Universal Properties a Categorical Look at Undergraduate Algebra and Topology

Home , Universal property

Category Theory Universal Properties

Universal Properties A categorical look at undergraduate algebra and topology

Julia Goedecke

Newnham College

23 February 2016, Adam’s Society

Julia Goedecke (Newnham) Universal Properties 23/02/2016 1 / 30 Category Theory Universal Properties

1 Category Theory Maths is Abstraction Category Theory: more abstraction

2 Universal Properties Within one category Mixing categories

Julia Goedecke (Newnham) Universal Properties 23/02/2016 2 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction

1 Category Theory Maths is Abstraction Category Theory: more abstraction

2 Universal Properties Within one category Mixing categories

Julia Goedecke (Newnham) Universal Properties 23/02/2016 3 / 30 Examples My pet and my friend’s pet are both cats. Cats, dogs, dolphins are all mamals. My home, my old school, the maths department are all buildings.

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction What is Abstraction?

Abstraction Take example/situation/idea. Determine some (important) properties. “Lift” those away from the example/situation/idea. Work with abstracted properties. Should get many more examples which also ﬁt these “lifted” properties.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 4 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction What is Abstraction?

Examples My pet and my friend’s pet are both cats. Cats, dogs, dolphins are all mamals. My home, my old school, the maths department are all buildings.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 4 / 30 After that also (not necessarily in this order) negative numbers (abstraction of debt?) rational numbers (abstraction of proportions) real numbers (abstraction of lengths)

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Numbers

The probably most important step of abstraction in the history of mathematics: “3 apples” −→ “3”

Julia Goedecke (Newnham) Universal Properties 23/02/2016 5 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Numbers

The probably most important step of abstraction in the history of mathematics: “3 apples” −→ “3” After that also (not necessarily in this order) negative numbers (abstraction of debt?) rational numbers (abstraction of proportions) real numbers (abstraction of lengths)

Julia Goedecke (Newnham) Universal Properties 23/02/2016 5 / 30 Equivalence relations Study equality, congruence (mod n) and “having same image under a function”. Isolate: reﬂexivity, symmetry, transitivity. Deﬁne equivalence relation. Work with the abstract idea rather than one example .....

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction More examples

Groups Addition in Z, “clock” addition (mod n) and composing symmetries have similar properties. Isolate the properties. Deﬁne an abstract group. Get lots more examples, and a whole area of mathematics.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 6 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction More examples

Equivalence relations Study equality, congruence (mod n) and “having same image under a function”. Isolate: reﬂexivity, symmetry, transitivity. Deﬁne equivalence relation. Work with the abstract idea rather than one example .....

Julia Goedecke (Newnham) Universal Properties 23/02/2016 6 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction One more level of abstraction

We notice throughout our studies that certain objects come with special maps:

objects “structure preserving” maps sets functions groups group homomorphisms rings ring homomorphisms modules/vector spaces linear maps topological spaces continuous maps

Julia Goedecke (Newnham) Universal Properties 23/02/2016 7 / 30 We can compose them:

A −→ B −→ C

There is an identity:

1A f f f 1B A ,2A ,2B = A ,2B = A ,2B ,2B

Composition is associative: (h◦g)◦f = h◦(g◦f )

f g h A ,2B ,2C ,2D

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction One more level of abstraction

What do they have in common?

Julia Goedecke (Newnham) Universal Properties 23/02/2016 8 / 30 There is an identity:

1A f f f 1B A ,2A ,2B = A ,2B = A ,2B ,2B

Composition is associative: (h◦g)◦f = h◦(g◦f )

f g h A ,2B ,2C ,2D

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction One more level of abstraction

What do they have in common? We can compose them:

A −→ B −→ C

Julia Goedecke (Newnham) Universal Properties 23/02/2016 8 / 30 Composition is associative: (h◦g)◦f = h◦(g◦f )

f g h A ,2B ,2C ,2D

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction One more level of abstraction

What do they have in common? We can compose them:

A −→ B −→ C

There is an identity:

1A f f f 1B A ,2A ,2B = A ,2B = A ,2B ,2B

Julia Goedecke (Newnham) Universal Properties 23/02/2016 8 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction One more level of abstraction

What do they have in common? We can compose them:

A −→ B −→ C

There is an identity:

1A f f f 1B A ,2A ,2B = A ,2B = A ,2B ,2B

Composition is associative: (h◦g)◦f = h◦(g◦f )

f g h A ,2B ,2C ,2D

Julia Goedecke (Newnham) Universal Properties 23/02/2016 8 / 30 for each A ∈ obC, a morphism1 A : A −→ A, the identity, for each tripel A, B, C ∈ obC, a composition

◦ : Hom(A, B) × Hom(B, C)−→ Hom(A, C) (f , g)7−→ g◦f

such that the following axioms hold:

1 Identity: For f : A −→ B we have f ◦1A = f = 1B◦f . 2 Associativity: For f : A −→ B, g : B −→ C and h: C −→ D we have h◦(g◦f ) = (h◦g)◦f .

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Deﬁnition of a category

A category C consists of a collection obC of objects A, B, C,... and for each pair of objects A, B ∈ obC, a collection C(A, B) = HomC(A, B) of morphisms f : A −→ B, equipped with

Julia Goedecke (Newnham) Universal Properties 23/02/2016 9 / 30 1 Identity: For f : A −→ B we have f ◦1A = f = 1B◦f . 2 Associativity: For f : A −→ B, g : B −→ C and h: C −→ D we have h◦(g◦f ) = (h◦g)◦f .

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Deﬁnition of a category

A category C consists of a collection obC of objects A, B, C,... and for each pair of objects A, B ∈ obC, a collection C(A, B) = HomC(A, B) of morphisms f : A −→ B, equipped with

for each A ∈ obC, a morphism1 A : A −→ A, the identity, for each tripel A, B, C ∈ obC, a composition

◦ : Hom(A, B) × Hom(B, C)−→ Hom(A, C) (f , g)7−→ g◦f

such that the following axioms hold:

Julia Goedecke (Newnham) Universal Properties 23/02/2016 9 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Deﬁnition of a category

A category C consists of a collection obC of objects A, B, C,... and for each pair of objects A, B ∈ obC, a collection C(A, B) = HomC(A, B) of morphisms f : A −→ B, equipped with

for each A ∈ obC, a morphism1 A : A −→ A, the identity, for each tripel A, B, C ∈ obC, a composition

◦ : Hom(A, B) × Hom(B, C)−→ Hom(A, C) (f , g)7−→ g◦f

such that the following axioms hold:

1 Identity: For f : A −→ B we have f ◦1A = f = 1B◦f . 2 Associativity: For f : A −→ B, g : B −→ C and h: C −→ D we have h◦(g◦f ) = (h◦g)◦f .

Julia Goedecke (Newnham) Universal Properties 23/02/2016 9 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction What is Category Theory?

One more level of abstraction. addition and symmetries of polyhedra −→ groups equality and congruence −→ equivalence relations integers −→ ring theory Category Theory is “mathematics about mathematics”. sets, groups, vectorspaces etc. −→ categories A language for mathematicians. A way of thinking.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 10 / 30 Motto of category theory We want to really understand how and why things work, so that we can present them in a way which makes everything “look obvious”.

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Categorical point of view

In category theory: We are not only interested in objects (such as sets, groups, ...), but how different objects of the same kind relate to each other. We are interested in global structures and connections.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 11 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Categorical point of view

Motto of category theory We want to really understand how and why things work, so that we can present them in a way which makes everything “look obvious”.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 11 / 30 A group G is a one-object category with the group elements as morphisms: e ∈ G is identity morphism. group multiplication is composition. A poset P is a category: The elements of P are the objects. Hom(x, y) has one element if x ≤ y, empty otherwise. Reﬂexivity gives identities. Transitivity gives composition.

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Examples of categories

Any collection of sets with a certain structure and structure-preserving maps will form a category. But also:

Julia Goedecke (Newnham) Universal Properties 23/02/2016 12 / 30 A poset P is a category: The elements of P are the objects. Hom(x, y) has one element if x ≤ y, empty otherwise. Reﬂexivity gives identities. Transitivity gives composition.

Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Examples of categories

Julia Goedecke (Newnham) Universal Properties 23/02/2016 12 / 30 Category Theory Maths is Abstraction Universal Properties Category Theory: more abstraction Examples of categories

Any collection of sets with a certain structure and structure-preserving maps will form a category. But also: A group G is a one-object category with the group elements as morphisms: e ∈ G is identity morphism. group multiplication is composition. A poset P is a category: The elements of P are the objects. Hom(x, y) has one element if x ≤ y, empty otherwise. Reﬂexivity gives identities. Transitivity gives composition.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 12 / 30 Category Theory Within one category Universal Properties Mixing categories

1 Category Theory Maths is Abstraction Category Theory: more abstraction

2 Universal Properties Within one category Mixing categories

Julia Goedecke (Newnham) Universal Properties 23/02/2016 13 / 30 Note: could be unique map X −→ Y or Y −→ X.

Category Theory Within one category Universal Properties Mixing categories Universal Property Template

Template X has property P, and if Y also has property P, then there is a unique map between X and Y which “ﬁts with the property P”.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 14 / 30 Category Theory Within one category Universal Properties Mixing categories Universal Property Template

Template X has property P, and if Y also has property P, then there is a unique map between X and Y which “ﬁts with the property P”.

Note: could be unique map X −→ Y or Y −→ X.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 14 / 30 Deﬁnition An object T ∈ obC is called terminal object when there is, for every A ∈ obC, a unique morphism A −→ T in C.

Examples Sets X: exactly one function X −→ {∗}. Groups G: exactly one group hom G −→ 0 = {e}. Vector spaces V : exactly one linear map V −→ 0. Top. spaces X: exactly one continuous map X −→ {∗}.

Category Theory Within one category Universal Properties Mixing categories Terminal objects

First example: property P = “is an object” (“empty property”).

Julia Goedecke (Newnham) Universal Properties 23/02/2016 15 / 30 Examples Sets X: exactly one function X −→ {∗}. Groups G: exactly one group hom G −→ 0 = {e}. Vector spaces V : exactly one linear map V −→ 0. Top. spaces X: exactly one continuous map X −→ {∗}.

Category Theory Within one category Universal Properties Mixing categories Terminal objects

First example: property P = “is an object” (“empty property”). Deﬁnition An object T ∈ obC is called terminal object when there is, for every A ∈ obC, a unique morphism A −→ T in C.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 15 / 30 Category Theory Within one category Universal Properties Mixing categories Terminal objects

First example: property P = “is an object” (“empty property”). Deﬁnition An object T ∈ obC is called terminal object when there is, for every A ∈ obC, a unique morphism A −→ T in C.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 15 / 30 Groups G: exactly one group hom0 −→ G. Vector spaces V : exactly one linear map0 −→ V . Rings R: exactly one ring homomorphism Z −→ R. Sets X: exactly one function ∅ −→ X. Topological spaces: also ∅.

Examples

Category Theory Within one category Universal Properties Mixing categories Initial objects

P = “is an object”, but “unique arrow from” rather than “unique arrow to”. Deﬁnition An object I ∈ obC is called initial object when there is, for every A ∈ obC, a unique morphism I −→ A in the category C.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 16 / 30 Sets X: exactly one function ∅ −→ X. Topological spaces: also ∅.

Category Theory Within one category Universal Properties Mixing categories Initial objects

Examples Groups G: exactly one group hom0 −→ G. Vector spaces V : exactly one linear map0 −→ V . Rings R: exactly one ring homomorphism Z −→ R.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 16 / 30 Topological spaces: also ∅.

Category Theory Within one category Universal Properties Mixing categories Initial objects

Examples Groups G: exactly one group hom0 −→ G. Vector spaces V : exactly one linear map0 −→ V . Rings R: exactly one ring homomorphism Z −→ R. Sets X: exactly one function ∅ −→ X.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 16 / 30 Category Theory Within one category Universal Properties Mixing categories Initial objects

Examples Groups G: exactly one group hom0 −→ G. Vector spaces V : exactly one linear map0 −→ V . Rings R: exactly one ring homomorphism Z −→ R. Sets X: exactly one function ∅ −→ X. Topological spaces: also ∅.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 16 / 30 cartesian product with the product topology.

Topological spaces:

∃!h which satisﬁes π1◦h = f and π2◦h = g.

Examples Sets: cartesian product A × B = {(a, b) | a ∈ A, b ∈ B}. Groups: cartesian product with pointwise group structure.

Category Theory Within one category Universal Properties Mixing categories Products

Universal property of a product

A × B

π2 π1 ABs{ '

Julia Goedecke (Newnham) Universal Properties 23/02/2016 17 / 30 cartesian product with the product topology.

Topological spaces:

Examples Sets: cartesian product A × B = {(a, b) | a ∈ A, b ∈ B}. Groups: cartesian product with pointwise group structure.

Category Theory Within one category Universal Properties Mixing categories Products

Universal property of a product

∃!h C ,2 A × B f π2 π1 g zÔ ABsz $,'

∃!h which satisﬁes π1◦h = f and π2◦h = g.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 17 / 30 cartesian product with the product topology.

Topological spaces:

Category Theory Within one category Universal Properties Mixing categories Products

Universal property of a product

∃!h C ,2 A × B f π2 π1 g zÔ ABsz $,'

∃!h which satisﬁes π1◦h = f and π2◦h = g.

Examples Sets: cartesian product A × B = {(a, b) | a ∈ A, b ∈ B}. Groups: cartesian product with pointwise group structure.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 17 / 30 cartesian product with the product topology.

Topological spaces:

Category Theory Within one category Universal Properties Mixing categories Products

Universal property of a product

∃!h C ,2 A × B f π2 π1 g zÔ ABsz $,'

∃!h which satisﬁes π1◦h = f and π2◦h = g.

Examples Sets: cartesian product A × B = {(a, b) | a ∈ A, b ∈ B}. Groups: cartesian product with pointwise group structure.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 17 / 30 cartesian product with the product topology.

Category Theory Within one category Universal Properties Mixing categories Products

Universal property of a product

∃!h C ,2 A × B f π2 π1 g zÔ ABsz $,'

∃!h which satisﬁes π1◦h = f and π2◦h = g.

Examples Sets: cartesian product A × B = {(a, b) | a ∈ A, b ∈ B}. Groups: cartesian product with pointwise group structure. Topological spaces:

Julia Goedecke (Newnham) Universal Properties 23/02/2016 17 / 30 Category Theory Within one category Universal Properties Mixing categories Products

Universal property of a product

∃!h C ,2 A × B f π2 π1 g zÔ ABsz $,'

∃!h which satisﬁes π1◦h = f and π2◦h = g.

Examples Sets: cartesian product A × B = {(a, b) | a ∈ A, b ∈ B}. Groups: cartesian product with pointwise group structure. Topological spaces: cartesian product with the product topology.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 17 / 30 ∃!h s.t. h◦ι1 = f , h◦ι2 = g

Examples disjoint union of sets A ` B. disjoint union of topological spaces. free product of groups G ∗ H. (external) direct sum of modules M ⊕ N = M × N.

Category Theory Within one category Universal Properties Mixing categories Coproducts

Universal property of a coproduct

A B

ι2 ι1 ' s{ A + B

Julia Goedecke (Newnham) Universal Properties 23/02/2016 18 / 30 Examples disjoint union of sets A ` B. disjoint union of topological spaces. free product of groups G ∗ H. (external) direct sum of modules M ⊕ N = M × N.

Category Theory Within one category Universal Properties Mixing categories Coproducts

Universal property of a coproduct

A B f ι2 ι1 g ' sz zÔ A + B ,2$, C ∃!h

∃!h s.t. h◦ι1 = f , h◦ι2 = g

Julia Goedecke (Newnham) Universal Properties 23/02/2016 18 / 30 Category Theory Within one category Universal Properties Mixing categories Coproducts

Universal property of a coproduct

A B f ι2 ι1 g ' sz zÔ A + B ,2$, C ∃!h

∃!h s.t. h◦ι1 = f , h◦ι2 = g

Examples disjoint union of sets A ` B. disjoint union of topological spaces. free product of groups G ∗ H. (external) direct sum of modules M ⊕ N = M × N.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 18 / 30 Category Theory Within one category Universal Properties Mixing categories A stranger example

Poset as category: Hom(x, y) has one element if x ≤ y, empty otherwise. Universal properties in a poset Terminal object is “top element” (if it exists). Initial object is “bottom element” (if it exists). Products are meets (e.g. in a powerset: intersection). Coproducts are joins (e.g. in a powerset: union).

Julia Goedecke (Newnham) Universal Properties 23/02/2016 19 / 30 ∃ unique f : X −→ Y and g : Y −→ X “commuting with P”. Then g◦f : X −→ X also “commutes with P”.

Identity1 X : X −→ X always “commutes with P”.

But have unique such: so1 X = g◦f .

Similarly1 Y = f ◦g. So X =∼ Y .

Category Theory Within one category Universal Properties Mixing categories Uniqueness

Any universal object is unique (up to iso) Suppose X, Y both universal for P.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 20 / 30 Then g◦f : X −→ X also “commutes with P”.

Identity1 X : X −→ X always “commutes with P”.

But have unique such: so1 X = g◦f .

Similarly1 Y = f ◦g. So X =∼ Y .

Category Theory Within one category Universal Properties Mixing categories Uniqueness

Any universal object is unique (up to iso) Suppose X, Y both universal for P. ∃ unique f : X −→ Y and g : Y −→ X “commuting with P”.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 20 / 30 Identity1 X : X −→ X always “commutes with P”.

But have unique such: so1 X = g◦f .

Similarly1 Y = f ◦g. So X =∼ Y .

Category Theory Within one category Universal Properties Mixing categories Uniqueness

Any universal object is unique (up to iso) Suppose X, Y both universal for P. ∃ unique f : X −→ Y and g : Y −→ X “commuting with P”. Then g◦f : X −→ X also “commutes with P”.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 20 / 30 But have unique such: so1 X = g◦f .

Similarly1 Y = f ◦g. So X =∼ Y .

Category Theory Within one category Universal Properties Mixing categories Uniqueness

Any universal object is unique (up to iso) Suppose X, Y both universal for P. ∃ unique f : X −→ Y and g : Y −→ X “commuting with P”. Then g◦f : X −→ X also “commutes with P”.

Identity1 X : X −→ X always “commutes with P”.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 20 / 30 Similarly1 Y = f ◦g. So X =∼ Y .

Category Theory Within one category Universal Properties Mixing categories Uniqueness

Any universal object is unique (up to iso) Suppose X, Y both universal for P. ∃ unique f : X −→ Y and g : Y −→ X “commuting with P”. Then g◦f : X −→ X also “commutes with P”.

Identity1 X : X −→ X always “commutes with P”.

But have unique such: so1 X = g◦f .

Julia Goedecke (Newnham) Universal Properties 23/02/2016 20 / 30 So X =∼ Y .

Category Theory Within one category Universal Properties Mixing categories Uniqueness

Any universal object is unique (up to iso) Suppose X, Y both universal for P. ∃ unique f : X −→ Y and g : Y −→ X “commuting with P”. Then g◦f : X −→ X also “commutes with P”.

Identity1 X : X −→ X always “commutes with P”.

But have unique such: so1 X = g◦f .

Similarly1 Y = f ◦g.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 20 / 30 Category Theory Within one category Universal Properties Mixing categories Uniqueness

Any universal object is unique (up to iso) Suppose X, Y both universal for P. ∃ unique f : X −→ Y and g : Y −→ X “commuting with P”. Then g◦f : X −→ X also “commutes with P”.

Identity1 X : X −→ X always “commutes with P”.

But have unique such: so1 X = g◦f .

Similarly1 Y = f ◦g. So X =∼ Y .

Julia Goedecke (Newnham) Universal Properties 23/02/2016 20 / 30 Coproduct is “opposite” of product

∃!h ∃!h C ,2 A × B C rl A + B D: ld :3 g\ f π2 f ι2 π1 g ι1 g zÔ ABsz $,' AB

Category Theory Within one category Universal Properties Mixing categories Turning around arrows

Initial is “opposite” of terminal Terminal T : for all A, ∃! map A −→ T . Initial I: for all A, ∃! map A ←− I.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 21 / 30 Category Theory Within one category Universal Properties Mixing categories Turning around arrows

Initial is “opposite” of terminal Terminal T : for all A, ∃! map A −→ T . Initial I: for all A, ∃! map A ←− I.

Coproduct is “opposite” of product

∃!h ∃!h C ,2 A × B C rl A + B D: ld :3 g\ f π2 f ι2 π1 g ι1 g zÔ ABsz $,' AB

Julia Goedecke (Newnham) Universal Properties 23/02/2016 21 / 30 Direct products Direct product is both product and coproduct. E.g. direct sum of modules (vector spaces, abelian groups...)

Category Theory Within one category Universal Properties Mixing categories Coinciding properties

Zero objects For groups and modules, initial = terminal. Deﬁne zero-object0 to be both initial and terminal. Gives at least one map between any two objects:

A −→ 0 −→ B

Julia Goedecke (Newnham) Universal Properties 23/02/2016 22 / 30 Category Theory Within one category Universal Properties Mixing categories Coinciding properties

Zero objects For groups and modules, initial = terminal. Deﬁne zero-object0 to be both initial and terminal. Gives at least one map between any two objects:

A −→ 0 −→ B

Direct products Direct product is both product and coproduct. E.g. direct sum of modules (vector spaces, abelian groups...)

Julia Goedecke (Newnham) Universal Properties 23/02/2016 22 / 30 In terms of elements K = {k ∈ A | f (k) = 0}, k the inclusion into A.

Category Theory Within one category Universal Properties Mixing categories Kernels

Universal property of a kernel

k f K ,2 ,2 A ,2 B LR :D 3; ∃!l g 0 C Kernel of f is universal map whose post-composition with f is zero.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 23 / 30 Category Theory Within one category Universal Properties Mixing categories Kernels

Universal property of a kernel

k f K ,2 ,2 A ,2 B LR :D 3; ∃!l g 0 C Kernel of f is universal map whose post-composition with f is zero.

In terms of elements K = {k ∈ A | f (k) = 0}, k the inclusion into A.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 23 / 30 In modules/vector spaces/abelian groups Q = B/Im(f ) = {b + Im(f )}, q the quotient map.

A ,2 Im(f ) ,2 ,2 B ,2 B/Im(f )

Category Theory Within one category Universal Properties Mixing categories Cokernels: “turn around the arrows”

Universal property of a cokernel

f q A ,2 B ,2 Q

g ∃!p 0 $ #+ C Cokernel of f is universal map whose pre-composition with f is zero.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 24 / 30 Category Theory Within one category Universal Properties Mixing categories Cokernels: “turn around the arrows”

Universal property of a cokernel

f q A ,2 B ,2 Q

g ∃!p 0 $ #+ C Cokernel of f is universal map whose pre-composition with f is zero.

In modules/vector spaces/abelian groups Q = B/Im(f ) = {b + Im(f )}, q the quotient map.

A ,2 Im(f ) ,2 ,2 B ,2 B/Im(f )

Julia Goedecke (Newnham) Universal Properties 23/02/2016 24 / 30 Construction Actual construction is complicated and slightly tedious. Working with universal property is often easier than with the elements.

Category Theory Within one category Universal Properties Mixing categories Tensor Product

Tensor Product of Vector Spaces/Modules ϕ V × W ,2 V ⊗ W

∃!h h !) U ϕ is universal bilinear map out of V × W , tensor product U ⊗ V “makes bilinear h into linear h”.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 25 / 30 Category Theory Within one category Universal Properties Mixing categories Tensor Product

Tensor Product of Vector Spaces/Modules ϕ V × W ,2 V ⊗ W

∃!h h !) U ϕ is universal bilinear map out of V × W , tensor product U ⊗ V “makes bilinear h into linear h”.

Construction Actual construction is complicated and slightly tedious. Working with universal property is often easier than with the elements.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 25 / 30 Construction ab G = G/[G, G] [G, G] is commutator: normal subgroup generated by all aba−1b−1.

Category Theory Within one category Universal Properties Mixing categories Abelianisation

Abelianisation of a group

G ,2 ab G

∃! & A Every group hom to an abelian group A factors uniquely through the abelianisation.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 26 / 30 Category Theory Within one category Universal Properties Mixing categories Abelianisation

Abelianisation of a group

G ,2 ab G

∃! & A Every group hom to an abelian group A factors uniquely through the abelianisation.

Construction ab G = G/[G, G] [G, G] is commutator: normal subgroup generated by all aba−1b−1.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 26 / 30 Construction F = {(a, b) ∈ R × R | b 6= 0}/ ∼ equivalence relation ∼ is (a, b) ∼ (c, d) iff ad = bc.

Category Theory Within one category Universal Properties Mixing categories Field of fractions

Field of fractions of an integral domain

R ,2 ,2 F % ∃! % K Every injective ring hom to a field K factors uniquely through the field of fractions. “Smallest field into which R can be embedded.”

Julia Goedecke (Newnham) Universal Properties 23/02/2016 27 / 30 Category Theory Within one category Universal Properties Mixing categories Field of fractions

Field of fractions of an integral domain

R ,2 ,2 F % ∃! % K Every injective ring hom to a field K factors uniquely through the field of fractions. “Smallest field into which R can be embedded.”

Construction F = {(a, b) ∈ R × R | b 6= 0}/ ∼ equivalence relation ∼ is (a, b) ∼ (c, d) iff ad = bc.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 27 / 30 Generalisation Abelianisation and Stone-Cechˇ compactiﬁcation are examples of adjunctions: very important concept in Category Theory.

Category Theory Within one category Universal Properties Mixing categories Stone-Cechˇ Compactiﬁcation

Compactiﬁcation of a topological space

X ,2 βX

∃! % K Every continuous map to a compact Hausdorff space K factors uniquely through the Stone-Cechˇ compactiﬁcation.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 28 / 30 Category Theory Within one category Universal Properties Mixing categories Stone-Cechˇ Compactiﬁcation

Compactiﬁcation of a topological space

X ,2 βX

∃! % K Every continuous map to a compact Hausdorff space K factors uniquely through the Stone-Cechˇ compactiﬁcation.

Generalisation Abelianisation and Stone-Cechˇ compactiﬁcation are examples of adjunctions: very important concept in Category Theory.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 28 / 30 Transferable: situations with different details may have same universal property: transfer ideas/proofs/... Functorial: deﬁning things via universal properties gives them good categorical properties (used all over maths). Useful: e.g. to show two objects are isomorphic, show they satisfy same universal property.

Category Theory Universal Properties Why bother?

Advantages of Universal Properties Tidyier: details may be messy, working with universal property can give clear and elegant proofs.

Julia Goedecke (Newnham) Universal Properties 23/02/2016 29 / 30 Functorial: deﬁning things via universal properties gives them good categorical properties (used all over maths). Useful: e.g. to show two objects are isomorphic, show they satisfy same universal property.

Category Theory Universal Properties Why bother?

Julia Goedecke (Newnham) Universal Properties 23/02/2016 29 / 30 Useful: e.g. to show two objects are isomorphic, show they satisfy same universal property.

Category Theory Universal Properties Why bother?

Advantages of Universal Properties Tidyier: details may be messy, working with universal property can give clear and elegant proofs. Transferable: situations with different details may have same universal property: transfer ideas/proofs/... Functorial: deﬁning things via universal properties gives them good categorical properties (used all over maths).

Julia Goedecke (Newnham) Universal Properties 23/02/2016 29 / 30 Category Theory Universal Properties Why bother?

Julia Goedecke (Newnham) Universal Properties 23/02/2016 29 / 30 Category Theory Universal Properties Thanks for listening!

Julia Goedecke (Newnham) Universal Properties 23/02/2016 30 / 30