Universal constructions: limits and colimits

Universal constructions:

limits and colimits

Consider and arbitrary but fixed category K for a while.

Universal constructions:

limits and colimits

Consider and arbitrary but fixed category K for a while.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

I !_I→J -J

 !_{J →I}





!_I→I = id_I ^





 id_J = !_{J →J}

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

Initial and terminal objects

An object I ∈ |K| is initial in K if for each object A ∈ |K| there is exactly one morphism from I to A.

Examples:

• ∅ is initial in Set.

• For any signature Σ ∈ |AlgSig|, T_Σ is initial in Alg(Σ).

• For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, the initial model of hΣ, Φi is initial in Mod(Σ, Φ), the full subcategory of Alg(Σ) determined by the class Mod (Σ, Φ) of all models of Φ.

Look for initial objects in other categories.

Theorem: Initial objects, if exist, are unique up to isomorphism:

• Any two initial objects in K are isomorphic.

I⁰  i = !_I→I⁰ I A

- i⁻¹

-

!_I→A

 6

!_I⁰_→A = i⁻¹;!_I→A

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

− any singleton set {∗} is terminal in Set.

− For any signature Σ ∈ |AlgSig|, “singleton” Σ-algebra 1_Σ is terminal in Alg(Σ).

− For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, “singleton”

Σ-algebra 1_Σ is terminal in Mod(Σ, Φ).

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

− any singleton set {∗} is terminal in Set.

− For any signature Σ ∈ |AlgSig|, “singleton” Σ-algebra 1_Σ is terminal in Alg(Σ).

− For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, “singleton”

Σ-algebra 1_Σ is terminal in Mod(Σ, Φ).

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

− any singleton set {∗} is terminal in Set.

− For any signature Σ ∈ |AlgSig|, “singleton” Σ-algebra 1_Σ is terminal in Alg(Σ).

− For any signature Σ ∈ |AlgSig| and set of Σ-equations Φ, “singleton”

Σ-algebra 1_Σ is terminal in Mod(Σ, Φ).

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

• Show that terminal objects are unique to within an isomorphism.

Terminal objects

An object T ∈ |K| is terminal in K if for each object A ∈ |K| there is exactly one morphism from A to T.

terminal = co-initial

Exercises: Dualise those for initial objects.

• Look for terminal objects in standard categories.

• Show that terminal objects are unique to within an isomorphism.

• Look for categories where there is an object which is both initial and terminal.

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁

 f₂

6 We write hf₁, f₂i for h defined as above. Then: ∃! h

hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁

 f₂

6 We write hf₁, f₂i for h defined as above. Then: ∃! h

hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Products

A product of two objects A, B ∈ |K| is any object A × B ∈ |K| with two morphisms (product projections) π₁ : A × B → A and π₂: A × B → B such that for any object C ∈ |K| with morphisms f₁: C → A and f₂: C → B there exists a unique morphism h : C → A × B such that h;π₁ = f₁ and h;π₂ = f₂.

In Set, Cartesian product is a product

A  π₁ A × B B

- π₂

C

@

@

@

@

@

@ I

f₁



f₂ 6

We write hf₁, f₂i for h defined as above. Then: ∃! h hf₁, f₂i;π₁ = f₁ and hf₁, f₂i;π₂ = f₂. Moreover, for any h into the product A × B: h = hh;π₁, h;π₂i.

Essentially, this equationally defines product!

Theorem: Products are defined to within an isomorphism (which commutes with projections).

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

− Now: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_A= hπ_B, π_Ai;(hπ_A⁰ , π_B⁰ i;π_A)= hπ_B, π_Ai;π_A⁰ = π_A

− Similarly: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_B = π_B

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

− Now: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_A = hπ_B, π_Ai;(hπ_A⁰ , π_B⁰ i;π_A)= hπ_B, π_Ai;π_A⁰ = π_A

− Similarly: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_B = π_B

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

− Now: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_A = hπ_B, π_Ai;(hπ_A⁰ , π_B⁰ i;π_A) = hπ_B, π_Ai;π_A⁰ = π_A

− Similarly: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_B = π_B

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

− Now: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_A = hπ_B, π_Ai;(hπ_A⁰ , π_B⁰ i;π_A) = hπ_B, π_Ai;π_A⁰ = π_A

− Similarly: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_B = π_B

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

− Now: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_A = hπ_B, π_Ai;(hπ_A⁰ , π_B⁰ i;π_A) = hπ_B, π_Ai;π_A⁰ = π_A

− Similarly: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_B = π_B

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B A × B

 π_A

@

@

@

@

@ R π_B

B × A

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_A×B = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_{B ×A} = hπ_B⁰ , π_A⁰ i

− Now: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_A = hπ_B, π_Ai;(hπ_A⁰ , π_B⁰ i;π_A) = hπ_B, π_Ai;π_A⁰ = π_A

− Similarly: (hπ_B, π_Ai;hπ_A⁰ , π_B⁰ i);π_B = π_B

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A A

B P

 π_A

@

@

@

@

@ R π_B

P⁰

@

@

@

@

@

I π_A⁰

π_B⁰



hπ_A⁰ , π_B⁰ i



 id_P = hπ_A, π_Bi ^

hπ_B, π_Ai -





 id_P⁰ = hπ_A⁰ , π_B⁰ i

− By much the same argument, any two products of A and B are isomorphic.

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

A B C

A × B

?

@

@

@

@

@ R (A × B) × C

?

@

@

@

@

@

@

@

@

@

@@R

B × C

?

A × (B × C)

?

HH

HH

HH

HH HHj -



















^





Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

A  π₁ A × B B

π₂ - H

HH HH

Y

f₁ *

f₂

∃? h6

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

A  π₁ A × B B

π₂ - Y

HH HH

HH

f₁

*





 f₂

∃? h6

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

A  π₁ A + (A × B) + B B π₂ -

Y

HH HH

HH

f₁

*





 f₂

∃! h6

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

• Define products in the category of relations, Rel, with sets (as objects) and binary relations as morphisms between them.

op

Exercises

• Product commutes (up to isomorphism): A × B ∼= B × A

• Product is associative (up to isomorphism): (A × B) × C ∼= A × (B × C)

• What is a product of two objects in a preorder category?

• Define the product of any family of objects. What is the product of the empty family?

• For any algebraic signature Σ ∈ |AlgSig|, try to define products in Alg(Σ), PAlg_s(Σ), PAlg(Σ). Expect troubles in the two latter cases. . .

• Define products in the category of partial functions, Pfn, with sets (as objects) and partial functions as morphisms between them.

• Define products in the category of relations, Rel, with sets (as objects) and binary relations as morphisms between them.

op

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Coproducts

coproduct = co-product

A coproduct of two objects A, B ∈ |K| is any object A + B ∈ |K| with two

morphisms (coproduct injections) ι₁: A → A + B and ι₂: B → A + B such that for any object C ∈ |K| with morphisms f₁: A → C and f₂: B → C there exists a

unique morphism h : A + B → C such that ι₁;h = f₁ and ι₂;h = f₂.

In Set, disjoint union is a coproduct A ι₁ - A + B B

 ι₂

C

@

@

@

@

@

@ R f₁

f₂

? We write [f₁, f₂] for h defined as above. Then: ∃! h

ι₁;[f₁, f₂] = f₁ and ι₂;[f₁, f₂] = f₂. Moreover, for any h from the coproduct A + B: h = [ι₁;h, ι₂;h].

Essentially, this equationally defines coproduct!

Theorem: Coproducts are defined to within an isomorphism (which commutes with

Equalisers

An equaliser of two “parallel” morphisms f, g : A → B is a morphism e : E → A

such that e;f = e;g, and such that for all h : H → A, if h;f = h;g then for a unique morphism k : H → E, k;e = h.

A - B

f g -

E e -

H 







3 h

∃! k6

• Equalisers are unique up to isomorphism.

• Every equaliser is mono.

• Every epi equaliser is iso.

Equalisers

An equaliser of two “parallel” morphisms f, g : A → B is a morphism e : E → A

such that e;f = e;g, and such that for all h : H → A, if h;f = h;g then for a unique morphism k : H → E, k;e = h.

A - B

f g -

E e -

H 







3 h

∃! k6

• Equalisers are unique up to isomorphism.

• Every equaliser is mono.

• Every epi equaliser is iso.

Equalisers

An equaliser of two “parallel” morphisms f, g : A → B is a morphism e : E → A

such that e;f = e;g, and such that for all h : H → A, if h;f = h;g then for a unique morphism k : H → E, k;e = h.

A - B

f g -

E e -

H 







3 h

∃! k6

• Equalisers are unique up to isomorphism.

• Every equaliser is mono.

• Every epi equaliser is iso.