Category Theory Primer for Beginners

A Category Theory primer, presented at the FPIndia Bangalore Sept 2025 Meetup

Transcript

1. CATEGORY THEORY PRIMER FOR BEGINNERS Anupam Jain FPIndia – Bangalore Meetup – Sep 2025 1 / 52
2. WHAT WE WILL ACCOMPLISH TODAY • Understand what category theory even is • Learn basics to build intuition. • Map concepts to functional programming • Do something useful with it 2 / 52
3. WHAT IS CATEGORY THEORY Theory of how to categorise things 3 / 52
4. focus on the things you really care about 4 / 52
5. Universe 5 / 52
6. Universe a b c 6 / 52
7. Universe a b c d e f g h 7 / 52
8. CATEGORY THEORY IS THE MATHEMATICS OF DEFINING CATEGORIES AND FOCUSING ON THEM Precisely, formally, define - • the operations that can be performed on things • the properties that we can access on things That’s what programming is 8 / 52
9. WHAT IS PROGRAMMING • Coming up with the algorithmic steps to accomplish something • Operations and properties • Do this. Do this. Check this. Do that. 9 / 52
10. BUT ALSO • Defining the structure on which we can do these ops • Some structure is usually provided • Some you create your own • Usually they are at different levels 10 / 52
11. THE STRUCTURE YOU CARE ABOUT CHANGES FOR DIFFERENT PARTS OF A PROGRAM • Dog.bark • Person.converse • LivingThing.makeSound Levels may change 11 / 52
12. BUT STRUCTURE DOESN’T CHANGE IN ISOLATION If you were focusing on LivingThings, perhaps you divide the world into living and non-living. cat dog whale chair car sun 12 / 52
13. DECIDING WHAT YOU WANT TO FOCUS ON WILL TELL YOU HOW TO SLICE UP THE WORLD • German Shepherd, Golden Retriever, Pug, ... • My German Shepherd, Your Golden Retriever, ... • Me, You, Them, ... 13 / 52
14. SO WHAT IS CATEGORY THEORY? The Big Idea Category theory is a very abstract branch of mathematics, concerned with: • Objects and their context • Their relationships with other objects • And how we can derive properties and operations on them • It gives us a framework to manage this complexity • Allows us to analyse the connections between multiple groups and levels at the same time • It’s very general, so applies in a lot of situations 14 / 52
15. CATEGORY THEORY IS VERY SIMPLE • However, extremely simple things, can be the very complicated • Because simple things generalise to a lot of usecases 15 / 52
16. SO IT’S IMPORTANT TO REMEMBER • Maths is made up and the points don’t matter • It’s supposed to be fun. emergent complexity • The human brain is good at this stuff. visual, patterns 16 / 52
17. A CATEGORY FROM FIRST PRINCIPLES a b c d e f 17 / 52
18. A CATEGORY AT ITS CORE a b c d e f 18 / 52
19. A CATEGORY AT ITS CORE a b c d e f 19 / 52
20. A PROPERTY! • a ∼ b ∧ b ∼ c ⇒ a ∼ c • Transitivity a b c 20 / 52
21. A PROPERTY! • a ∼ b ∧ b ∼ c ⇒ a ∼ c • Transitivity a b c 20 / 52
22. A PROPERTY! • a ∼ a • Reflexivity a a 21 / 52
23. A PROPERTY! • a ∼ a • Reflexivity a a 21 / 52
24. MORE STRUCTURE How do we represent relations in programming? 1. A common way to relate A and B is via a predicate f(A,B) 10 11 12 ≤ ≤ ≤ 22 / 52
25. MORE STRUCTURE How do we represent relations in programming? 1. A common way to relate A and B is via a predicate f(A,B) 10 11 12 ≤ ≤ ≤ ≤ 23 / 52
26. MORE STRUCTURE 1. Perhaps a more useful way to relate A and B is via a function A → B 10 11 12 +1 +1 24 / 52
27. MORE STRUCTURE 1. Perhaps a more useful way to relate A and B is via a function A → B 10 11 12 +1 +1 +1? 25 / 52
28. MORE STRUCTURE 1. Perhaps a more useful way to relate A and B is via a function A → B 10 11 12 +1 +1 +1? +1? 26 / 52
29. PROBLEM • Seems like we have lost associativity • And reflexivity • But remember the part about it all being made up? • We can just make up new structure that fits! 27 / 52
30. PROBLEM • Seems like we have lost associativity • And reflexivity • But remember the part about it all being made up? • We can just make up new structure that fits! 27 / 52
31. PROBLEM • Seems like we have lost associativity • And reflexivity • But remember the part about it all being made up? • We can just make up new structure that fits! 27 / 52
32. COMPOSITION OF FUNCTIONS 1. Composition of two functions f and g is written f ◦ g 2. Pronounced after 3. Composition of +1 and +1 is +1 ◦ +1 = +2 28 / 52
33. THROW IT IN THE MIX! Axiom 1: Rules of Composition If you have a f : A → B and g : B → C, there must be an arrow g ◦ f : A → C. A f → B g → C And composition must be associative. If f : A → B, g : B → C, and h : C → D: h ◦ (g ◦ f) = (h ◦ g) ◦ f A f → B g → C h → D 29 / 52
34. IDENTITY FUNCTIONS 1. An identity function is one that leaves its argument unchanged 2. The identity function for integers could be +0 30 / 52
35. THROW IT IN THE MIX! Axiom 2: Identity For every object A, there must exist a special identity morphism, idA : A → A. A idA→ A Or more commonly: A idA This identity morphism acts as a neutral element for composition. For any f : A → B: f ◦ idA = f and idB ◦ f = f 31 / 52
36. RECOVERY • Transitivity • In our structure we must have an arrow from 10 to 12, and that arrow must be the same as +1 ◦ +1 = +2, 10 11 12 +1 +1 +2 32 / 52
37. RECOVERY • Reflexivity • In our structure we must have an arrow from 10 to 10, and 11 to 11, and so on, • And that arrow must act as the identity element for function composition 10 11 12 +1 +1 +2 id10 33 / 52
38. A CATEGORY AT ITS CORE A category C is - 1. A bunch of Objects: denoted with capital letters A,B,C,... 2. A bunch of Arrows: connections between objects. Denoted with lower case letters f,g,h,... • A • B • C • D • E • F • G • H i j f g h k l 34 / 52
39. THE CORE COMPONENTS: MORPHISMS • The Arrows are more formally called Morphisms. • Categories focus more on morphisms rather than objects. • A morphism represents a relationship between any two objects A and B in a category. • There could be any number of distinct morphisms between any two objects. • A morphism from A to B is written as f : A → B. 35 / 52
40. THE CORE COMPONENTS: MORPHISMS We can represent objects and morphisms diagramatically as follows: A f → B Morphisms compose. The composition is denoted by the symbol (◦). So the composition of g after f is written as: g ◦ f Diagramatically: A B C f g 36 / 52
41. THE RULES: AXIOMS OF A CATEGORY (1/2) Morphisms must obey two rules (axioms): Axiom 1: Rules of Composition If you have a f : A → B and g : B → C, there must be an arrow g ◦ f : A → C. A f → B g → C And composition must be associative. If f : A → B, g : B → C, and h : C → D: h ◦ (g ◦ f) = (h ◦ g) ◦ f A f → B g → C h → D 37 / 52
42. THE RULES: AXIOMS OF A CATEGORY (2/2) Axiom 2: Identity For every object A, there must exist a special identity morphism, idA : A → A. A idA→ A Or more commonly: A idA This identity morphism acts as a neutral element for composition. For any f : A → B: f ◦ idA = f and idB ◦ f = f 38 / 52
43. EXAMPLE: EMPTY CATEGORY (0) • The empty category, denoted as 0, is the unique category with: • No objects. • No morphisms. • It vacuously satisfies all the category axioms because there is nothing to check! 39 / 52
44. EXAMPLE: THE TERMINAL CATEGORY (1) • The terminal or final category, denoted as 1, consists of: • One object, let’s call it •. • One morphism, which must be the identity morphism id• : • → •. • id• 40 / 52
45. EXAMPLE: THE TERMINAL CATEGORY (1) • The terminal or final category, denoted as 1, consists of: • One object, let’s call it •. • One morphism, which must be the identity morphism id• : • → •. • id• 40 / 52
46. EXAMPLE: THE TERMINAL CATEGORY (1) • The terminal or final category, denoted as 1, consists of: • One object, let’s call it •. • One morphism, which must be the identity morphism id• : • → •. • id• 40 / 52
47. EXAMPLE: TWO CATEGORY Let’s define a category with just two objects, A and B. • Objects: {A,B} • Morphisms: {idA,idB,f : A → B,g : A → B} A f,g→ B • Composition is not defined for f and g since their domains and codomains don’t align. The identity morphisms are also omitted from the diagram for clarity. • All axioms hold! 41 / 52
48. EXAMPLE: THE CATEGORY SET This is the most common example of a category. • Objects: All sets. (e.g., {1,2,3}, R, {cat,dog}) • Morphisms: All functions between sets. • Composition: Standard function composition. • Identity: The identity function for each set, idA (x) = x. 42 / 52
49. EXAMPLES: PREORDER CATEGORY Any preordered set (P,≤) can be viewed as a thin category (at most one morphism between any two objects). • Objects: The elements of the set P. • Morphisms: A unique morphism exists from x to y if and only if x ≤ y. ∣Hom(x, y)∣ ≤ 1 for all x, y ∈ P • Composition: If x ≤ y and y ≤ z, the composite morphism exists because transitivity ensures x ≤ z. • Identity: For any object x, the identity morphism exists because reflexivity ensures x ≤ x. 43 / 52
50. EXAMPLES: PREORDER CATEGORY Any preordered set (P,≤) can be viewed as a thin category (at most one morphism between any two objects). • Objects: The elements of the set P. • Morphisms: A unique morphism exists from x to y if and only if x ≤ y. ∣Hom(x, y)∣ ≤ 1 for all x, y ∈ P • Composition: If x ≤ y and y ≤ z, the composite morphism exists because transitivity ensures x ≤ z. • Identity: For any object x, the identity morphism exists because reflexivity ensures x ≤ x. 43 / 52
51. EXAMPLES: PREORDER CATEGORY Any preordered set (P,≤) can be viewed as a thin category (at most one morphism between any two objects). • Objects: The elements of the set P. • Morphisms: A unique morphism exists from x to y if and only if x ≤ y. ∣Hom(x, y)∣ ≤ 1 for all x, y ∈ P • Composition: If x ≤ y and y ≤ z, the composite morphism exists because transitivity ensures x ≤ z. • Identity: For any object x, the identity morphism exists because reflexivity ensures x ≤ x. 43 / 52
52. OBJECT IDENTITY Is this a pipe? 44 / 52
53. LET’S IDENTIFY TWO OBJECTS Initial Object (I) An object I is initial if for every object X in the category, there exists a unique morphism from I to X. I X Y Z Terminal Object (T) An object T is terminal if for every object X in the category, there exists a unique morphism from X to T. T X Z Y 45 / 52
54. PRODUCT OF OBJECTS Definition Given two objects A and B, their product 1. Is written A × B 2. Has two morphisms (projections): – π1 : A × B → A – π2 : A × B → B A × B A π1 π2 46 / 52
55. PRODUCT OF OBJECTS Construction For any other object Z with morphisms f : Z → A and g : Z → B, there exists a unique morphism h : Z → A × B such that the diagram commutes. π1 ◦ h = f and π2 ◦ h = g A × B A Z π1 π2 f g h 47 / 52
56. SUM OF OBJECTS Definition: Sum of A and B Given two objects A and B, their sum (or coproduct): 1. Is written A + B 2. Has two morphisms (injections): – ι1 : A → A + B – ι2 : B → A + B A + B A ι1 ι2 48 / 52
57. SUM OF OBJECTS Construction For any other object Z with morphisms f : A → Z and g : B → Z, there exists a unique morphism h : A + B → Z such that the diagram commutes. h ◦ ι1 = f and h ◦ ι2 = g A + B A Z ι1 ι2 f g h 49 / 52
58. EXERCISE Given the definitions of SUM, PRODUCT, INITIAL, and TERMINAL objects? PROVE INITIAL is an identity for SUM TERMINAL is an identity for PRODUCT 50 / 52
59. THANK YOU Questions? 51 / 52
60. 52 / 52