and G is acyclic, then for all level comonoids X underJ and for all natural transformations τ : UF → UG, thereexists a multiplicative natural transformation
θX : T(ΩF ) X → T(ΩG)
extending s−1τ .
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Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Proof of the existence theorem
Proof.Since Φ(X) admits a particularly nice differential filtration,acyclic models methods suffice to prove the existence ofa (transposed tensor) natural transformation
τX : T(F ) A
Φ(X) → T(G),
extending τ .
We can then apply the Cobar Duality Theorem and setθX = Ind(τX).
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Existence of DCSH maps
Theorem (H.-P.-S.)Let D be a small category, and let F ,G : D → C befunctors. Let U : C → Ch be the forgetful functor.
If there is a set of models in D with respect to which F isfree and G is acyclic, then for all natural transformationsτ : UF → UG, there exists a natural transformation offunctors into A
θX : ΩF → ΩG
extending s−1τ
Thus, for all d ∈ Ob D, the chain map
τd : UF (d) → UG(d)
is naturally a DCSH-map.
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Topological consequences I
Theorem (H.-Parent-Scott-Tonks)There is a natural, coassociative coproduct ψK on ΩC∗K ,given by the composite
ΩC∗KϕK−−→ Ω
(C∗K ⊗ C∗K
) q−→ ΩC∗K ⊗ ΩC∗K ,
where q is Milgram’s natural transformation.
Furthermore, Szczarba’s natural equivalence of chainalgebras
Sz : ΩC∗K'−→ C∗GK
is a DCSH-map with respect to ψK and to the naturalcoproduct ∆GK on C∗GK .
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Topological consequences I
Theorem (H.-Parent-Scott-Tonks)There is a natural, coassociative coproduct ψK on ΩC∗K ,given by the composite
ΩC∗KϕK−−→ Ω
(C∗K ⊗ C∗K
) q−→ ΩC∗K ⊗ ΩC∗K ,
where q is Milgram’s natural transformation.
Furthermore, Szczarba’s natural equivalence of chainalgebras
Sz : ΩC∗K'−→ C∗GK
is a DCSH-map with respect to ψK and to the naturalcoproduct ∆GK on C∗GK .
AlgebraicRelativism
Kathryn Hess
Algebra inmonoidalcategories
Operads and“relativeoperads”
ExistencetheoremsSetting the stage
Diffraction and cobarduality
Existence of parametrizedmorphisms
Topological consequences II
The DCSH-existence theorem has also been applied inconstructions of models of homotopy orbit spaces and ofdouble loop spaces.