Category theory for computer science

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Category theory for computer science

• generality • abstraction • convenience • constructiveness •

Overall idea

look at all objects exclusively through relationships between them

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Category theory for computer science

• generality • abstraction • convenience • constructiveness •

Overall idea

look at all objects exclusively through relationships between them

capture relationships between objects as appropriate morphisms between them

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(Cartesian) product

• Cartesian product of two sets A and B, is the set

A × B = {ha, bi | a ∈ A, b ∈ B} with projections π₁: A × B → A and π₂: A × B → B given by π₁(ha, bi) = a and π₂(ha, bi) = b.

• A product of two sets A and B, is any set P with projections π₁: P → A and π₂: P → B such that for any set C with functions f₁: C → A and f₂: C → B there exists a unique function h : C → P such that h;π₁ = f₁ and h;π₂ = f₂.

A P B

C

π₁

π₂ -

@

@ I

f₁

f₂ 6

∃! h Theorem: Cartesian product (of sets A and B)

is a product (of A and B).

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(Cartesian) product

A P B

C

π₁

π₂ -

@

@ I

f₁

f₂ 6

Recall the definition of (Cartesian) product of Σ-algebras.

Define product of Σ-algebras as above. What have you changed?

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(Cartesian) product

A P B

C

π₁

π₂ -

@

@ I

f₁

f₂ 6

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(Cartesian) product

A P B

C

π₁

π₂ -

@

@ I

f₁

f₂ 6

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(Cartesian) product

A P B

C

π₁

π₂ -

@

@ I

f₁

f₂ 6

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(Cartesian) product

A P B

C

π₁

π₂ -

@

@ I

f₁

f₂ 6

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(Cartesian) product

A P B

C

π₁

π₂ -

@

@ I

f₁

f₂ 6

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Pitfalls of generalization

the same concrete definition ; distinct abstract generalizations

Given a function f : A → B, the following conditions are equivalent:

• f is a surjection: ∀b ∈ B·∃a ∈ A·f (a) = b.

• f is an epimorphism: for all h₁, h₂ : B → C, if f ;h₁ = f ;h₂ then h₁ = h₂.

• f is a retraction: there exists g : B → A such that g;f = id_B. BUT: Given a Σ-homomorphism f : A → B for A, B ∈ Alg(Σ):

f is retraction =⇒ f is surjection ⇐⇒ f is epimorphism BUT: Given a (weak) Σ-homomorphism f : A → B for A, B ∈ PAlg(Σ):

f is retraction =⇒ f is surjection =⇒ f is epimorphism

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Pitfalls of generalization

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Pitfalls of generalization

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Pitfalls of generalization

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Pitfalls of generalization

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Pitfalls of generalization

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Pitfalls of generalization

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Pitfalls of generalization

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Pitfalls of generalization

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Categories

Definition: Category K consists of:

• a collection of objects: |K|

• mutually disjoint collections of morphisms: K(A, B), for all A, B ∈ |K|;

m : A → B stands for m ∈ K(A, B)

• morphism composition: for m : A → B and m⁰: B → C, we have m; m⁰ : A → C;

− the composition is associative: for m₁: A₀ → A₁, m₂: A₁ → A₂ and m₃: A₂ → A₃, (m₁;m₂);m₃ = m₁;(m₂;m₃)

− the composition has identities: for A ∈ |K|, there is id_A: A → A such that for all m : A → A, m ;id = m , and m : A → A , id ;m = m .

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Categories

− the composition has identities: for A ∈ |K|, there is id_A: A → A such that for all m₁: A₁ → A, m₁;id_A = m₁, and m₂: A → A₂, id_A;m₂ = m₂.

BTW: “collection” means “set”, “class”, etc, as appropriate.

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Categories

− the composition has identities: for A ∈ |K|, there is id_A: A → A such that for all m : A → A, m ;id = m , and m : A → A , id ;m = m .

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Categories

• morphism composition: for m : A → B and m⁰: B → C, we have m; m⁰: A → C;

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Categories

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Categories

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Categories

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Categories

K is locally small if for all A, B ∈ |K|, K(A, B) is a set.

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Presenting finite categories

0:

1: •

2: • - •

3: • - • - •

?

4: • - • - • - •

?

6

#

? . . .

identities omitted

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Presenting finite categories

0:

1: •

2: • - •

3: • - • - •

?

4: • - • - • - •

?

6

#

?

identities omitted

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Presenting finite categories

. . .0:

1: •

2: • - •

3: • - • - •

?

4: • - • - • - •

?

6

#

?

identities omitted

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Presenting finite categories

. . .0:

1: •

2: • - •

3: • - • - •

?

4: • - • - • - •

?

6

#

?

identities omitted

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Presenting finite categories

. . .0:

1: •

2: • - •

3: • - • - •

?

4: • - • - • - •

?

6

#

? . . .

identities omitted

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Generic examples

Discrete categories: A category K is discrete if all K(A, B) are empty, for distinct A, B ∈ |K|, and K(A, A) = {id_A} for all A ∈ |K|.

Preorders: A category K is thin if for all A, B ∈ |K|, K(A, B) contains at most one element.

Every preorder ≤ ⊆ X × X

Every (small) category K determines a preorder ≤_K ⊆ |K| × |K|, where for A, B ∈ |K|, A ≤_K B iff K(A, B) is nonempty.

Monoids: A category K is a monoid if |K| is a singleton.

Every monoid X = hX, ;, idi, where ; : X × X → X and id ∈ X,

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Generic examples

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Generic examples

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Generic examples

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Generic examples

− reflexivity: x ≤ x

− transitivity: if x ≤ y and y ≤ z then x ≤ z Every (small) category K determines a preorder ≤_K ⊆ |K| × |K|, where for A, B ∈ |K|, A ≤_K B iff K(A, B) is nonempty.

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Generic examples

Every preorder ≤ ⊆ X × X determines a thin category K_≤ with |K_≤| = X and for x, y ∈ |K_≤|, K_≤(x, y) is nonempty iff x ≤ y.

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Generic examples

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Generic examples

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Generic examples

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Generic examples

Every preorder ≤ ⊆ X × X determines a thin category K≤ with |K_≤| = X and for x, y ∈ |K_≤|, K_≤(x, y) is nonempty iff x ≤ y.

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Generic examples

Every monoid X = hX, ;, idi, where ; : X × X → X and id ∈ X, determines a (monoid) category K_X with |K_X| = {∗}, K(∗, ∗) = X and the composition

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Examples

• Sets (as objects) and functions between them (as morphisms) with the usual composition form the category Set.

Functions have to be considered with their sources and targets

• For any set S, S-sorted sets (as objects) and S-functions between them (as morphisms) with the usual composition form the category Set^S.

• For any signature Σ, Σ-algebras (as objects) and their homomorphisms (as morphisms) form the category Alg(Σ).

• For any signature Σ, partial Σ-algebras (as objects) and their weak homomorphisms (as morphisms) form the category PAlg(Σ).

• For any signature Σ, partial Σ-algebras (as objects) and their strong homomorphisms (as morphisms) form the category PAlg_s(Σ).

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Examples

• Algebraic signatures (as objects) and their morphisms (as morphisms) with the

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Examples

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Examples

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Examples

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Examples

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Examples

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Examples

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Substitutions

For any signature Σ = (S, Ω), the category of Σ-substitutions Subst_Σ is defined as follows:

− objects of Subst_Σ are S-sorted sets (of variables);

− morphisms in Subst_Σ(X, Y ) are substitutions θ : X → |T_Σ(Y )|,

− composition is defined in the obvious way:

for θ₁: X → Y and θ₂ : Y → Z, that is functions θ₁ : X → |T_Σ(Y )| and θ₂: Y → |T_Σ(Z)|, their composition θ₁;θ₂: X → Z in Subst_Σ is a function θ₁;θ₂: X → |T_Σ(Z)| such that for each x ∈ X, (θ₁;θ₂)(x) = θ₂^#(θ₁(x)).

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Substitutions

for θ₁: X → Y and θ₂ : Y → Z, that is functions θ₁ : X → |T_Σ(Y )| and θ₂: Y → |T_Σ(Z)|, their composition θ₁;θ₂: X → Z in Subst_Σ is a function θ₁;θ₂: X → |T_Σ(Z)| . . . such that for each x ∈ X, (θ₁;θ₂)(x) = θ₂^#(θ₁(x)).

− the composition θ₁;θ₂: X → Z, which is a function θ₁;θ₂ : X → |T_Σ(Z)|, is not the function composition of θ₁ : X → |T_Σ(Y )| and θ₂: Y → |T_Σ(Z)|

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Substitutions

for θ₁: X → Y and θ₂ : Y → Z, that is functions θ₁ : X → |T_Σ(Y )| and θ₂: Y → |T_Σ(Z)|, their composition θ₁;θ₂: X → Z in Subst_Σ is a function θ₁;θ₂: X → |T_Σ(Z)| such that for each x ∈ X, (θ₁;θ₂)(x) = θ₂^#(θ₁(x)).

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Subcategories

Given a category K, a subcategory of K is any category K⁰ such that

• |K⁰| ⊆ |K|,

• K⁰(A, B) ⊆ K(A, B), for all A, B ∈ |K⁰|,

• composition in K⁰ coincides with the composition in K on morphisms in K⁰, and

• identities in K⁰ coincide with identities in K on objects in |K⁰|.

A subcategory K⁰ of K is full if K⁰(A, B) = K(A, B) for all A, B ∈ |K⁰|.

Any collection X ⊆ |K| gives the full subcategory K _X of K by |K _X| = X.

• The category FinSet of finite sets is a full subcategory of Set.

• The discrete category of sets is a subcategory of the category of sets with inclusions as morphisms, which is a subcategory of the category of sets with injective functions as morphisms, which is a subcategory of Set.

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Subcategories

• |K⁰| ⊆ |K|,

• The discrete category of sets is a subcategory of the category of sets with inclusions as morphisms, which is a subcategory of the category of sets with

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Subcategories

• |K⁰| ⊆ |K|,

• The discrete category of sets is a subcategory of the category of sets with inclusions as morphisms, which is a subcategory of the category of sets with injective functions as morphisms, which is a subcategory of Set.

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Subcategories

• |K⁰| ⊆ |K|,

• The discrete category of sets is a subcategory of the category of sets with inclusions as morphisms, which is a subcategory of the category of sets with

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Reversing arrows

Given a category K, its opposite category K^op is defined as follows:

− objects: |K^op| = |K|

− morphisms: K^op(A, B) = K(B, A) for all A, B ∈ |K^op| = |K|

− composition: given m₁: A → B and m₂ : B → C in K^op, that is, m₁: B → A and m₂: C → B in K, their composition in K^op, m₁;m₂ : A → C, is set to be their composition m₂;m₁ : C → A in K.

Theorem: The identities in K^op coincide with the identities in K.

Theorem: Every category is opposite to some category:

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Reversing arrows

(K^op)^op = K

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Duality principle

If W is a categorical concept (notion, property, statement, . . . ) then its dual , co-W , is obtained by reversing all the morphisms in W . Example:

P (X): “for any object Y there exists a morphism f : X → Y ” co-P (X): “for any object Y there exists a morphism f : Y → X”

NOTE: co-P (X) in K coincides with P (X) in K^op.

Theorem: If a property W holds for all categories then co-W holds for all categories as well.

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Duality principle

If W is a categorical concept (notion, property, statement, . . . ) then its dual, co-W, is obtained by reversing all the morphisms in W. Example:

P (X): “for any object Y there exists a morphism f : X → Y ” co-P (X): “for any object Y there exists a morphism f : Y → X”

NOTE: co-P (X) in K coincides with P (X) in K^op.

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Duality principle

P (X): “for any object Y there exists a morphism f : X → Y ” co-P (X): “for any object Y there exists a morphism f : Y → X” NOTE: co-P (X) in K coincides with P (X) in K^op.

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Duality principle

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Duality principle

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Product categories

Given categories K and K⁰, their product K × K⁰ is the category defined as follows:

− objects: |K × K⁰| = |K| × |K⁰|

− morphisms: (K × K⁰)(hA, A⁰i, hB, B⁰i) = K(A, B) × K⁰(A⁰, B⁰) for all A, B ∈ |K| and A⁰, B⁰ ∈ |K⁰|

− composition: for hm₁, m⁰₁i : hA, A⁰i → hB, B⁰i and hm₂, m⁰₂i : hB, B⁰i → hC, C⁰i in K × K⁰, their composition in K × K⁰ is

hm₁, m⁰₁i;hm₂, m⁰₂i = hm₁;m₂, m⁰₁;m⁰₂i .

A

B

C

m₁ -

m₂ -

? m₁;m₂

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Product categories

− objects: |K × K⁰| = |K| × |K⁰|

hm₁, m⁰₁i;hm₂, m⁰₂i = hm₁;m₂, m⁰₁;m⁰₂i .

A

B

C

m₁ - m⁰

m₂ - m⁰

? m₁;m₂

Define Kⁿ, where K is a category and n ≥ 1.

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Product categories

− objects: |K × K⁰| = |K| × |K⁰|

hm₁, m⁰₁i;hm₂, m⁰₂i = hm₁;m₂, m⁰₁;m⁰₂i .

A

B

C

m₁ -

m₂ -

? m₁;m₂