algebraic relativism: encoding the higher structure of ...sma.epfl.ch/~hessbell/crm.pdf ·...

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

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

Existencetheorems

Slogan

AlgebraicRelativism

Kathryn Hess

Existencetheorems

Slogan

AlgebraicRelativism

Kathryn Hess

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

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

Co-rings

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

Co-rings

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

Co-rings

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

A A⊗ Aµ //µoo A

A⊗ Aµ // A A⊗ I

IdA⊗η

::vvvvvvvvv∼=

I ⊗ Aη⊗IdA

ddHHHHHHHHH∼=

commute.

AlgebraicRelativism

Kathryn Hess

Co-rings

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

Co-rings

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

f⊗f // A′ ⊗ A′

Iη′

Af // A′ A

f // A′

commute.

AlgebraicRelativism

Kathryn Hess

Co-rings

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

M M ⊗ Aρoo

M ⊗ Aρ // A M ⊗ I

IdM⊗η

::uuuuuuuuu∼=

commute.

Dually, (M, λ), with λ : A⊗M → M is a left A-module ifthe analogous diagrams commute.

AlgebraicRelativism

Kathryn Hess

Co-rings

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

M M ⊗ Aρoo

M ⊗ Aρ // A M ⊗ I

IdM⊗η

::uuuuuuuuu∼=

commute.

Dually, (M, λ), with λ : A⊗M → M is a left A-module ifthe analogous diagrams commute.

AlgebraicRelativism

Kathryn Hess

Co-rings

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

g⊗IdA // M ′ ⊗ A

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

Co-rings

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

g⊗IdA // M ′ ⊗ A

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

Co-rings

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

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

Co-rings

Existencetheorems

Bimodules

A⊗M ⊗ A

λ⊗IdA

IdA⊗ρ // A⊗M

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

Co-rings

Existencetheorems

Bimodules

A⊗M ⊗ A

λ⊗IdA

IdA⊗ρ // A⊗M

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

Co-rings

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 ⊗

AlgebraicRelativism

Kathryn Hess

Co-rings

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

is a coassociative morphism of A-bimodules.

CoRingA is the category of A-co-rings and theirmorphisms.

AlgebraicRelativism

Kathryn Hess

Co-rings

Existencetheorems

Example: the canonical co-ringIf ϕ : B → A is a monoid morphism, then

R = A⊗B

is an A-co-ring, where

ψ : R → R ⊗A

is the following composite of A-bimodule maps.

A ∼= //

B ⊗B

A⊗Bϕ⊗

BA)⊗

A(A⊗

BA) A⊗

BA∼=oo

This example arose in Galois theory.

AlgebraicRelativism

Kathryn Hess

Co-rings

Existencetheorems

(A,R)Mod

Ob (A,R)Mod = Ob AMod

(A,R)Mod(M,N) = AMod(R ⊗A

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⊗

−−−→ R ⊗A

R ⊗A

−−−→ R ⊗A

M ′ ϕ′−→ M ′′.

AlgebraicRelativism

Kathryn Hess

Co-rings

Existencetheorems

(A,R)Mod

Ob (A,R)Mod = Ob AMod

(A,R)Mod(M,N) = AMod(R ⊗A

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⊗

−−−→ R ⊗A

R ⊗A

−−−→ R ⊗A

M ′ ϕ′−→ M ′′.

AlgebraicRelativism

Kathryn Hess

Co-rings

Existencetheorems

Mod(A,R)

Ob Mod(A,R) = Ob ModA

Mod(A,R)(M,N) = ModA(M ⊗A

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

−−−→ M ⊗A

R ⊗A

Rϕ⊗

−−−→ M ′ ⊗A

Rϕ′−→ M ′′.

AlgebraicRelativism

Kathryn Hess

Co-rings

Existencetheorems

Mod(A,R)

Ob Mod(A,R) = Ob ModA

Mod(A,R)(M,N) = ModA(M ⊗A

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

−−−→ M ⊗A

R ⊗A

Rϕ⊗

−−−→ M ′ ⊗A

Rϕ′−→ M ′′.

AlgebraicRelativism

Kathryn Hess

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),

AlgebraicRelativism

Kathryn Hess

Existencetheorems

A binary operation

µ : X × X → X : (x , y) 7→ x · y

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),

AlgebraicRelativism

Kathryn Hess

Existencetheorems

A binary operation

µ : X × X → X : (x , y) 7→ x · y

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),

AlgebraicRelativism

Kathryn Hess

Existencetheorems

A binary operation

µ : X × X → X : (x , y) 7→ x · y

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),

AlgebraicRelativism

Kathryn Hess

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

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

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

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

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

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

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

Existencetheorems

The graded monoidal structure

Let−− : MΣ ×MΣ −→ MΣ

)(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

Existencetheorems

The composition monoidal structure

Let− − : MΣ ×MΣ −→ MΣ

)(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

Existencetheorems

The composition monoidal structure

Let− − : MΣ ×MΣ −→ MΣ

)(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

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

for all k ≥ 0 and all ~n = (n1, ...,nk ) ∈ Nk , that areappropriately equivariant, associative and unital.

AlgebraicRelativism

Kathryn Hess

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

for all k ≥ 0 and all ~n = (n1, ...,nk ) ∈ Nk , that areappropriately equivariant, associative and unital.

AlgebraicRelativism

Kathryn Hess

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

is given by “block permutation.”

AlgebraicRelativism

Kathryn Hess

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

is given by “composition of functions”.

AlgebraicRelativism

Kathryn Hess

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

is given by “composition of functions”.

AlgebraicRelativism

Kathryn Hess

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

AlgebraicRelativism

Kathryn Hess

Existencetheorems

⇐⇒ ∃ µn : P(n)⊗ A⊗n → An≥0

⇐⇒ ∃ δn : C ⊗ P(n) → C⊗nn≥0

AlgebraicRelativism

Kathryn Hess

Existencetheorems

⇐⇒ ∃ µn : P(n)⊗ A⊗n → An≥0

⇐⇒ ∃ δn : C ⊗ P(n) → C⊗nn≥0

AlgebraicRelativism

Kathryn Hess

Existencetheorems

Example

A-algebras are monoids in M.

A-coalgebras are comonoids in M.

AlgebraicRelativism

Kathryn Hess

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

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)

ϕ⊗n

C′ ⊗ P(n)

δ′n // (C′)⊗n

AlgebraicRelativism

Kathryn Hess

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

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)

ϕ⊗n

C′ ⊗ P(n)

δ′n // (C′)⊗n

AlgebraicRelativism

Kathryn Hess

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

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

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

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

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

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

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

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 =⊕

Ci ⊗R C′j

andd ′′ = d ⊗R C′ + C ⊗R d ′.

AlgebraicRelativism

Kathryn Hess

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

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

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

The category DCSH

AlgebraicRelativism

Kathryn Hess

The category DCSH

AlgebraicRelativism

Kathryn Hess

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

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

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

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

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

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′)

ChΣmult

(T(ΩC) X,T(ΩC′)

) Lin−−→ (A,Φ(X))-Coalg(C,C′)

AlgebraicRelativism

Kathryn Hess

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

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

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

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

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

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 .

AlgebraicRelativism

Kathryn Hess

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

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

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

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

Topological consequences II

The DCSH-existence theorem has also been applied inconstructions of models of homotopy orbit spaces and ofdouble loop spaces.

algebraic relativism: encoding the higher structure of ...sma.epfl.ch/~hessbell/crm.pdf ·...

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