[PPT] - Incidence bialgebras of monoidal categories Lucia Rotheray TU PowerPoint Presentation

SLIDE 1

Incidence bialgebras of monoidal categories

Lucia Rotheray

TU Dresden

Oslo, 30.07.2019

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 1 / 13

SLIDE 2

Idea

For a (nice) monoidal category (C, ⊗, 1) and a field k we will construct a bialgebra on the k-vector space spank(MorC). Monoidal product multiplication Unit object unit element Composition coproduct Identity morphisms counit.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 2 / 13

SLIDE 3

Motivation

There are several combinatorial Hopf algebras with coproducts of the form ∆(Γ) =

γ⊆Γ

Γ/γ

some sort of

contraction

⊗γ. For example: ∆( ) = ⊗ 1 + 1 ⊗ + ⊗ ∆( ) = ⊗ 1 + 1 ⊗ + ⊗ Slight change of perspective: ∆(Γ) =

γ1 ◦ γ2

some sort of composition =Γ

γ1 ⊗ γ2.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 3 / 13

SLIDE 4

The algebra structure

Lemma

Let (C, ·, 1) be a (small, strict) monoidal category, k a field and kC the k-vector space spanned by MorC. Then (kC, ·, i1) defines an associative unital k-algebra.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 4 / 13

SLIDE 5

The coalgebra structure

Definition (Decompositions and length)

Given f ∈ MorC and n ∈ N, we define: Nn(f ) = {(f1, . . . , fn) ∈ C×n | f1 ◦ . . . ◦ fn = f } ˆ Nn(f ) = {(f1, . . . , fn) ∈ (C \ IdC)×n | f1 ◦ . . . ◦ fn = f } ℓ(f ) = sup{n ∈ N | ˆ Nn(f ) = ∅}.

Definition (Locally finite category)

C is called locally finite if |N2(f )| is finite for every f ∈ MorC.

Definition (M¨

bius category)

C is called M¨

bius if it is locally finite and every morphism has finite length.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 5 / 13

SLIDE 6

The coalgebra structure

Theorem (Joni and Rota ’79)

Let C be a category, k a field and kC the k-vector space spanned by MorC. The following defines a coassociative counital k-coalgebra structure on kC iff C is locally finite: ∆(f ) =

(f1,f2)∈N2(f )

f1 ⊗ f2, ε(f ) =

1 f ∈ IdC

0, f / ∈ IdC

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 6 / 13

SLIDE 7

Examples

Example

Let (P, ) be a poset and CP the category defined by ObCP = P, CP(x, y) =

(x, y), x y

∅, else . CP is M¨

bius and locally finite iff every interval [x, y] := {z ∈ P | x z y} is finite.

Example

The path category of a quiver is locally finite and M¨

bius.

Example

The PROP whose morphisms are (operadic) rooted forests is locally finite.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 7 / 13

SLIDE 8

The bialgebra structure

Definition (ULF property)

A functor F : C → D has the unique lifting of factorisations property if the map N2(f ) → N2(Ff ) (f1, f2) → (Ff1, Ff2) is bijective for all f ∈ C.

Lemma

ULF functors reflect identity morphisms, i.e. f ∈ IdC ⇔ Ff ∈ IdD. We are interested in the case F = ·, i.e. we want a bijection N2(f ) × N2(g) → N2(f · g) ((f1, f2), (g1, g2)) → (f1 · g1, f2 · g2) for all f , g ∈ C.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 8 / 13

SLIDE 9

The bialgebra structure

Theorem

Let (C, ·, 1) be a monoidal category, k a field and kC the k-vector space spanned by MorC and (kC, ∆, ε) denote the incidence coalgebra structure and (kC, ·, i1) the k-algebra structure defined above. If C is M¨

bius and · has the ULF property, the following hold:

1 (kC, ·, i1, ∆, ε) is a k-bialgebra. 2 This bialgebra is a Hopf algebra if and only if (ObC, ·) forms a group.

Proof.

ε(i1) = 1 as i1 is an identity. ULF functors reflect identities ⇒ ǫ(f · g) = ǫ(f )ǫ(g). C M¨

bius ⇒ ∆(ix) = ix ⊗ ix∀x ∈ ObC, in particular ∆(1) = 1 ⊗ 1.

· ULF ⇒ ∆(f · g) = ∆(f )∆(g) and ε(f · g) = ε(f )ε(g). If C is M¨

bius, (kC, ∆, ε) is pointed.Then we can apply Theorem: A pointed bialgebra is a Hopf algebra

if and only if every group-like element is invertible.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 9 / 13

SLIDE 10

Example

Let (M, ·, 1) be a monoid with a partial relation satisfying x y ∧ z t ⇒ x · z y · t and CM the category defined by ObCM = M, CM(x, y) =

(x, y), x y

0, else . The ULF condition for · becomes “[x, y] × [z, t] → [x · z, y · t] is a bijection”.

Example

The path category of a quiver does not admit a ULF monoidal product.

Example

The disjoint union of planar rooted forests is ULF.

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 10 / 13

SLIDE 11

A more in-depth example

Let S be the monoid of (0, 1) strings of finite length where the product is tacking one string onto the end of another, e.g. · = We impose the following partial order on S: If p, q have the same number of 1s and 0s, we define q p ⇔

i

j=1

pj ≥

i

j=1

qj ∀i = 1, . . . , h + w This gives us a category CS as previously described. Viewing the strings as paths lets us view the morphisms as skew-shapes (with extra group like ”legs”).

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 11 / 13

SLIDE 12

Skew shape category/bialgebra

Composition:

=

monoidal product: · = Coproduct: ∆( ) = ⊗ + ⊗ + ⊗ + ⊗ + ⊗

Lucia Rotheray (TU Dresden) Incidence bialgebras of monoidal categories Oslo, 30.07.2019 12 / 13

SLIDE 13

Incidence bialgebras of monoidal categories

Lucia Rotheray

TU Dresden

Oslo, 30.07.2019

Idea

For a (nice) monoidal category (C, ⊗, 1) and a field k we will construct a bialgebra on the k-vector space spank(MorC). Monoidal product multiplication Unit object unit element Composition coproduct Identity morphisms counit.

Motivation

There are several combinatorial Hopf algebras with coproducts of the form ∆(Γ) =

Γ/γ

contraction

⊗γ. For example: ∆( ) = ⊗ 1 + 1 ⊗ + ⊗ ∆( ) = ⊗ 1 + 1 ⊗ + ⊗ Slight change of perspective: ∆(Γ) =

some sort of composition =Γ

γ1 ⊗ γ2.

The algebra structure

Lemma

Let (C, ·, 1) be a (small, strict) monoidal category, k a field and kC the k-vector space spanned by MorC. Then (kC, ·, i1) defines an associative unital k-algebra.

The coalgebra structure

Definition (Decompositions and length)

Given f ∈ MorC and n ∈ N, we define: Nn(f ) = {(f1, . . . , fn) ∈ C×n | f1 ◦ . . . ◦ fn = f } ˆ Nn(f ) = {(f1, . . . , fn) ∈ (C \ IdC)×n | f1 ◦ . . . ◦ fn = f } ℓ(f ) = sup{n ∈ N | ˆ Nn(f ) = ∅}.

Definition (Locally finite category)

C is called locally finite if |N2(f )| is finite for every f ∈ MorC.

Definition (M¨

C is called M¨

The coalgebra structure

Theorem (Joni and Rota ’79)

Let C be a category, k a field and kC the k-vector space spanned by MorC. The following defines a coassociative counital k-coalgebra structure on kC iff C is locally finite: ∆(f ) =

f1 ⊗ f2, ε(f ) =

0, f / ∈ IdC

Examples

Example

Let (P, ) be a poset and CP the category defined by ObCP = P, CP(x, y) =

∅, else . CP is M¨

Example

The path category of a quiver is locally finite and M¨

Example

The PROP whose morphisms are (operadic) rooted forests is locally finite.

The bialgebra structure

Definition (ULF property)

A functor F : C → D has the unique lifting of factorisations property if the map N2(f ) → N2(Ff ) (f1, f2) → (Ff1, Ff2) is bijective for all f ∈ C.

Lemma

ULF functors reflect identity morphisms, i.e. f ∈ IdC ⇔ Ff ∈ IdD. We are interested in the case F = ·, i.e. we want a bijection N2(f ) × N2(g) → N2(f · g) ((f1, f2), (g1, g2)) → (f1 · g1, f2 · g2) for all f , g ∈ C.

The bialgebra structure

Theorem

Let (C, ·, 1) be a monoidal category, k a field and kC the k-vector space spanned by MorC and (kC, ∆, ε) denote the incidence coalgebra structure and (kC, ·, i1) the k-algebra structure defined above. If C is M¨

Proof.

ε(i1) = 1 as i1 is an identity. ULF functors reflect identities ⇒ ǫ(f · g) = ǫ(f )ǫ(g). C M¨

· ULF ⇒ ∆(f · g) = ∆(f )∆(g) and ε(f · g) = ε(f )ε(g). If C is M¨

if and only if every group-like element is invertible.

Example

Let (M, ·, 1) be a monoid with a partial relation satisfying x y ∧ z t ⇒ x · z y · t and CM the category defined by ObCM = M, CM(x, y) =

0, else . The ULF condition for · becomes “[x, y] × [z, t] → [x · z, y · t] is a bijection”.

Example

The path category of a quiver does not admit a ULF monoidal product.

Example

The disjoint union of planar rooted forests is ULF.

A more in-depth example

Let S be the monoid of (0, 1) strings of finite length where the product is tacking one string onto the end of another, e.g. · = We impose the following partial order on S: If p, q have the same number of 1s and 0s, we define q p ⇔

i

pj ≥

i

qj ∀i = 1, . . . , h + w This gives us a category CS as previously described. Viewing the strings as paths lets us view the morphisms as skew-shapes (with extra group like ”legs”).

Skew shape category/bialgebra

Composition:

monoidal product: · = Coproduct: ∆( ) = ⊗ + ⊗ + ⊗ + ⊗ + ⊗

Further reading

(Weak) incidence bialgebras of monoidal categories (Ulrich Kraehmer, L.R.), arXiv:1803.07897v4.

93–139.

eometrie Diff` erentielle Cat` egoriques, 16(3):280–282, 1975.

Imma G` alvez-Carillo, Joachim Kock and Andrew Tonks, Decomposition Spaces, incidence algebras and M¨

2015 arXiv:1512.07580 Ralph M. Kaufmann, Benjamin C. Ward Feyman Categories arXiv:1312.1269v3 Claude Cibils and Marc Rosso, Hopf quivers, 2000 arXiv:math/0009106.