Category Theory Course Notes

Ettore Forigo

ChapterΒ 1

1.1 Definition of Category

A category (1-category) π’ž consists of:

1 - A class Ob(π’ž) of objects of π’ž

2 - βˆ€X,Y ∈ Ob(π’ž).
a class Homπ’ž(X,Y ) of morphisms from X to Y

3 - βˆ€X ∈ Ob(π’ž).
an identity morphism idX ∈ Homπ’ž(X,X)

4 - βˆ€X,Y,Z ∈ Ob(π’ž).
a composition rule:

Homπ’ž(Y,Z) Γ— Homπ’ž(X,Y ) β†’ Homπ’ž(X,Z)
(g,f)↦→g ∘ f

Such that it satisfies the following axioms:

1 - Associativity of composition:

βˆ€X,Y,Z,W ∈ Ob(π’ž).
βˆ€f ∈ Homπ’ž(X,Y ),g ∈ Homπ’ž(Y,Z),h ∈ Homπ’ž(Z,W).
h ∘ (g ∘ f) = (h ∘ g) ∘ f

2 - Neutrality:

βˆ€X,Y ∈ Ob(π’ž).
βˆ€f ∈ Homπ’ž(X,Y ).
idY ∘ f = f ∧ f ∘ idX = f

1.2 Thin Categories

A category is thin if parallel morphisms are always the same, meaning that there is only one morphism between two objects.

In a thin category all morphisms are monic and epic.

1.3 Definition of Initial Object

An object I of a category π’ž is initial (dual of terminal, special case of a colimit (of a functor from π’ž to the empty category))
⇕
βˆ€X ∈ Ob(π’ž).
βˆƒ!f ∈ Homπ’ž(I,X)

1.4 Definition of Terminal Object

An object T of a category π’ž is terminal (dual of initial, special case of limit (of a functor from the empty category to π’ž))
⇕
βˆ€X ∈ Ob(π’ž).
βˆƒ!f ∈ Homπ’ž(X,T)

1.5 Definition of Monomorphism

A morphism f : X β†’ Y in a category π’ž (f ∈ Homπ’ž(X,Y )) is a monomorphism (or monic in π’ž) (dual of epimorphism)
⇕
βˆ€Z ∈ Ob(π’ž).βˆ€p,q ∈ Homπ’ž(Z,X).
f ∘ p = f ∘ q =β‡’ p = q

Example:
In Set monomorphisms are precisely the injective maps.

Monomorphisms β€œcan be cancelled” from the left.

1.6 Definition of Split Monomorphism

A split monomorphism (dual of split epi) is a morphism f : X β†’ Y such that there exists a morphism g : Y β†’ X such that:

g ∘ f = idX

Proposition: every split mono is a mono.
Proposition: in Set, every mono f : X β†’ Y where X is inhabited is a split mono, assuming LEM holds.

1.7 Definition of Epimorphism

A morphism f : X β†’ Y in a category π’ž (f ∈ Homπ’ž(X,Y )) is an epimorphism (or epic in π’ž) (dual of monomorphism)
⇕
βˆ€Z ∈ Ob(π’ž).βˆ€p,q ∈ Homπ’ž(Y,Z).
p ∘ f = q ∘ f =β‡’ p = q

Example:
In Set epimorphisms are precisely the surjective maps.

Epimorphisms β€œcan be cancelled” from the right.

1.8 Definition of Split Epimorphism

A split epimorphism (dual of split mono) is a morphism f : X β†’ Y such that there exists a morphism g : Y β†’ X such that:

f ∘ g = idY

Proposition: every split epi is an epi.
Proposition: in Set, every epi is a split epi ⇐⇒ assuming LEM holds.

1.9 Definition of Isomorphism

A morphism f : X β†’ Y in a category π’ž (f ∈ Homπ’ž(X,Y )) is an isomorphism
⇕
βˆƒg ∈ Homπ’ž(Y,X).
f ∘ g = idY ∧ g ∘ f = idX

idXβˆ€X ∈ Ob(π’ž) is always an isomorphisms for every category π’ž.

Objects X and Y in a category π’ž are isomorphic
⇕
there exists an isomorphism between X and Y (X∼ =Y )

In Set, if there exists an isomorphism between X and Y , X and Y are called eqinumerous.

1.10 Definition of Opposite Category

β€œThe mother of all dualities”

Let π’ž be a category. Then its opposite category π’žop is the following category:

- Ob(π’žop) : = Ob(π’ž)

- Homπ’žop(X,Y ) : = Homπ’ž(Y,X)

- identities and composition inherited from π’ž
idX ∈ Homπ’ž(X,X) = idXop ∈ Homπ’žop(X,X)
f ∘ g : = gop ∘ fop

Observations / Remarks:

- An object I of π’ž is initial in π’ž

⇕

I is terminal when regarded as an object of π’žop

- A morphism in π’ž is a monomorphism

⇕

it is an epimorphism in π’žop

1.11 Dualities

injective maps in Set (monomorphism in Set) ↔ surjective maps in Set (epimorphism in Set)

≀ ↔ β‰₯

∩ ↔ βˆͺ

{x} ↔ βˆ…

βŠ‚ ↔ quotient set

Γ— (cartesian product) ↔ disjoint union (tagged)

f ∘ g ↔ g ∘ f

1.12 Definition of Product

A product (dual of coproduct, special case of limit) of two objects X and Y in a category π’ž consists of:

- an object P of π’ž

- a morphism Ο€X : P β†’ X in π’ž

- a morphism Ο€Y : P β†’ Y in π’ž

such that for every object Q of π’ž together with morphisms Ο†X : Q β†’ X,Ο†Y : Q β†’ Y there is exactly one morphism Q β†’ P such that the following diagram commutes:

Ο†X = Ο€X∘!
Ο†Y = Ο€Y ∘!

Remarks:

- Ο€X and Ο€Y are called projection morphisms (also in limits).

- Products are always associative and commutative up to isomorphism.

- There is also the notion of the (co) product of zero, one, three, four, ... objects.

- The zero case of a product is just a terminal object, an object with exactly one morphism from each object.

1.13 Definition of Coproducts

A coproduct (dual of product, special case of colimit) of two objects X and Y in a category π’ž consists of:

- an object C of π’ž

- a morphism ΞΉX : X β†’ C in π’ž

- a morphism ΞΉY : Y β†’ C in π’ž

such that for every object D of π’ž together with morphisms Ο‡X : X β†’ D,Ο‡Y : Y β†’ D there is exactly one morphism C β†’ D which renders the following diagram commutative:

Ο‡X =! ∘ ΞΉX
Ο‡Y =! ∘ ΞΉY

Remarks:

- Products in π’žop are precisely coproducts in π’ž

- The zero case of a coproduct is the same as an initial object.

1.14 Definition of Functor

A (covariant) functor F : π’žβ†’π’Ÿ from a category π’ž to a category π’Ÿ consists of:

- an object F(X) ∈ Ob(π’Ÿ) for each object X ∈ Ob(π’ž)

- a morphism F(f) : F(X) β†’ F(Y ) in π’Ÿ for each morphism f : X β†’ Y in π’ž

such that:

- βˆ€X ∈ Ob(π’ž).F(idX) = idF(X)

- βˆ€X,Y,Z ∈ Ob(π’ž).βˆ€f : X β†’ Y βˆˆπ’ž,g : Y β†’ Z in π’ž.F(g ∘ f) = F(g) ∘ F(f)

Motto:
Functors β„β†’π’ž are ℐ-shaped diagrams in π’ž

Functors preserve commutative diagrams
Functors preserve isomorphisms

1.15 Definition of Contravariant Functor

A contravariant functor π’žβ†’π’Ÿ is a covariant functor π’žop β†’π’Ÿ

1.16 Definition of Identity Functor

The identity functor Idπ’ž on a category π’ž is the following functor:

Idπ’ž : π’žβ†’π’ž
X↦→X
f↦→f

1.17 Definition of Constant Functor

Let X0 be an object of a category π’ž.
The constant functor Idπ’ž on X0 is the following functor:

Idπ’ž : π’žβ†’π’ž
X↦→X
f↦→f

1.18 Forgetful Functors

A forgetful functor ’forgets’ or drops some or all of the input’s structure or properties ’before’ mapping to the output.

Examples:

- From vector space category to group category

- From vector space category to set category

- From abelian group category to group category

1.19 Definition of Discrete Category

The discrete category associated with a set X, written π’Ÿ(X), is a category containing all the objects of X as objects, and no morphisms between different objects, just the identity morphisms.

1.20 Definition of Induced Functors

Claim:
Any map between sets can be turned into a functor.

Let f : X β†’ Y be a map between sets.

Consider the discrete categories π’Ÿ(X),π’Ÿ(Y ).

Then f induces the following functor π’Ÿ(X) β†’ D(Y ):
x↦→f(x)
idx↦→idf(x)

1.21 Definition of Essentially Surjective Functor

A functor F : π’žβ†’π’Ÿ is essentially surjective iff:

βˆ€Y ∈ Ob(π’Ÿ).βˆƒX ∈ Ob(π’ž)|F(X)∼=Y

1.22 Definition of Faithful Functor

A functor F : π’žβ†’π’Ÿ is faithful iff:

βˆ€X,Y ∈ Ob(π’ž).
βˆ€f,g : X β†’ Y in π’ž
F(f) = F(g) = β‡’ f = g

Reformulation: iff

βˆ€X,Y ∈ Ob(π’ž).
Homπ’ž(X,Y ) β†’ Homπ’Ÿ(F(X),F(Y ))
f↦→F(f)

is injective.

1.23 Definition of Full Functor

A functor F : π’žβ†’π’Ÿ is full iff:

βˆ€X,Y ∈ Ob(π’ž).
βˆ€g : F(X) β†’ F(Y ) in π’Ÿ
βˆƒf : X β†’ Y in π’ž|F(f) = g

Reformulation: iff

βˆ€X,Y ∈ Ob(π’ž).
Homπ’ž(X,Y ) β†’ Homπ’Ÿ(F(X),F(Y ))
f↦→F(f)

is surjective.

1.24 Definition of Fully Faithful Functor

A functor is fully faithful iff it is full and faithful.

Reformulation: iff

βˆ€X,Y ∈ Ob(π’ž).
Homπ’ž(X,Y ) β†’ Homπ’Ÿ(F(X),F(Y ))
f↦→F(f)

is bijective.

1.25 Definition of Elementary Equivalence

An elementary equivalence is a fully faithful, essentially surjective functor.

1.26 Definition of Equivalence of Categories

Categories are called equivalent iff there is an elementary equivalence between them.

Remark:
Equivalent categories have exactly the same categorical properties.

1.27 Definition of Natural Transformation

A natural transformation Ξ· : F β‡’ G between two functors F,G : C β†’ D consists of:

- for each object X ∈ Ob(π’ž) a morphism Ξ·X : F(X) β†’ G(X) in π’Ÿ

such that for all morphisms f : X β†’ Y in π’ž, the naturality square commutes:

G(f) ∘ ηX = ηY ∘ F(f)

Motto:
Natural transformations are uniform families of morphisms.

1.28 Definition of Functor Category

Let π’ž,π’Ÿ be categories.
The functor category [π’ž,π’Ÿ] has:

- as objects: all functors π’žβ†’π’Ÿ

- as morphisms: Hom[π’ž,π’Ÿ](F,G) : = {h : F β‡’ G |h is a natural transformation }

- as identity: for the object F, the identity idF : F β‡’ F (idF )X : F(X) β†’ F(X)
given by idF(X)

- as composition rule:
(Ο‰ ∘ Ξ·)X : = Ο‰X ∘ Ξ·X

Ξ·X : F(X) β†’ G(X)
Ο‰X : G(X) β†’ H(X)
(Ο‰ ∘ Ξ·)X : F(X) β†’ H(X)

and Ο‰ ∘ Ξ· should be natural.

1.29 Definition of Small Category

A category π’ž is small when Ob(π’ž) is just a set and not a proper class.

1.30 Definition of Category of Categories

The 1-category of 1-categories, Cat has:

- as objects: all categories

- as morphisms: HomCat(π’ž,π’Ÿ) : = {F : π’ž β†’ π’Ÿ |F is a functor}

- as identities Idπ’ž (the identity functor)

- as composition rule:
F : π’žβ†’π’Ÿ
G : π’Ÿβ†’β„°

G ∘ F : π’žβ†’β„°
X↦→G(F(X))
f↦→G(F(f))

There are two issues with this definition:

- Size issue (in ZFC). (it’s too big, the objects don’t fit in a proper class?)
Remedies:

- just consider the category of small categories

- switch foundations

- It ignores natural transformations
Remedy:
Consider the 2-category of 1-categories

The 2-category of 1-categories has:

- as objects: all 1-categories

- as morphisms: functors

- as -2-morphisms / 2-cells: natural transformations

1.31 Definition of Cone

A cone (dual of cocone) of a diagram (functor) F : β„β†’π’ž in a category π’ž consists of:

- an object A of π’ž (the ”tip” of the cone)

- for each object X ∈ Ob(π’ž), a morphism Ο€X : A β†’ F(X)

such that for all morphisms f : X β†’ Y in ℐ, the triangle:

Ο€Y = Ο€X ∘ F(f)

commutes.

1.32 Definition of Cocone

A cocone (dual of cone) of a diagram (functor) F : β„β†’π’ž in a category π’ž consists of:

- an object A of π’ž (the ”tip” of the cocone)

- for each object X ∈ Ob(π’ž), a morphism Ο€X : F(X) β†’ A

such that for all morphisms f : X β†’ Y in ℐ, the triangle:

Ο€X = Ο€Y ∘ F(f)

commutes.

1.33 Definition of Morphism Between Cones

A morphism between a cone (A,(Ο€X)X) and a further cone (B,(Ο•X)X) of a diagram F : ℐ β†’π’ž consists of a morphism f : A β†’ B in π’ž such that:

Ο€X = Ο€Y ∘ f

commutes.

1.34 Definition of Limit

A limit (dual of colimit) of a diagram F : β„β†’π’ž is a terminal cone of F, that is, a terminal object in the category of of cones of cones of F.

Remark:
A terminal object of π’ž is the limit of the unique functor from the empty category to π’ž.

1.35 Definition of Colimit

A colimit (dual of limit) of a diagram F : β„β†’π’ž is an initial cocone of F.

Remark:
An initial object of π’ž is the colimit of the unique functor from the empty category to π’ž.

1.36 Definition of Equalizer of Two Set-Theoretic Maps

Let f,g : X β†’ Y . Then the equalizer of f and g is the following function:

Eq(f,g) = x ∈ X|f(x) = g(x)

1.37 Definition of Pullback

A pullback P (also called fiber product of the domains over the codomain) (dual of pushout) is the limit of a diagram consisting of two morphisms f : X β†’ Z and g : Y β†’ Z with a common codomain.

It comes equipped with two natural morphisms P β†’ X and P β†’ Y .

1.38 Definition of Pushout

A pushout P (also called fibered coproduct) (dual of pullback) is the colimit of a diagram consisting of two morphisms f : Z β†’ X and g : Z β†’ Y with a common domain.

It comes equipped with two morphisms X β†’ P and Y β†’ P.

1.39 Definition of Small Diagram

A small diagram in π’ž is a diagram β„β†’π’ž where ℐ is a small category.

1.40 Definition of Complete Cateogory

A category π’ž is complete (dual of cocomplete) iff every small diagram in π’ž has a limit (it has all small limits).

Assuming LEM, the only categories which have all limits or all colimits are (some) thin categories.

1.41 Definition of Cocomplete Category

A category π’ž is cocomplete (dual of complete) iff every small diagram in π’ž has a colimit (it has all small colimits).

π’ž complete β‡β‡’π’žop cocomplete.

1.42 Definition of Presheaf

A presheaf (plural presheaves) on a category π’ž is a functor π’žop β†’ Set

Motto:
we picture a presheaf F on π’ž as an β€œideal, fictional, object of π’žβ€ in that we know its relation to actual objects of π’ž

1.43 Definition of Λ† X

Xˆ (X hat) is a presheaf:

π’žop β†’ Set
T↦→Homπ’ž(T,X)

1.44 Definition of Representable Presheaf

A presheaf F : π’žop β†’ Set is representable iff:

βˆƒX ∈ Ob(π’ž) : F∼=XΛ†

1.45 Definition of Adjoint Functors

Let F : C β†’ D, G : D β†’ C

Then, F ⊣ G β€œF is left adjoint to G”
(or G ⊒ F (β€œG is right adjoint to F”))

iff for every object X ∈ Ob(π’ž),Y ∈ Ob(π’Ÿ) there is an isomorphism:

Homπ’Ÿ(F(X),Y )∼=Homπ’ž(X,G(Y ))

naturally in X and Y .

Every adjunction L ⊣ R gives rise to a monad:

The monad functor will be: M : = R ∘ L

The natural transformation:
Ξ· : Id β‡’ M

will be given by:
Ξ·X : X β†’ R(L(X))

which is in 1:1 correspondence with:
idRL(X) : RL(X) β†’ RL(X)

since:
Hom(LA,B)∼=Hom(A,RB)

which means that:
LA β†’ B

is in 1:1 correspondence with:
A β†’ RB

The natural transformation:
ΞΌ : M ∘ M β‡’ M

will be given by:
ΞΌX : RLRL(X) β†’ RL(X)

induced from:
LRL(X) β†’ L(X)

which is in 1:1 correspondence with:
idRL(X) : RL(X) β†’ RL(X)

Remark:
The monad axioms should also be checked.

1.46 Currying Adjunction

The β€œproduct-Hom adjunction” or currying adjunction is the following:

_ Γ— S ⊣ HomSet(S, _)

Hom Set(X Γ— S,Y )∼=HomSet(X,HomSet(S,Y ))

1.47 Adjunction of Logical Connectives

β€œβˆƒβ€ ⊣ β€œextending the context” ⊣ β€œβˆ€β€

The left adjunctions means that it is possible to freely convert between proofs of the following kind:

β€œAssume βˆƒx ∈ X : A(x) ... Hence B.” (βˆƒx ∈ X : A(x) ⊒ B)

and

β€œLet x ∈ X be arbitrary. Assume A(x) ... Hence B.” (A(x) ⊒x∈XB)

The right adjunction means that it is possible to freely convert between proofs of the following kind:

β€œLet x ∈ X be arbitrary. Assume A ... Hence B(x).” (A ⊒x∈XB(x))

and

β€œAssume A. ... Hence βˆ€x ∈ X : B(x).” (A ⊒ (βˆ€x ∈ X.B(x)))

1.48 Monoids

A monoid consists of:

- a set M

- an element e ∈ M

- an operation ∘ : M Γ— M β†’ M

such that:

- βˆ€x ∈ M.x ∘ e = x = e ∘ x

- βˆ€x,y,z ∈ M.(x ∘ y) ∘ z = x ∘ (y ∘ z)

1.49 Monoids Categorically

Equivalently, a monoid consists of:

- an object M

- a morphism 1 from a terminal object to every other object.

- a map M Γ— M β†’ M

such that certain diagrams commute.

1.50 Definition of Monoidal Category

A monoidal category (sometimes called tensor category) consists of:

- a category π’ž

- a functor βˆ— : π’žΓ—π’žβ†’π’ž

- an object 1 ∈ Ob(π’ž)

- natural isomorphisms:

- 1 βˆ— X∼=X

- X βˆ— 1∼ =X

- X βˆ— (Y βˆ— Z)∼=(X βˆ— Y ) βˆ— Z

such that certain coherence conditions are satisfied.

Remark:
In any monoidal category one can speak of monoid objects.

1.51 Definition of Monad

A monad over a category π’ž consists of:

- a functor M : π’žβ†’π’ž

- a natural transformation Ξ· : Idπ’žβ‡’ M

- a natural transformation ΞΌ : M ∘ M β‡’ M

such that certain diagrams commute.

Every monad is given rise to by an adjunction (always of a free and forgetful functor pair).
There are two ways of factorizing a monad into adjoint functors, one is the Kleisli category.

1.52 Definition of Kleisli Category

The Kleisli category π’žM of a monad M in a category π’ž is the following category:

- objects: objects of π’ž

- morphisms: Homπ’žM(X,Y ) : = Homπ’ž(X,M(Y ))

1.53 Definition of Cobordism Category

The category nCob (β€œthe cobordism category”) has:

- as objects (n βˆ’ 1)-dimensional oriented manifolds

- as morphisms: n-dimesional cobordisms between those

1.54 Definition of Category of Hilbert Spaces

Hilb is the category of Hilbert spaces (vector spaces with additional structure).

Hilbert spaces are important in quantum physics, because they can be used to model β€œslices” of spacetime.

1.55 Definition of Topological Quantum Field Theory

A topological quantum field theory (in spacetime dimension n) is a monoidal functor between the monoidal categories nCob and Hilb:

Z : nCob β†’ Hilb

Z maps each (n βˆ’ 1)-dimensional slice of n-dimensional spacetime to the Hilbert space modelling that slice, and Z maps a morphism X β†’ Y in nCob to the β€œpropagator” Z(X) β†’ Z(Y ).