From “combinatorial” monoids to bialgebras and Hopf algebras, functorially

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From “combinatorial” monoids to bialgebras and Hopf algebras, functorially

Laurent Poinsot

LIPN & CReA

University Paris XIII & École de l’Air France

June, 20th 2014

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“Combinatorial” monoids

• Finite decomposition monoid: For eachx ∈M, there are only finitely many y,z ∈M such that x =y ∗z.

• Filtered monoid: A monoid together with a decreasing filtration . . .⊆M₂ ⊆M₁ ⊆M₀⊆M such that x_m∗x_n∈M_m+n and1∈M₀.

• Locally finite monoid: For eachx ∈M, there are onlyfinitely many x₁,· · · ,x_n ∈M\ {1}such that x =x₁∗ · · · ∗x_n.

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“Combinatorial” monoids

• Filtered monoid: A monoid together with a decreasing filtration . . .⊆M₂ ⊆M₁ ⊆M₀ ⊆M such that x_m∗x_n∈M_m+n and1∈M₀.

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“Combinatorial” monoids

• Filtered monoid: A monoid together with a decreasing filtration . . .⊆M₂ ⊆M₁ ⊆M₀ ⊆M such that x_m∗x_n∈M_m+n and1∈M₀.

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Motivations

Large algebra

The class of finite decomposition monoids is the larger class for which convolution of functions is possible.

Let R be a commutative ring with a unit. LetM be a finite decomposition monoid. Then one can define theR-coalgebra R^(M) (free module with basis M)

∆(x) = X

x=y∗z

y ⊗z and

(x) =1 .

It follows that one can consider its dual R-algebraR[[M]], called the large algebra of M, of all functions fromM toR. Its multiplication is given by convolution

(f ∗g)(x) = X

x=y∗z

f(y)g(z).

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Motivations

Large algebra

∆(x) = X

x=y∗z

y ⊗z and

(x) =1 .

(f ∗g)(x) = X

x=y∗z

f(y)g(z).

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Motivations

Large algebra

∆(x) = X

x=y∗z

y ⊗z and

(x) =1 .

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Motivations

Möbius inversion formula

When M is a locally finite monoid, thenR[[M]]admits a structure of a (complete) filtered algebra.

That makes it possible to consider a star operation. Given f ∈R[[M]] such that f(1) =0, then one definesf^?=P

n≥0fⁿ .

It follows that {f ∈R[[M]] : f(1) =1}is asubgroup of invertible elements of R[[M]]. The inverseof f is given by (f −δ₁)^?.

Möbius inversion formula: letζ =P

x∈Mx (called thezêta function of M), and let µ=ζ⁻¹ (called the Möbius functionof M). Then for all

f,g ∈R[[M]],

g(x) = X

x=y∗z

f(y)⇔f(x) = X

x=y∗z

g(y)µ(z).

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Motivations

n≥0fⁿ .

It follows that {f ∈R[[M]] : f(1) =1}is asubgroup of invertible elements of R[[M]]. The inverseof f is given by (f −δ₁)^?.

f,g ∈R[[M]],

g(x) = X

x=y∗z

f(y)⇔f(x) = X

x=y∗z

g(y)µ(z).

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Motivations

n≥0fⁿ .

It follows that {f ∈R[[M]] : f(1) =1}is asubgroup of invertible elements of R[[M]]. The inverseof f is given by(f −δ₁)^?.

f,g ∈R[[M]],

g(x) = X

x=y∗z

f(y)⇔f(x) = X

x=y∗z

g(y)µ(z).

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Motivations

n≥0fⁿ .

x∈Mx (called thezêta function of M), and let µ=ζ⁻¹ (called the Möbius functionofM).

Then for all f,g ∈R[[M]],

g(x) = X

x=y∗z

f(y)⇔f(x) = X

x=y∗z

g(y)µ(z).

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Motivations

n≥0fⁿ .

x∈Mx (called thezêta function of M), and let µ=ζ⁻¹ (called the Möbius functionofM). Then for all

f,g ∈R[[M]],

g(x) = X

x=y∗z

f(y)⇔f(x) = X

x=y∗z

g(y)µ(z).

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Purpose of the talk

• Give a category-theoretic interpretation of these combinatorial monoids as monoid objects in a monoidal category.

• Explain some known and new results using monoidal functors.

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Purpose of the talk

• Give a category-theoretic interpretation of these combinatorial monoids as monoid objects in a monoidal category.

• Explain some known and new results using monoidal functors.

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Finite decomposition monoid

Let M be a monoid. It is said to be afinite decomposition monoidif its product ∗has finite fibers.

In details this means that for each x ∈M, there are onlyfinitely many y,z ∈M such that x =y ∗z.

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Finite decomposition monoid

Let M be a monoid. It is said to be afinite decomposition monoidif its product ∗has finite fibers.

In details this means that for each x ∈M, there are onlyfinitely many y,z ∈M such that x =y ∗z.

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Category-theoretic interpretation

Let us consider the category FinFibSet of all sets with finite-fiber maps. It admits a structure of a symmetric monoidal category inherited from the set-theoretic product.

The category of monoid objects in FinFibSetin then the category of finite decomposition monoids (homomorphisms of monoids with finite fibers).

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Category-theoretic interpretation

Let us consider the category FinFibSet of all sets with finite-fiber maps. It admits a structure of a symmetric monoidal category inherited from the set-theoretic product.

The category of monoid objects in FinFibSetin then the category of finite decomposition monoids (homomorphisms of monoids with finite fibers).

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Large algebra

A R-module is said to be a topologically free R-module whenever it is isomorphic to a module of the form R^X for some setX.

Each such module admits a linear topologywhose basis of open neighborhoods of zero is given byR^(X^\A) for finite subsetsA⊆X. Clearlylim←−A∈P_fin(X)R^X/R^(X^\A)∼=lim←−A∈P_fin(X)R^A ∼=R^X, hence R^X is complete in the inverse limit topology (where allR^A are discrete), this topology is equivalent to the product topology(with R discrete). Let us denote by _RTopFreeModthe category of all topologically free modules with continuouslinear maps.

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Large algebra

Each such module admits a linear topologywhose basis of open neighborhoods of zero is given byR^(X^\A) for finite subsetsA⊆X.

Clearlylim←−A∈P_fin(X)R^X/R^(X^\A)∼=lim←−A∈P_fin(X)R^A ∼=R^X, hence R^X is complete in the inverse limit topology (where allR^A are discrete), this topology is equivalent to the product topology(with R discrete). Let us denote by _RTopFreeModthe category of all topologically free modules with continuouslinear maps.

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Large algebra

Each such module admits a linear topologywhose basis of open neighborhoods of zero is given byR^(X^\A) for finite subsetsA⊆X. Clearlylim←−A∈P_fin(X)R^X/R^(X^\A)∼=lim←−A∈P_fin(X)R^A ∼=R^X, hence R^X is complete in the inverse limit topology (where allR^A are discrete), this topology is equivalent to the product topology(with R discrete).

Let us denote by _RTopFreeModthe category of all topologically free modules with continuouslinear maps.

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Large algebra

Each such module admits a linear topologywhose basis of open neighborhoods of zero is given byR^(X^\A) for finite subsetsA⊆X. Clearlylim←−A∈P_fin(X)R^X/R^(X^\A)∼=lim←−A∈P_fin(X)R^A ∼=R^X, hence R^X is complete in the inverse limit topology (where allR^A are discrete), this topology is equivalent to the product topology(with R discrete).

Let us denote by _RTopFreeModthe category of all topologically free modules with continuouslinear maps.

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Completed tensor product

Let us provide to the algebraic tensor product R^X ⊗_R R^Y a linear topology as follows.

For each A∈P_fin(X)andB ∈P_fin(Y), let us consider the canonical map R^X ⊗_R R^Y →R^A⊗_R R^B ∼=R^A×B.

The kernels, say K_A,B, of these maps form the basis of the topology. And

R^X^×Y ∼=lim←−

A,B

R^A×B ∼=lim←−

A,B

(R^X ⊗_RR^Y)/K_A,B .

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Completed tensor product

The kernels, say K_A,B, of these maps form the basis of the topology. And

A,B

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Completed tensor product

Thekernels, say K_A,B, of these maps form the basis of the topology.

And

A,B

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Completed tensor product

Thekernels, say K_A,B, of these maps form the basis of the topology.

And

A,B

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Completed tensor product

One thus defines R^X⊗ˆ_RR^Y =R^X^×Y (⊗ˆ is a bifunctor), so that R^X⊗ˆ_RR^Y is the completionof R^X ⊗_RR^Y (in the linear topology).

There exists a continuous R-bilinear mapcan:R^X ×R^Y →R^X⊗Rˆ ^Y. Theorem (Universal property of ⊗)ˆ

Let φ:R^X ×R^Y →R^Z be a continuousR-bilinear map. Then, there exists a unique continuous R-linear mapφ0:R^X⊗ˆ_RR^Y →R^Z such that

φ₀◦can=φ.

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Completed tensor product

There exists a continuous R-bilinear mapcan:R^X ×R^Y →R^X⊗Rˆ ^Y.

Theorem (Universal property of ⊗)ˆ

φ₀◦can=φ.

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Completed tensor product

There exists a continuous R-bilinear mapcan:R^X ×R^Y →R^X⊗Rˆ ^Y. Theorem (Universal property of ⊗)ˆ

φ₀◦can=φ.

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Monoidal category

RTopFreeModwith ⊗ˆ becomes a symmetric monoidal category.

Let us define a functor R⁻:FinFibSet→_RTopFreeMod such that X 7→R^X

and for φ:X →Y, letR^φ:R^X →R^Y be given by (R^φ)(f)(y) = X

x∈φ⁻¹({y})

f(x)

f ∈R^X,y ∈Y.

R⁻ is a monoidal functor, hence it lifts to a functor between categories of monoid objects (it is a property of monoidal functors). One recovers M 7→R[[M]], where M is a finite decomposition monoid, and this corrects the lack of functoriality of the large algebra as defined usually.

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Monoidal category

and for φ:X →Y, letR^φ:R^X →R^Y be given by (R^φ)(f)(y) = X

x∈φ⁻¹({y})

f(x)

f ∈R^X,y ∈Y.

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Monoidal category

and for φ:X →Y, let R^φ:R^X →R^Y be given by (R^φ)(f)(y) = X

x∈φ⁻¹({y})

f(x)

f ∈R^X,y ∈Y.

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Monoidal category

x∈φ⁻¹({y})

f(x)

f ∈R^X,y ∈Y.

R⁻ is a monoidal functor,

hence it lifts to a functor between categories of monoid objects (it is a property of monoidal functors). One recovers M 7→R[[M]], where M is a finite decomposition monoid, and this corrects the lack of functoriality of the large algebra as defined usually.

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Monoidal category

x∈φ⁻¹({y})

f(x)

f ∈R^X,y ∈Y.

R⁻ is a monoidal functor, hence it lifts to a functor between categories of monoid objects (it is a property of monoidal functors).

One recovers M 7→R[[M]], where M is a finite decomposition monoid, and this corrects the lack of functoriality of the large algebra as defined usually.

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Monoidal category

x∈φ⁻¹({y})

f(x)

f ∈R^X,y ∈Y.

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Filtered sets

A (decreasing) filtrationon a set X is a decreasing sequence (X_n)nof finite subsets of X. (X,(X_n)_n)is thus called a filtered set.

A morphism f: (X,(X_n)n)→(Y,(Y_n)n) is a set-theoretic mapf:X →Y such that for each n,f(X_n)⊆Y_n. Such a map is said to be a

filtration-preserving map.

The category of all filtered sets admits a monoidal tensor (X,(X_n)n)⊗(Y,(Y_n)n) = (X ×Y,(Tⁿ(X,Y))n)with

Tⁿ(X,Y) =

n

[

i=0

X_i ×Yn−i .

The unit is the one-point set ∗ with filtration ∗_n=∅ for alln>0 and

∗₀=∗.

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Filtered sets

Tⁿ(X,Y) =

n

[

i=0

X_i ×Yn−i .

∗₀=∗.

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Filtered sets

Tⁿ(X,Y) =

n

[

i=0

X_i ×Yn−i .

∗₀=∗.

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Filtered sets

Tⁿ(X,Y) =

n

[

i=0

X_i ×Yn−i .

The unit is the one-point set ∗ with filtration∗_n=∅ for alln>0 and

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Sub-monoidal categories

A filtered set(X,(X_n)n)is

• Exhausted ifX =X₀;

• SeparatedifT

n≥0X_n=∅;

• Connected if it is both separated and exhausted, andX₀\X₁ =∗.

A setX with an exhausted and separated filtration is equivalent to a setX with a length function `:X →N. (X_n:={x ∈X:`(x)≥n}and

`(x) :=sup{n∈N:x ∈Xn}.)

A filtered set is connected if, and only if, there is aunique element of length zero.

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Sub-monoidal categories

A filtered set(X,(X_n)n)is

• Exhausted ifX =X₀;

• SeparatedifT

n≥0X_n=∅;

• Connected if it is both separated and exhausted, andX₀\X₁ =∗.

A setX with an exhausted and separated filtration is equivalent to a setX with a length function `:X →N. (X_n:={x ∈X:`(x)≥n}and

`(x) :=sup{n∈N:x ∈X_n}.)

A filtered set is connected if, and only if, there is aunique element of length zero.

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Locally finite monoid

A monoid M is said to be alocally finite monoidif for each x ∈M, there are only finitely many x₁,· · · ,x_n∈M\ {1} such thatx =x₁∗ · · · ∗x_n.

Such a monoid is necessarily a finite decomposition monoid. It may be equipped with alength function

`(x) =sup{n∈N:∃(x₁,· · · ,x_n)∈M\ {1}, x =x₁∗ · · · ∗x_n}that satisfies `(x ∗y)≥`(x) +`(y) and`(x) =0 if, and only if,x =1. It is called the canonical length function.

Hence a locally finite monoid is also a monoid object in the monoidal category of connected filtered sets.

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Locally finite monoid

A monoid M is said to be alocally finite monoidif for each x ∈M, there are only finitely many x₁,· · · ,x_n∈M\ {1} such thatx =x₁∗ · · · ∗x_n. Such a monoid is necessarily a finite decomposition monoid.

It may be equipped with alength function

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Locally finite monoid

`(x) =sup{n∈N:∃(x₁,· · · ,x_n)∈M\ {1}, x =x₁∗ · · · ∗x_n}

that satisfies `(x ∗y)≥`(x) +`(y) and`(x) =0 if, and only if,x =1. It is called the canonical length function.

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Locally finite monoid

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Locally finite monoid

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Monoid objects

One now considers the category cSetof all connected filtered setswith finite-fiberandfiltration-preserving maps. It is a monoidal category.

Theorem

A monoid object in cSetis precisely a locally finite monoid. Proof: A monoid object in cSetis thus a usual monoidM with a connected filtration (M_n)n of (two-sided) ideals of M. Let` be its

associated length function. It thus satisfies `(x∗y)≥`(x) +`(y). Since it is connected, `⁻¹({0}) ={1}. Let us assume that there exists some x ∈M with arbitrary long non-trivial decompositions. Then, for every n,

`(x)≥n (sincex =x₁∗ · · · ∗x_m,m≥n,x_i 6=1) which is impossible since the filtration is separated.

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Monoid objects

Theorem

A monoid object in cSetis precisely a locally finite monoid.

Proof: A monoid object in cSetis thus a usual monoidM with a connected filtration (M_n)n of (two-sided) ideals of M. Let` be its

associated length function. It thus satisfies `(x∗y)≥`(x) +`(y). Since it is connected, `⁻¹({0}) ={1}. Let us assume that there exists some x ∈M with arbitrary long non-trivial decompositions. Then, for every n,

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Monoid objects

Theorem

A monoid object in cSetis precisely a locally finite monoid.

Proof: A monoid object in cSetis thus a usual monoidM with a connected filtration (M_n)n of (two-sided) ideals of M. Let` be its

associated length function. It thus satisfies`(x∗y)≥`(x) +`(y). Since it is connected, `⁻¹({0}) ={1}. Let us assume that there exists some x ∈M with arbitrary long non-trivial decompositions. Then, for every n,

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Filtered modules

Filtered module: AR-moduleM endowed with a (decreasing) filtrationM_k of submodules.

Filtered maps: Linear maps that respect the filtrations.

Complete filtered module: M ∼=lim←−nM/M_k. Any filtered moduleM admits a completion, namely Mˆ =lim←−nM/M_k. LetMˆ_n be the kernel of the projection Mˆ →M/Mn. Then Mˆ is filtered (with( ˆMn)n) and Mˆ ∼=lim←−nMˆ/Mˆ_n.

Filtered tensor product: The algebraic tensor productM⊗_R N together with the filtration P

i+j=nM_i ⊗_RN_j.

Completed tensor product: M⊗Nˆ =M\⊗_R N. Monoid objects: Filtered (complete)R-algebras.

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Filtered modules

Complete filtered module: M ∼=lim←−nM/M_k. Any filtered moduleM admits a completion, namely Mˆ =lim←−nM/M_k. LetMˆ_n be the kernel of the projection Mˆ →M/Mn. Then Mˆ is filtered (with( ˆMn)n) and Mˆ ∼=lim←−nMˆ/Mˆ_n.

i+j=nM_i ⊗_RN_j.

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Filtered modules

Complete filtered module: M ∼=lim←−nM/M_k.

Any filtered moduleM admits a completion, namely Mˆ =lim←−nM/M_k. LetMˆ_n be the kernel of the projection Mˆ →M/Mn. Then Mˆ is filtered (with( ˆMn)n) and Mˆ ∼=lim←−nMˆ/Mˆ_n.

i+j=nM_i ⊗_RN_j.

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Filtered modules

Complete filtered module: M ∼=lim←−nM/M_k. Any filtered moduleM admits a completion, namely Mˆ =lim←−nM/M_k.

LetMˆ_n be the kernel of the projection Mˆ →M/Mn. Then Mˆ is filtered (with( ˆMn)n) and Mˆ ∼=lim←−nMˆ/Mˆ_n.

i+j=nM_i ⊗_RN_j.

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Filtered modules

Complete filtered module: M ∼=lim←−nM/M_k. Any filtered moduleM admits a completion, namely Mˆ =lim←−nM/M_k. LetMˆ_n be the kernel of the projection Mˆ →M/Mn. Then Mˆ is filtered (with ( ˆMn)n) and Mˆ ∼=lim←−nMˆ/Mˆ_n.

i+j=nM_i ⊗_RN_j.

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Filtered modules

i+j=nM_i ⊗_RN_j.

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Filtered modules

i+j=nM_i ⊗_RN_j.

Completed tensor product: M⊗Nˆ =M\⊗_R N.

Monoid objects: Filtered (complete)R-algebras.

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Filtered modules

i+j=nM_i ⊗_RN_j.

Completed tensor product: M⊗Nˆ =M\⊗_R N.

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Large algebra

Let M be a locally finite monoid. Then, its canonical filtration induces (functorially) a structure of an exhausted and separated filtered algebraon R[[M]].

It is given by I_n={f ∈R^M:∀x(`(x)<n⇒f(x) =0)}.

The associated (linear) topology is always stronger than the product topology (i.e., the canonical projections are continuous), and can be even strictly stronger.

R[[M]] is completein this topology but is not necessarily the completion of R[M] with the induced topology.

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Large algebra

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Large algebra

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Large algebra

R[[M]] iscomplete in this topology but is not necessarily the completion of R[M] with the induced topology.

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Remark

Of course R⁻ is again amonoidal functorfrom the category of connected filtered sets (with finite-fiber and filtration-preserving maps) to that of complete filtered modules.

Hence it lifts to a functor R[[−]] from the category of locally finite monoids to that of complete filtered algebras.

Remark

R[[M]] is anaugmented algebra with augmentation idealI₁ (this is due to the fact that M is connected as a filtered set).

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Remark

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Remark

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From large algebra to representable functor

Let M be a finite decomposition monoid.

Let us define a functor (−)^M:cAlg_R →Setby A7→A^M.

It is representablewith coordinate ring R[x_a:a∈M](polynomial ring in the indeterminates xa,a∈M).

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Ring scheme (or Hopf ring)

Actually the multiplicative and additive structures ofA[[M]] are naturalin the commutative algebra A. HenceA7→A[[M]] forms a ring object in the category of representable functors.

By Yoneda lemma it induces a structure of a Hopf ringon R[x_a:a∈M] (i.e., a ring object in the category of cocommutative coalgebras or a monoid object in the category of “abelian” Hopf algebras).

The additive partdefines the abelian Hopf algebra structure with coalgebra structure maps ∆_prim(x_a) =x_a⊗1+1⊗x_a,_prim(x_a) =0 and

S_prim(x_a) =−x_a,a∈M.

The multiplicative partinduces a bialgebra with∆(x_a) =P

b∗c=ax_b⊗x_c and(x_a) =1.

Of course both structures are related so that ring axioms hold.

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Ring scheme (or Hopf ring)

By Yoneda lemma it induces a structure of aHopf ring on R[x_a:a∈M]

(i.e., a ring object in the category of cocommutative coalgebras or a monoid object in the category of “abelian” Hopf algebras).

The additive partdefines the abelian Hopf algebra structure with coalgebra structure maps ∆_prim(x_a) =x_a⊗1+1⊗x_a,_prim(x_a) =0 and

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Ring scheme (or Hopf ring)

Theadditive part defines the abelian Hopf algebra structure with coalgebra structure maps ∆_prim(x_a) =x_a⊗1+1⊗x_a,_prim(x_a) =0 and

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Ring scheme (or Hopf ring)

Themultiplicative part induces a bialgebra with∆(x_a) =P

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Ring scheme (or Hopf ring)

Themultiplicative part induces a bialgebra with∆(x_a) =P

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Reconstruction theorem

Theorem

The large algebra R[[M]] is isomorphic to the ring ofR-rational points of the ring scheme (−)[[M]].

Proof: This comes from _cAlg_R(R[x_a:a∈M],R)∼=R[[M]] (of course as sets but also as rings).

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Reconstruction theorem

Theorem

The large algebra R[[M]] is isomorphic to the ring ofR-rational points of the ring scheme (−)[[M]].

Proof: This comes from _cAlg_R(R[x_a:a∈M],R)∼=R[[M]] (of course as sets but also as rings).

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Locally finite monoids to Hopf algebras

Let M be a locally finite monoid.

Let Abe a commutative R-algebra with a unit. Let us define

1+I₁(A)={f:M →A:f(1) =1}. It is a subgroup of the group of invertible elements of A[[M]].

It defines a group schemeA7→1+I₁(A) withrepresenting (orcoordinate) Hopf algebraR[x_a:a∈M\ {1}].

The antipode S is given by S(x_a) =µ(a)for each a∈M\ {1}, where µis the Möbius functionof M.

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Locally finite monoids to Hopf algebras

Let Abe a commutativeR-algebra with a unit. Let us define

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Locally finite monoids to Hopf algebras

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Locally finite monoids to Hopf algebras

The antipode S is given by S(x_a) =µ(a) for each a∈M\ {1}, where µis the Möbius functionof M.

From “combinatorial” monoids to bialgebras and Hopf algebras, functorially