Distributive laws for Lawvere

theories

Eugenia Cheng

University of SheffieldCT2011

1.

Plan

1. Introduction

2.

Plan

1. Introduction

2. Lawvere theories

2.

Plan

1. Introduction

2. Lawvere theories

3. Distributive laws for monads

2.

Plan

1. Introduction

2. Lawvere theories

3. Distributive laws for monads

4. Three ways to do it

2.

Plan

1. Introduction

2. Lawvere theories

3. Distributive laws for monads

4. Three ways to do it

5. Comparison.

2.

1. Introduction

Distributive laws give us a way of combining two or more

types of algebraic structure expressed as monads.

3.

1. Introduction

Distributive laws give us a way of combining two or more

types of algebraic structure expressed as monads.

E.g. monoids and abelian groups rings

3.

1. Introduction

Distributive laws give us a way of combining two or more

types of algebraic structure expressed as monads.

E.g. monoids and abelian groups rings

Question

What’s a distributive law for Lawvere theories?

3.

1. Introduction

Distributive laws give us a way of combining two or more

types of algebraic structure expressed as monads.

E.g. monoids and abelian groups rings

Question

What’s a distributive law for Lawvere theories?

• Lawvere theories correspond to finitary monads on Set.

3.

1. Introduction

Distributive laws give us a way of combining two or more

types of algebraic structure expressed as monads.

E.g. monoids and abelian groups rings

Question

What’s a distributive law for Lawvere theories?

• Lawvere theories correspond to finitary monads on Set.

• Lawvere theories are themselves monads in a certainbicategory.

3.

1. Introduction

Distributive laws give us a way of combining two or more

types of algebraic structure expressed as monads.

E.g. monoids and abelian groups rings

Question

What’s a distributive law for Lawvere theories?

• Lawvere theories correspond to finitary monads on Set.

• Lawvere theories are themselves monads in a certainbicategory.

—So we can look for distributive laws between these monads.

3.

1. Introduction

• A monad on V only gives algebras in V.

• A Lawvere theory gives models in any finite-product category.

4.

1. Introduction

• A monad on V only gives algebras in V.

• A Lawvere theory gives models in any finite-product category.

monadsLTs

4.

1. Introduction

• A monad on V only gives algebras in V.

• A Lawvere theory gives models in any finite-product category.

monadsLTs

algebras

4.

1. Introduction

• A monad on V only gives algebras in V.

• A Lawvere theory gives models in any finite-product category.

monadsLTs

algebras

4.

1. Introduction

• A monad on V only gives algebras in V.

• A Lawvere theory gives models in any finite-product category.

monadsLTs

algebrasmodels

4.

1. Introduction

• A monad on V only gives algebras in V.

• A Lawvere theory gives models in any finite-product category.

monadsLTs

algebrasmodels

Example

Distributive law for monoids over abelian groupsrings internal to any finite-product category V.

4.

2. Lawvere theories

5.

2. Lawvere theories

Idea

Encapsulate an algebraic theory in a category L.

5.

2. Lawvere theories

Idea

Encapsulate an algebraic theory in a category L.

• The objects of L are the natural numbers, our arities.

• A morphism k 1 is an operation of arity k.

• A morphism k m is m operations of arity k.

5.

2. Lawvere theories

Idea

Encapsulate an algebraic theory in a category L.

• The objects of L are the natural numbers, our arities.

• A morphism k 1 is an operation of arity k.

• A morphism k m is m operations of arity k.

We use F a skeleton of FinSet (finite sets and functions).

5.

2. Lawvere theories

Idea

Encapsulate an algebraic theory in a category L.

• The objects of L are the natural numbers, our arities.

• A morphism k 1 is an operation of arity k.

• A morphism k m is m operations of arity k.

We use F a skeleton of FinSet (finite sets and functions).

Definition

A Lawvere theory is a small category L with finite products,equipped with a strict identity-on-objects functor

Fop

L.

5.

2. Lawvere theories

Idea

Encapsulate an algebraic theory in a category L.

• The objects of L are the natural numbers, our arities.

• A morphism k 1 is an operation of arity k.

• A morphism k m is m operations of arity k.

We use F a skeleton of FinSet (finite sets and functions).

Definition

A Lawvere theory is a small category L with finite products,equipped with a strict identity-on-objects functor

Fop

L.

Note: in Fop the object m is the product of m copies of 1.

5.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

6.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

Example2-ary operations in the theory of monoids

6.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

Example2-ary operations in the theory of monoids

• (non-Σ) operads: only one i.e. ab

6.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

Example2-ary operations in the theory of monoids

• (non-Σ) operads: only one i.e. ab

• Lawvere theory: ab, a, a2, b, b2, aba, ab3a5, . . .

6.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

Example2-ary operations in the theory of monoids

• (non-Σ) operads: only one i.e. ab

• Lawvere theory: ab, a, a2, b, b2, aba, ab3a5, . . .

i.e. everything in the free monad on {a, b}.

6.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

Example2-ary operations in the theory of monoids

• (non-Σ) operads: only one i.e. ab

• Lawvere theory: ab, a, a2, b, b2, aba, ab3a5, . . .

i.e. everything in the free monad on {a, b}.

A morphism 3 2 is two 3-ary operations e.g.

(ab, a3), (a2b, abc), . . .

6.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

Example2-ary operations in the theory of monoids

• (non-Σ) operads: only one i.e. ab

• Lawvere theory: ab, a, a2, b, b2, aba, ab3a5, . . .

i.e. everything in the free monad on {a, b}.

A morphism 3 2 is two 3-ary operations e.g.

(ab, a3), (a2b, abc), . . .

Composition: 3 2 1{ab, a3} x2y

6.

2. Lawvere theories

Note

We are allowed to forget and repeat variables.

Example2-ary operations in the theory of monoids

• (non-Σ) operads: only one i.e. ab

• Lawvere theory: ab, a, a2, b, b2, aba, ab3a5, . . .

i.e. everything in the free monad on {a, b}.

A morphism 3 2 is two 3-ary operations e.g.

(ab, a3), (a2b, abc), . . .

Composition: 3 2 1{ab, a3} x2y

︸ ︷︷ ︸

ab.ab.a3

6.

2. Lawvere theories

We have many arities for the “same” operation.

arity operation

3 a, b, c abc

4 a, b, c, d abc

5 a, b, c, d, e abc...

7.

2. Lawvere theories

We have many arities for the “same” operation.

arity operation

3 a, b, c abc

4 a, b, c, d abc

5 a, b, c, d, e abc...

These are all related by forgetting variablesi.e. via projections in Fop.

13

4

5

6...

7.

2. Lawvere theories

Generalisations

8.

2. Lawvere theories

Generalisations

• use F = FinSet instead of a skeleton

8.

2. Lawvere theories

Generalisations

• use F = FinSet instead of a skeleton

• put P = “free finite product category” 2-monadnote that FinSetop is P1—could use PA to get “typed” theory

8.

2. Lawvere theories

Generalisations

• use F = FinSet instead of a skeleton

• put P = “free finite product category” 2-monadnote that FinSetop is P1—could use PA to get “typed” theory

• could just say a Lawvere theory is any finite productcategory C

8.

2. Lawvere theories

Generalisations

• use F = FinSet instead of a skeleton

• put P = “free finite product category” 2-monadnote that FinSetop is P1—could use PA to get “typed” theory

• could just say a Lawvere theory is any finite productcategory C

• could do finite limits instead of just products.

8.

2. Lawvere theories

Models ≡ algebras

A model for L in a finite-product category C is afinite-product preserving functor

L C

9.

2. Lawvere theories

Models ≡ algebras

A model for L in a finite-product category C is afinite-product preserving functor

L C

Idea1 A ∈ C underlying data

9.

2. Lawvere theories

Models ≡ algebras

A model for L in a finite-product category C is afinite-product preserving functor

L C

Idea1 A ∈ C underlying data

k Ak

9.

2. Lawvere theories

Models ≡ algebras

A model for L in a finite-product category C is afinite-product preserving functor

L C

Idea1 A ∈ C underlying data

k Ak

operation of arity k

k

1

Ak

A

9.

2. Lawvere theories

Lawvere theories vs monads

10.

2. Lawvere theories

Lawvere theories vs monads

Lawvere theory monad

10.

2. Lawvere theories

Lawvere theories vs monads

Lawvere theory monad

morphism

k 1

“k-ary

operation”

10.

2. Lawvere theories

Lawvere theories vs monads

set of k elements

Lawvere theory monad

morphism

k 1

“k-ary

operation”

element of T ([k])

10.

2. Lawvere theories

Lawvere theories vs monads

set of k elements

Lawvere theory monad

morphism

k 1

“k-ary

operation”

element of T ([k])

i.e. 1 T ([k]) ∈ Set

10.

2. Lawvere theories

Lawvere theories vs monads

set of k elements

Lawvere theory monad

morphism

k 1

“k-ary

operation”

element of T ([k])

i.e. 1 T ([k]) ∈ Set

morphism

k m

“m operations

of arity k”

10.

2. Lawvere theories

Lawvere theories vs monads

set of k elements

Lawvere theory monad

morphism

k 1

“k-ary

operation”

element of T ([k])

i.e. 1 T ([k]) ∈ Set

morphism

k m

“m operations

of arity k”

m elements of T ([k])

10.

2. Lawvere theories

Lawvere theories vs monads

set of k elements

Lawvere theory monad

morphism

k 1

“k-ary

operation”

element of T ([k])

i.e. 1 T ([k]) ∈ Set

morphism

k m

“m operations

of arity k”

m elements of T ([k])

i.e. [m] T ([k]) ∈ Set

10.

2. Lawvere theories

Lawvere theories vs monads

set of k elements

Lawvere theory monad

morphism

k 1

“k-ary

operation”

element of T ([k])

i.e. 1 T ([k]) ∈ Set

morphism

k m

“m operations

of arity k”

m elements of T ([k])

i.e. [m] T ([k]) ∈ Set

i.e. [m] [k] ∈ KlT

10.

2. Lawvere theories

Lawvere theories vs monads

set of k elements

Lawvere theory monad

morphism

k 1

“k-ary

operation”

element of T ([k])

i.e. 1 T ([k]) ∈ Set

morphism

k m

“m operations

of arity k”

m elements of T ([k])

i.e. [m] T ([k]) ∈ Set

i.e. [m] [k] ∈ KlT

Idea

Lawvere theories are related to monads via the Kleisli category.

10.

2. Lawvere theories

Definition

Monad T on Set Lawvere theory LT

LT = full subcategory of (KlT )op

whose objects are finite sets.

11.

2. Lawvere theories

Definition

Monad T on Set Lawvere theory LT

LT = full subcategory of (KlT )op

whose objects are finite sets.

Lawvere theory L monad TL on Set

TLX =

n∈Fop

∫

L(n, 1) × Xn.

11.

2. Lawvere theories

Definition

Monad T on Set Lawvere theory LT

LT = full subcategory of (KlT )op

whose objects are finite sets.

Lawvere theory L monad TL on Set

TLX =

n∈Fop

∫

L(n, 1) × Xn.

Theorem

This gives a correspondence between Lawvere theories andfinitary monads on Set.

11.

3. Distributive laws for monads

12.

3. Distributive laws for monadsIdea

Given monads S and T on C, can we make TS into a monad?

12.

3. Distributive laws for monadsIdea

Given monads S and T on C, can we make TS into a monad?

TSTS TTSS TS? µT µS

12.

3. Distributive laws for monadsIdea

Given monads S and T on C, can we make TS into a monad?

TSTS TTSS TS? µT µS

Definition (Beck)

A distributive law of monads S over T consists of a naturaltransformation

λ : ST ⇒ TS

satisfying some axioms.

12.

3. Distributive laws for monadsIdea

Given monads S and T on C, can we make TS into a monad?

TSTS TTSS TS? µT µS

Definition (Beck)

A distributive law of monads S over T consists of a naturaltransformation

λ : ST ⇒ TS

satisfying some axioms.

• Formal theory of monads (Street)Do this inside any bicategory, not just Cat.

12.

3. Distributive laws for monadsIdea

Given monads S and T on C, can we make TS into a monad?

TSTS TTSS TS? µT µS

Definition (Beck)

A distributive law of monads S over T consists of a naturaltransformation

λ : ST ⇒ TS

satisfying some axioms.

• Formal theory of monads (Street)Do this inside any bicategory, not just Cat.

• Iterated distributive laws (Cheng)Combine n monads with distributive laws andYang-Baxter condition.

12.

3. Distributive laws for monads

Examples

monoid + abelian group ring

horizontalcomposition

+verticalcomposition

2-category

13.

3. Distributive laws for monads

Examples

monoid + abelian group ring

horizontalcomposition

+verticalcomposition

2-category

Or combining more structures:

0-composition+

1-composition+...+

(n − 1)-composition

n-category

13.

3. Distributive laws for monads

A point of view

• The monad TS says we can express all structure as“S-structure followed by T -structure”.

14.

3. Distributive laws for monads

A point of view

• The monad TS says we can express all structure as“S-structure followed by T -structure”.

• The distributive law ST TS says “if we had it theother way round we could switch it over”.

14.

3. Distributive laws for monads

A point of view

• The monad TS says we can express all structure as“S-structure followed by T -structure”.

• The distributive law ST TS says “if we had it theother way round we could switch it over”.

For Lawvere theories

We want a way of combining A and B to give BA

corresponding to a distributive law of monads

TATB TBTA

14.

3. Distributive laws for monads

A point of view

• The monad TS says we can express all structure as“S-structure followed by T -structure”.

• The distributive law ST TS says “if we had it theother way round we could switch it over”.

For Lawvere theories

We want a way of combining A and B to give BA

corresponding to a distributive law of monads

TATB TBTA

withTBTA = TBA

14.

4. Three ways to do it

1. Factorisation systems over Fop.

—Rosebrugh and Wood, Distributive laws andfactorization (JPAA 2002)

15.

4. Three ways to do it

1. Factorisation systems over Fop.

—Rosebrugh and Wood, Distributive laws andfactorization (JPAA 2002)

2. Profunctors internal to Mon.

—Lack, Composing props (TAC 2004)

—Akhvlediani, Composing Lawvere theories (CT2010)

15.

4. Three ways to do it

1. Factorisation systems over Fop.

—Rosebrugh and Wood, Distributive laws andfactorization (JPAA 2002)

2. Profunctors internal to Mon.

—Lack, Composing props (TAC 2004)

—Akhvlediani, Composing Lawvere theories (CT2010)

3. Kleisli bicategory of P on profunctors

—Hyland, Distributive laws (CLP 2010)

15.

4. Three ways to do it

1. Factorisation systems over Fop

16.

4. Three ways to do it

1. Factorisation systems over Fop

In the composite theory BA every morphism can beexpressed as a composite

∈A ∈B

16.

4. Three ways to do it

1. Factorisation systems over Fop

In the composite theory BA every morphism can beexpressed as a composite

∈A ∈B

For example: × and +

The composite 3-ary operation a(b + c) can be expressed as

16.

4. Three ways to do it

1. Factorisation systems over Fop

In the composite theory BA every morphism can beexpressed as a composite

∈A ∈B

For example: × and +

The composite 3-ary operation a(b + c) can be expressed as

3 1

2ab, ac x + y

16.

4. Three ways to do it

1. Factorisation systems over Fop

In the composite theory BA every morphism can beexpressed as a composite

∈A ∈B

For example: × and +

The composite 3-ary operation a(b + c) can be expressed as

3 1

2ab, ac x + y

3ab, ac, ab2c x + y

16.

4. Three ways to do it

1. Factorisation systems over Fop

In the composite theory BA every morphism can beexpressed as a composite

∈A ∈B

For example: × and +

The composite 3-ary operation a(b + c) can be expressed as

3 1

2ab, ac x + y

3ab, ac, ab2c x + y

p1, p2

16.

4. Three ways to do it

1. Factorisation systems over Fop

In the composite theory BA every morphism can beexpressed as a composite

∈A ∈B

For example: × and +

The composite 3-ary operation a(b + c) can be expressed as

3 1

2ab, ac x + y

3ab, ac, ab2c x + y

p1, p2

—factorisations are only unique up to morphisms in Fop.

16.

4. Three ways to do it

Appealing fact (Rosebrugh and Wood)

Strict factorisation systems are distributive laws in Span.

17.

4. Three ways to do it

Appealing fact (Rosebrugh and Wood)

Strict factorisation systems are distributive laws in Span.

• A and B are categories i.e. monads in Span.

• AB BA makes BA into a monad in Spani.e. a category.

17.

4. Three ways to do it

Appealing fact (Rosebrugh and Wood)

Strict factorisation systems are distributive laws in Span.

• A and B are categories i.e. monads in Span.

• AB BA makes BA into a monad in Spani.e. a category.

It is the pullback

obFobF

A

17.

4. Three ways to do it

Appealing fact (Rosebrugh and Wood)

Strict factorisation systems are distributive laws in Span.

• A and B are categories i.e. monads in Span.

• AB BA makes BA into a monad in Spani.e. a category.

It is the pullback

obFobF

A B

obF

17.

4. Three ways to do it

Appealing fact (Rosebrugh and Wood)

Strict factorisation systems are distributive laws in Span.

• A and B are categories i.e. monads in Span.

• AB BA makes BA into a monad in Spani.e. a category.

It is the pullback

obFobF

A B

obF

.

17.

4. Three ways to do it

Appealing fact (Rosebrugh and Wood)

Strict factorisation systems are distributive laws in Span.

• A and B are categories i.e. monads in Span.

• AB BA makes BA into a monad in Spani.e. a category.

It is the pullback

obFobF

A B

obF

.composable pairs

k∈A

l∈B

m

17.

4. Three ways to do it

Appealing fact (Rosebrugh and Wood)

Strict factorisation systems are distributive laws in Span.

• A and B are categories i.e. monads in Span.

• AB BA makes BA into a monad in Spani.e. a category.

It is the pullback

obFobF

A B

obF

.composable pairs

k∈A

l∈B

m

The distributive law tells us how to re-express a pair

k∈B

l∈A

m as k∈A

l′∈B

m

17.

4. Three ways to do it

RW define distributive laws over I for I a groupoid—ensures equivalence relation on composable pairs.

18.

4. Three ways to do it

RW define distributive laws over I for I a groupoid—ensures equivalence relation on composable pairs.

However instead we can generate an equivalence relation.

18.

4. Three ways to do it

RW define distributive laws over I for I a groupoid—ensures equivalence relation on composable pairs.

However instead we can generate an equivalence relation.

Idea

18.

4. Three ways to do it

RW define distributive laws over I for I a groupoid—ensures equivalence relation on composable pairs.

However instead we can generate an equivalence relation.

Idea

Our original pullbackB ⊗ A

A B

obFobF obF

ignored the fact that Fop is in both A and B.

18.

4. Three ways to do it

RW define distributive laws over I for I a groupoid—ensures equivalence relation on composable pairs.

However instead we can generate an equivalence relation.

Idea

Our original pullbackB ⊗ A

A B

obFobF obF

ignored the fact that Fop is in both A and B.

So we want a coequaliser

B ⊗ Fop ⊗ A B ⊗ A B ⊗Fop Aabsorb F

op into A

absorb Fop into B

—looks like bimodules.

18.

4. Three ways to do it

Definition 1

A distributive law of Lawvere theories A over B is a factorisationsystem over Fop on the composite B ⊗ A in Span.

19.

4. Three ways to do it

Definition 1

A distributive law of Lawvere theories A over B is a factorisationsystem over Fop on the composite B ⊗ A in Span.

≡ a distributive law of A over B expressed as monads in

Bim(Span)

19.

4. Three ways to do it

Definition 1

A distributive law of Lawvere theories A over B is a factorisationsystem over Fop on the composite B ⊗ A in Span.

≡ a distributive law of A over B expressed as monads in

Bim(Span) ≃ Prof.

19.

4. Three ways to do it

Definition 1

A distributive law of Lawvere theories A over B is a factorisationsystem over Fop on the composite B ⊗ A in Span.

≡ a distributive law of A over B expressed as monads in

Bim(Span) ≃ Prof.

Aside on profunctors

19.

4. Three ways to do it

Definition 1

A distributive law of Lawvere theories A over B is a factorisationsystem over Fop on the composite B ⊗ A in Span.

≡ a distributive law of A over B expressed as monads in

Bim(Span) ≃ Prof.

Aside on profunctors

Given categories C and D, a profunctor CF

D is a functor

Dop × C

FSet.

19.

4. Three ways to do it

Definition 1

A distributive law of Lawvere theories A over B is a factorisationsystem over Fop on the composite B ⊗ A in Span.

≡ a distributive law of A over B expressed as monads in

Bim(Span) ≃ Prof.

Aside on profunctors

Given categories C and D, a profunctor CF

D is a functor

Dop × C

FSet.

A monad CA

C ∈ Prof corresponds to a category A

equipped with an identity-on-objects functor

C A.

19.

4. Three ways to do it

Definition 1

A distributive law of Lawvere theories A over B is a factorisationsystem over Fop on the composite B ⊗ A in Span.

≡ a distributive law of A over B expressed as monads in

Bim(Span) ≃ Prof.

Aside on profunctors

Given categories C and D, a profunctor CF

D is a functor

Dop × C

FSet.

A monad CA

C ∈ Prof corresponds to a category A

equipped with an identity-on-objects functor

C A.

So Lawvere theories arise as particular monads on Fop.

19.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

20.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

A monad C C is now a monoidal category A equippedwith an identity-on-objects monoidal functor C A.

20.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

A monad C C is now a monoidal category A equippedwith an identity-on-objects monoidal functor C A.

So again Lawvere theories arise as particular monads on Fop.

20.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

A monad C C is now a monoidal category A equippedwith an identity-on-objects monoidal functor C A.

So again Lawvere theories arise as particular monads on Fop.

Definition 2

A distributive law of Lawvere theories A over B is adistributive law in the bicategory Prof(Mon).

20.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

A monad C C is now a monoidal category A equippedwith an identity-on-objects monoidal functor C A.

So again Lawvere theories arise as particular monads on Fop.

Definition 2

A distributive law of Lawvere theories A over B is adistributive law in the bicategory Prof(Mon).

Theorem

Such a distributive law makes B⊗FopA into a Lawvere theory.

i.e. if A and B are finite-product categories, so is B ⊗Fop A.

20.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

A monad C C is now a monoidal category A equippedwith an identity-on-objects monoidal functor C A.

So again Lawvere theories arise as particular monads on Fop.

Definition 2

A distributive law of Lawvere theories A over B is adistributive law in the bicategory Prof(Mon).

Theorem

Such a distributive law makes B⊗FopA into a Lawvere theory.

i.e. if A and B are finite-product categories, so is B ⊗Fop A.

A ⊗Fop B

λB ⊗F

op A

20.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

A monad C C is now a monoidal category A equippedwith an identity-on-objects monoidal functor C A.

So again Lawvere theories arise as particular monads on Fop.

Definition 2

A distributive law of Lawvere theories A over B is adistributive law in the bicategory Prof(Mon).

Theorem

Such a distributive law makes B⊗FopA into a Lawvere theory.

i.e. if A and B are finite-product categories, so is B ⊗Fop A.

A ⊗Fop B

λB ⊗F

op A

Proof • Bare hands, or

20.

4. Three ways to do it

2. Prof(Mon) —internal profunctors in monoids

A monad C C is now a monoidal category A equippedwith an identity-on-objects monoidal functor C A.

So again Lawvere theories arise as particular monads on Fop.

Definition 2

A distributive law of Lawvere theories A over B is adistributive law in the bicategory Prof(Mon).

Theorem

Such a distributive law makes B⊗FopA into a Lawvere theory.

i.e. if A and B are finite-product categories, so is B ⊗Fop A.

A ⊗Fop B

λB ⊗F

op A

Proof • Bare hands, or

• The free finite-product category 2-monad on Prof.20.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

21.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

• Let P be the free finite-product category 2-monad on Cat.

• P extends to Prof via a distributive law.

• Let ProfP be the Kleisli bicategory for the extended P.

21.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

• Let P be the free finite-product category 2-monad on Cat.

• P extends to Prof via a distributive law.

• Let ProfP be the Kleisli bicategory for the extended P.

Then monads on 1 in ProfP are precisely Lawvere theories.

21.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

• Let P be the free finite-product category 2-monad on Cat.

• P extends to Prof via a distributive law.

• Let ProfP be the Kleisli bicategory for the extended P.

Then monads on 1 in ProfP are precisely Lawvere theories.

• A monad in ProfP is a profunctor 1 P1

21.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

• Let P be the free finite-product category 2-monad on Cat.

• P extends to Prof via a distributive law.

• Let ProfP be the Kleisli bicategory for the extended P.

Then monads on 1 in ProfP are precisely Lawvere theories.

• A monad in ProfP is a profunctor 1 P1

i.e. P1op × 1 Set

21.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

• Let P be the free finite-product category 2-monad on Cat.

• P extends to Prof via a distributive law.

• Let ProfP be the Kleisli bicategory for the extended P.

Then monads on 1 in ProfP are precisely Lawvere theories.

• A monad in ProfP is a profunctor 1 P1

i.e. P1op × 1 Set

i.e. FinSet Set a finitary monad.

21.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

• Let P be the free finite-product category 2-monad on Cat.

• P extends to Prof via a distributive law.

• Let ProfP be the Kleisli bicategory for the extended P.

Then monads on 1 in ProfP are precisely Lawvere theories.

• A monad in ProfP is a profunctor 1 P1

i.e. P1op × 1 Set

i.e. FinSet Set a finitary monad.

Definition 3

21.

4. Three ways to do it

3. Free finite-product category 2-monad approach.

• Let P be the free finite-product category 2-monad on Cat.

• P extends to Prof via a distributive law.

• Let ProfP be the Kleisli bicategory for the extended P.

Then monads on 1 in ProfP are precisely Lawvere theories.

• A monad in ProfP is a profunctor 1 P1

i.e. P1op × 1 Set

i.e. FinSet Set a finitary monad.

Definition 3

A distributive law of Lawvere theories A over B is a distributivelaw in the bicategory ProfP.

21.

5. Comparison

22.

5. Comparison

ClaimThese three methods all give the same answer as a distributivelaw between the associated monads.

22.

5. Comparison

ClaimThese three methods all give the same answer as a distributivelaw between the associated monads.

Idea

22.

5. Comparison

ClaimThese three methods all give the same answer as a distributivelaw between the associated monads.

Idea

Compare

• Finitary monads Set Set in CAT

22.

5. Comparison

ClaimThese three methods all give the same answer as a distributivelaw between the associated monads.

Idea

Compare

• Finitary monads Set Set in CAT

• Lawvere theories as

1. monads F F in Prof

2. monads F F in Prof(Mon)

3. monads 1 1 in ProfP.

22.

5. Comparison

ClaimThese three methods all give the same answer as a distributivelaw between the associated monads.

Idea

Compare

• Finitary monads Set Set in CAT

• Lawvere theories as

1. monads F F in Prof

2. monads F F in Prof(Mon)

3. monads 1 1 in ProfP.

SetT

Set

22.

5. Comparison

ClaimThese three methods all give the same answer as a distributivelaw between the associated monads.

Idea

Compare

• Finitary monads Set Set in CAT

• Lawvere theories as

1. monads F F in Prof

2. monads F F in Prof(Mon)

3. monads 1 1 in ProfP.

SetT

Set

FI∗

SetT∗

SetI∗

F

22.

5. Comparison

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

id-on-objects

monoidal functors

F −→ A

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

id-on-objects

monoidal functors

F −→ A

CATf (Set,Set)

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

id-on-objects

monoidal functors

F −→ A

CATf (Set,Set)f+f

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

id-on-objects

monoidal functors

F −→ A

CATf (Set,Set)f+f

equivalence

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

id-on-objects

monoidal functors

F −→ A

CATf (Set,Set)f+f

equivalence

≃

Prof(P1, P1)

forgetful

23.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

id-on-objects

monoidal functors

F −→ A

CATf (Set,Set)f+f

equivalence

≃

Prof(P1, P1)

forgetful

forgetfulf+f

23.

5. Comparison

Key points

24.

5. Comparison

Key points

• The functors send T to LT .

24.

5. Comparison

Key points

• The functors send T to LT .

• By pseudo-functoriality distributive laws map todistributive laws, and

LT ◦ LS∼= LTS

24.

5. Comparison

Key points

• The functors send T to LT .

• By pseudo-functoriality distributive laws map todistributive laws, and

LT ◦ LS∼= LTS

• Moreover the functors are full and faithful, so givenLawvere theories on the right, any distributive lawbetween them corresponds to one on the left.

So we have three equivalent notions of distributive laws for

Lawvere theories, which correspond to distributive laws

between the associated monads.

24.

5. Comparison

Prof(F, F)

ProfP(1, 1)

Prof(Mon)(F, F)

Monads

Lawvere theories

id-on-objects functors

F −→ A

id-on-objects

monoidal functors

F −→ A

CATf (Set,Set)

equivalence

f+f ≃

Prof(P1, P1)

forgetful

forgetfulf+f

25.

5. Comparison

Key points

• The functors are monoidal, and send T to LT .

• By pseudo-functoriality distributive laws map todistributive laws, and

LT ◦ LS∼= LTS

• Moreover the functors are full and faithful, so givenLawvere theories on the right, any distributive lawbetween them corresponds to one on the left.

26.

5. Comparison

Key points

• The functors are monoidal, and send T to LT .

• By pseudo-functoriality distributive laws map todistributive laws, and

LT ◦ LS∼= LTS

• Moreover the functors are full and faithful, so givenLawvere theories on the right, any distributive lawbetween them corresponds to one on the left.

So we have three equivalent notions of distributive laws for

Lawvere theories, which correspond to distributive laws

between the associated monads.

26.