Free objects

Adjunctions

Recall:

Term algebras

Fact: For any S-sorted set X of variables, Σ-algebra A and valuation v : X → |A|, there is a unique Σ-homomorphism v^#: T_Σ(X) → A that extends v, so that

id_{X ,→|TΣ(X )|};v^# = v

X |T_Σ(X)|

|A|

T_Σ(X)

A -

HH

HH

HH

HH H

j ? ?

id_{X ,→|TΣ(X )|}

v |v^#| ∃! v^#

Set^S Alg(Σ)

Free objects

Consider any functor G : K⁰ → K

Definition: Given an object A ∈ |K|, a free object over A w.r.t. G is a K⁰-object A⁰ ∈ |K⁰| together with a K-morphism η_A: A → G(A⁰) (called unit morphism) such that given any K⁰-object B⁰ ∈ |K⁰| with K-morphism f : A → G(B⁰), for a unique K⁰-morphism f^#: A⁰ → B⁰ we have

η_A;G(f^#) = f

Paradigmatic example:

Term algebra T_Σ(X) with unit id_{X ,→|TΣ(X )|}: X → |T_Σ(X)| is free over X ∈ |Set^S| w.r.t. the carrier functor | | : Alg(Σ) → Set^S

A G(A⁰)

G(B⁰)

A⁰

B⁰ -

HH

HH

HH

HH H

j ? ?

η_A

f G(f^#) ∃! f^#

 G

K K⁰

Examples

• Consider inclusion i : Int ,→ Real, viewing Int and Real as (thin) categories, and i as a functor between them. For any real r ∈ Real, the ceiling of r,

dre ∈ Int is free over r w.r.t. i.

What about free objects w.r.t. the inclusion of rationals into reals?

• For any set X ∈ |Set|, the “free monoid” List(X) = hX∗, b, i is free over X w.r.t. | | : Monoid → Set.

• For any graph G ∈ |Graph|, the category of its paths, Path(G) ∈ |Cat|, is free over G w.r.t. the graph functor G : Cat → Graph.

• Discrete topologies, completion of metric spaces, free groups, ideal completion of partial orders, ideal completion of free partial algebras, . . .

Makes precise these and other similar examples Indicate unit morphisms!

Free equational models

• Recall: for any algebraic signature Σ = hS, Ωi, term algebra T_Σ(X) is free over X ∈ |Set^S| w.r.t. the carrier functor | | : Alg(Σ) → Set^S.

• For any set of Σ-equations Φ, for any set X ∈ |Set^S|, there exist a model F_Φ(X) ∈ Mod (Φ) that is free over X w.r.t. the carrier functor

| | : Mod(hΣ, Φi) → Set^S, where Mod(hΣ, Φi) is the full subcategory of Alg(Σ) given by the models of Φ.

• For any algebraic signature morphism σ : Σ → Σ⁰, for any Σ-algebra

A ∈ |Alg(Σ)|, there exist a Σ⁰-algebra F_σ(A) ∈ |Alg(Σ⁰)| that is free over A w.r.t. the reduct functor _σ : Alg(Σ⁰) → Alg(Σ).

• For any equational specification morphism σ : hΣ, Φi → hΣ⁰, Φ⁰i, for any model A ∈ Mod (Φ), there exist a model F_σ(A) ∈ Mod (Φ⁰) that is free over A w.r.t.

the reduct functor _σ : Mod(hΣ⁰, Φ⁰i) → Mod(hΣ, Φi).

Prove the above.

Facts

Consider a functor G : K⁰ → K, and object A ∈ |K|, and an object A⁰ ∈ |K⁰| free over A w.r.t. G with unit η_A: A → G(A⁰).

• A free objects over A w.r.t. G the initial objects in the comma category (C_A, G), where C_A: 1 → K is the constant functor.

• A free object over A w.r.t. G, if exists, is unique up to isomorphism.

• The function ( )^#: K(A, G(B⁰)) → K⁰(A⁰, B⁰) is bijective for each B⁰ ∈ |K⁰|.

• For any morphisms g₁, g₂ : A⁰ → B⁰ in K⁰, g₁ = g₂ iff η_A;G(g₁) = η_A;G(g₂).

Colimits as free objects

Fact: In a category K, given a diagram D of shape G(D), the colimit of D in K is a free object over D w.r.t. the diagonal functor ∆^G(D)_K : K → Diag^G(D)_K .

Spell this out for initial objects, coproducts, coequalisers, and pushouts

Left adjoints

Consider a functor G : K⁰ → K.

Fact: Assume that for each object A ∈ |K| there is a free object over A w.r.t. G, say F(A) ∈ |K⁰| is free over A with unit η_A: A → G(F(A)). Then the mapping:

− (A ∈ |K|) 7→ (F(A) ∈ |K⁰|)

− (f : A → B) 7→ ((f ;η_B)^#: F(A) → F(B))

form a functor F : K → K⁰. Moreover, η : Id_K → F;G is a natural transformation.

A G(F(A))

B G(F(B))

F(A)

F(B) -

-

? ? ?

η_A

η_B

f G(F(f )) F(f ) =

(f ;η_B)^#

 G

K K⁰

Proof

F preserves identities: A G(F(A))

A G(F(A))

F(A)

F(A) -

-

? ? ?

η_A

η_A

id_A G(id_F(A))

= id_G(F(A)) id_F(A) F(id_A) = (id_A;η_A)^# = id_F(A)

F preserves composition:

F(f ;g) = (f ;g;η_C)^# = F(f );F(g)

A G(F(A))

B G(F(B))

C G(F(C))

F(A)

F(B)

F(C) -

-

-

? ? ?

? ? ?

η_A

η_B

η_C f

g

G(F(f ))

G(F(g))

!









F(f );F(g) G(F(f );F(g)) =

G(F(f ));G(F(g))

F(f )

F(g)

Left adjoints

Definition: A functor F : K → K⁰ is left adjoint to (a functor) G : K⁰ → K with unit (natural transformation) η : Id_K → F;G if for all objects A ∈ |K|, F(A) ∈ |K⁰| is free over A with unit morphism η_A: A → G(F(A)).

Examples

• The term-algebra functor T_Σ: Set^S → Alg(Σ) is left adjoint to the carrier functor | | : Alg(Σ) → Set^S, for any algebraic signature Σ = hS, Ωi.

• The ceiling d e : Real → Int is left adjoint to the inclusion i : Int ,→ Real of integers into reals.

• The path-category functor Path : Graph → Cat is left adjoint to the graph functor G : Cat → Graph.

• . . . other examples given by the examples of free objects above . . .

Uniqueness of left adjoints

Fact: A left adjoint to any functor G : K⁰ → K, if exists, is determined uniquely up to a natural isomorphism: if F : K → K⁰ and F⁰: K → K⁰ are left adjoint to G with units η : Id_K → F;G and η⁰: Id_K → F⁰;G, respectively, then there exists a natural isomorphism τ : F → F⁰ such that η;(τ ·G) = η⁰.

A

G(F(A))

G(F⁰(A))

F(A)

F⁰(A)

1 PP

PP

PPq ? ? η_A

η_A⁰

G(τ_A) =

(τ ·G)_A τ_A

Proof: For each A ∈ |K|, τ_A = (η_A⁰ )^#. Put also τ_A⁻¹ = (η_A)^#0.

Then show:

− τ_A;τ_A⁻¹ = id_F(A) and τ_A⁻¹;τ_A = id_F⁰_(A)

− τ : F → F⁰ is indeed a natural transfor- mation

− For f : A → B, F(f ) = (f ;η_B)^#.

− For g₁, g₂ : F(A) → •, if η_A;G(g₁) = η_A;G(g₂) then g₁ = g₂.

Left adjoints and colimits

Let F : K → K⁰ be left adjoint to G : K⁰ → K with unit η : Id_K → F;G.

Fact: F is cocontinuous (preserves colimits).

Proof:

K

•

D_n• •

•

A - AA K

 D

'

&

$

%

X









 6

J J J J J J J JJ ]

Z Z Z Z Z Z Z }

αn

G(Y )















>

 6

C C C C CCO

ηD_n;G(βn)

∃!h -

K⁰

•

F(D_n)• •

•

A - AA K



F(D) '

&

$

%

F(X)









 6

J J J J J J J JJ ]

Z Z Z Z Z Z Z }

F(α_n)

Y















>

 6

C C C C CCO

βn

h^# -

Left adjoints and limits

Let F : K → K⁰ be left adjoint to G : K⁰ → K with unit η : Id_K → F;G.

Fact: G is continuous (preserves limits).

Proof:

K

•

G(D_n)• •

•

A - AA K

 G(D)

'

&

$

%

G(X)











? J

J J

J J

J J

JJ^ Z

Z Z

Z Z

Z Z

~

G(α_n)

Y















=

 ?

C C

C C

CCW

βn

^{ηY ;G(h)}

K⁰

•

D_n• •

•

A - AA K



D '

&

$

%

X











? J

J J

J J

J J

JJ^ Z

Z Z

Z Z

Z Z

~

αn

F(Y )















=

 ?

C C

C C

CCW

β_n^#

 ^∃!h

Existence of left adjoints

Fact: Let K⁰ be a locally small complete category. Then a functor G : K⁰ → K has a left adjoint iff

1. G is continuous, and

2. for each A ∈ |K| there exists a set {f_i: A → G(X_i) | i ∈ I} (of objects

X_i ∈ |K⁰| with morphisms f_i: A → G(X_i), i ∈ I) such that for each B ∈ |K⁰| and h : A → G(B), for some f : X_i → B, i ∈ I, we have h = f_i;f.

Proof:

“ ⇒”: Let F : K → K⁰ be left adjoint to G with unit η : Id_K → F;G. Then 1 follows by the previous fact, and for 2 just put I = {∗}, X∗ = F(A), and f∗ = η^A: A → G(F(A))

“ ⇐”: It is enough to show that for each A ∈ |K| the comma category (C_A, G) has an initial object. Under our assumptions, (C_A, G) is complete. The rest follows by the next fact.

On the existence of initial objects

Fact: A locally small complete category K has an initial object if there exists a set of objects I ⊆ |K| such that for all B ∈ |K|, for some X ∈ I there is f : X → B.

Proof: Let P ∈ |K| be a product of I, with projections p_X : P → X for X ∈ I. Let e : E → P be an “equaliser” (limit) of all morphisms in K(P, P ). Then E is initial in K, since for any B ∈ |K|:

• e;p_X;f : E → B, where f : X → B for some X ∈ I.

• Given g₁, g₂: E → B, take their equaliser e⁰ : E⁰ → E. As in the previous item, we have h : P → E⁰. Then h;e;e⁰ : P → P, and by the construction of

e : E → P, e;h;e⁰;e = e;id_P = id_E;e. Now, since e is mono, e;h;e⁰ = id_E, and so e⁰ is a mono retraction, hence an isomorphism, which proves g₁ = g₂.

Cofree objects

Consider any functor F : K → K⁰

Definition: Given an object A⁰ ∈ |K⁰|, a cofree object under A⁰ w.r.t. F is a K-object A ∈ |K| together with a K-morphism ε_A⁰ : F(A) → A⁰ (called counit

morphism) such that given any K-object B ∈ |K| with K⁰-morphism g : F(B) → A⁰, for a unique K-morphism g^#: B → A we have

F(g^#);ε_A⁰ = g

Paradigmatic example:

Function spaces, coming soon

A

B

F(A)

F(B)

A⁰ -

*

6 6

ε_A⁰

∃! g^# F(g^#) g F -

K K⁰

Examples

• Consider inclusion i : Int ,→ Real, viewing Int and Real as (thin) categories, and i as a functor between them. For any real r ∈ Real, the floor of r,

brc ∈ Int is cofree under r w.r.t. i.

What about cofree objects w.r.t. the inclusion of rationals into reals?

• Fix a set X ∈ |Set|. Consider functor F_X : Set → Set defined by:

− for any set A ∈ |Set|, F_X(A) = A × X

− for any function f : A → B, F_X(f ) : A × X → B × X is a function given by F_X(f )(ha, xi) = hf (a), xi.

Then for any set A ∈ |Set|, the powerset A^X ∈ |Set| (i.e., the set of all functions from X to A) is a cofree objects under A w.r.t. F_X. The counit morphism

ε_A: F_X(A^X) = A^X × X → A is the evaluation function: ε_A(hf, xi) = f (x).

A generalisation to deal with exponential objects will (not) be discussed later

Facts

Dual to those for free objects: Consider a functor F : K → K⁰, object A⁰ ∈ |K⁰|, and an object A ∈ |K| cofree under A⁰ w.r.t. F with counit ε_A⁰ : F(A) → A⁰.

• Cofree objects under A⁰ w.r.t. F are the terminal objects in the comma category (F, C_A⁰), where C_A⁰ : 1 → K⁰ is the constant functor.

• A cofree object under A⁰ w.r.t. F, if exists, is unique up to isomorphism.

• The function ( )^#: K⁰(F(B), A⁰) → K(B, A) is bijective for each B ∈ |K|.

• For any morphisms g₁, g₂ : B → A in K, g₁ = g₂ iff F(g₁);ε_A⁰ = F(g₂);ε_A⁰.

Limits as cofree objects

Fact: In a category K, given a diagram D of shape G(D), the limit of D in K is a cofree object under D w.r.t. the diagonal functor ∆^G(D)_K : K → Diag^G(D)_K .

Spell this out for terminal objects, products, equalisers, and pullbacks

Right adjoints

Consider a functor F : K → K⁰.

Fact: Assume that for each object A⁰ ∈ |K⁰| there is a cofree object under A⁰ w.r.t.

F, say G(A⁰) ∈ |K⁰| is cofree under A⁰ with counit ε_A⁰ : F(G(A⁰)) → A⁰. Then the mapping:

− (A⁰ ∈ |K⁰|) 7→ (G(A⁰) ∈ |K|)

− (g : B⁰ → A⁰) 7→ ((ε_B⁰;g)^#: G(B⁰) → G(A⁰))

form a functor G : K⁰ → K. Moreover, ε : G;F → Id_K⁰ is a natural transformation.

A⁰ F(G(A⁰))

B⁰ F(G(B⁰))

G(A⁰)

G(B⁰)

-

-

6 6

6

ε_A⁰

ε_B⁰ F(G(g)) g

G(g) = (ε_B⁰;g)^#

 G

K K⁰

Right adjoints

Definition: A functor G : K⁰ → K is right adjoint to (a functor) F : K → K⁰ with counit (natural transformation) ε : G;F → Id_K⁰ if for all objects A⁰ ∈ |K⁰|,

G(A⁰) ∈ |K| is cofree under A⁰ with counit morphism ε_A⁰ : F(G(A⁰)) → A⁰.

Fact: A right adjoint to any functor F : K → K⁰, if exists, is determined uniquely up to a natural isomorphism: if G : K⁰ → K and G⁰: K⁰ → K are right adjoint to F with counits ε : G;F and ε⁰: G⁰;F, respectively, then there exists a natural

isomorphism τ : G → G⁰ such that (τ ·F);ε⁰ = ε.

Fact: Let G : K⁰ → K be right adjoint to F : K → K⁰ with counit ε : G;F → Id_K⁰. Then G is continuous (preserves limits) and F is cocontinuous (preserves colimits).

From left adjoints to adjunctions

Fact: Let F : K → K⁰ be left adjoint to G : K⁰ → K with unit η : Id_K → F;G.

Then there is a natural transformation ε : G;F → Id_K⁰ such that:

• (G·η);(ε·G) = id_G G(A⁰) G(F(G(A⁰)))

G(A⁰)

F(G(A⁰))

A⁰ -

HH

HH

HH

HHj ? ? η_G(A⁰₎

id_G(A⁰₎ G(ε_A⁰) ε_A⁰

• (η·F);(F·ε) = id_F G(F(A))

A

F(G(F(A)))

F(A)

F(A) -

*

6 6

ε_F(A)

id_F(A) η_A F(η_A)

Proof (idea):

Put ε_A⁰ = (id_G(A⁰₎)^#.

From right adjoints to adjunctions

Fact: Let G : K⁰ → K be right adjoint to F : K → K⁰ with counit ε : G;F → Id_K⁰. Then there is a natural transformation η : Id_K → F;G such that:

• (G·η);(ε·G) = id_G G(A⁰) G(F(G(A⁰)))

G(A⁰)

F(G(A⁰))

A⁰ -

HH

HH

HH

HHj ? ? η_G(A⁰₎

id_G(A⁰₎ G(ε_A⁰) ε_A⁰

• (η·F);(F·ε) = id_F G(F(A))

A

F(G(F(A)))

F(A)

F(A) -

*

6 6

ε_F(A)

id_F(A) η_A F(η_A)

Proof (idea):

Put η_A = (id_F(A))^#.

From adjunctions to left and right adjoints

Fact: Consider two functors F : K → K⁰ and G : K⁰ → K with natural transformations η : Id_K → F;G and ε : G;F → Id_K⁰ such that:

• (G·η);(ε·G) = id_G

• (η·F);(F·ε) = id_F Then:

• F is left adjoint to G with unit η.

• G is right adjoint to F with counit ε.

Proof: For A ∈ |K|, B⁰ ∈ |K⁰| and f : A → G(B⁰), define f^# = F(f );ε_B⁰. Then f^#: F(A) → B⁰ satisfies η_A;G(f^#) = f and is the only such morphism in

K⁰(F(A), B⁰). This proves that F(A) is free over A with unit η_A, and so indeed, F is left adjoint to G with unit η.

The proof that G is right adjoint to F with counit ε is similar.

Adjunctions

Definition: An adjunction between categories K and K⁰ is hF, G, η, εi

where F : K → K⁰ and G : K⁰ → K are functors, and η : Id_K → F;G and ε : G;F → Id_K⁰ natural transformations such that:

• (G·η);(ε·G) = id_G

• (η·F);(F·ε) = id_F

Equivalently, such an adjunction may be given by:

• Functor G : K⁰ → K and all A ∈ |K|, a free object over A w.r.t. G.

• Functor G : K⁰ → K and its left adjoint.

• Functor F : K → K⁰ and all A⁰ ∈ |K⁰|, a cofree object under A⁰ w.r.t. F.

• Functor F : K → K⁰ and its right adjoint.