Category Theory
Universal Properties
Universal Properties
A categorical look at undergraduate algebra and topology
Julia Goedecke
Newnham College
24 February 2017, Archimedeans
Julia Goedecke (Newnham) Universal Properties 24/02/2017 1 / 30
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Universal Properties. Universal Properties. A categorical look at undergraduate algebra and topology. Julia Goedecke. Newnham College.
Typology: Lecture notes
2021/2022
1 / 71
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Universal Properties and more Lecture notes Algebra in PDF only on Docsity!
Category Theory Universal Properties
Universal Properties
A categorical look at undergraduate algebra and topology
Julia Goedecke
Newnham College
24 February 2017, Archimedeans
Category Theory Universal Properties
(^1) Category Theory
Maths is Abstraction Category Theory: more abstraction
(^2) Universal Properties
Within one category Mixing categories
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.
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.
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.
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)
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.
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
Universal Properties Category Theory: more abstraction
One more level of abstraction
What do they have in common?
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:
A
(^1) A (^) , 2 A f^ ,^2 B = A f^ ,^2 B = A f^ ,^2 B
(^1) B (^) , 2 B
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:
A
(^1) A (^) , 2 A f^ ,^2 B = A f^ ,^2 B = A f^ ,^2 B
(^1) B (^) , 2 B
Composition is associative: (h◦g)◦f = h◦(g◦f )
A f^ ,^2 B
g (^) , 2 C h^ ,^2 D
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 morphism 1A : 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:
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 morphism 1A : 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 ◦ (^1) A = f = (^1) B ◦f. (^2) Associativity: For f : A −→ B, g : B −→ C and h : C −→ D we have h◦(g◦f ) = (h◦g)◦f.
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.
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.
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”.