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.
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