This document provides an introduction to homotopy type theory, including: - Types represent objects, propositions, functions, and proofs. - Equalities are represented as paths between types, with properties like reflexivity and transitivity. - Dependent types allow the output of a function to depend on the input. - Identity types represent proofs of equality between elements of a type. - Function extensionality and univalence are axioms in homotopy type theory.

1. A very brief introduction to
Homotopy Type Theory

2. The Institute for Advanced Study in Princeton, in my opinion, has ruined more good scientists than any institution has
created, judged by what they did before they came and judged by what they did after. Not that they weren't good
afterwards, but they were superb before they got there and were only good afterwards.
-- Richard Hamming

3. Vladimir Voevodsky
• Cutting-edge Math so complex probability
of mistake in any given proof very high
• Automated proof checking the present
and future of Math
• Video:
Univalent Foundations: New
Foundations of Mathematics
http://video.ias.edu/node/6395

4. Frege  Martin-Löf  Voevodsky
• Gottlob Frege, active 1879 – 1923
works in logic, foundations of arithmetic, and philosophy
• Per Martin-Löf
An Intuitionistic Theory of Types, 1975
• Vladimir Voevodsky
A very short note on the homotopy λ-calculus, 2006
• Institute for Advanced Study
Homotopy Type Theory: Univalent Foundation of Mathematics, 2013
a.k.a. The HoTT Book

6. Every-thing has a type
• Objects are types
• Propositions are types
• Functions are types
• Proofs are types

7. A theory of rules and no axioms
(For our purposes. There are 2 axioms in higher homotopy type theory)
• “axioms” appear when we introduce concrete types
• The game is to manipulate concrete types and elements with the rules
• i.e. Type Theory provides an algebra over types

8. a:A ”a is of type A ”
• Type theory is a deductive system based on 2 forms of judgment
• ”a is of type A ” is the most basic form of judgment
• Construction of a proposition is a proof in intuitionistic mathematics

9. Equality has special cases
• Judgmental equality a :≡ b is a “witness”
• p : a =A b is a proposition (and of course a type)
• p-1 : b =A a is a different type (we will come back to this)

10. Universes and families
• Universe : a type whose elements are types
• U0 : U1 : U2 : …
• Every type belongs to some universe A : Ui
• A family of types: type B varying over a type A
B : A  U
indicates the universe U is the codomain
note this is a simple function signature

11. 0 : U the type that is not
• False, ⊥, bottom
• You cannot construct it
• Hence it does not exist. Anything can derive from it.
ex falso quodlibit

12. 1 : U unit
• True, ⊤, top, ()
• Always the same
one-valued logic
• C family of languages oddly names this type “void”

13. 2 : U boolean
• 2-valued logic
highest n-valued fully consistent logic
• Can be derived as a special case of coproduct / summation type

14. Infinite types
• E.g. Natural numbers
which have a special place in practical application
• Defined recursively

15. A  B function type
• Signature matters to define type
not the body of function

16. Exercise:
• Construct a theorem of function equivalence
( f = g )

17. Exercise:
• Construct a theorem of function equivalence
( f = g )
• Trick exercise…we will come back to this

18. A + B : U Coproduct type
• a.k.a. Summation type
F# discriminated union
• To construct the function A + B  C
requires the functions
A  C
B  C

19. A x B : U Product type
• Tuple, Cartesian product

20. Introducing types
• Formation rule
e.g. you can form A  B when A is a type and B is a type
• Introduction rule (a.k.a. constructor)
e.g. functions have one constructor, λ-abstraction
• Elimination rule
how to use elements of type, e.g. function application
• Uniqueness principle (optional, a.k.a. η-expansion)
unique maps into and out of type (e.g. coproduct)

21. Lots of other types
• E.g. Lists
which of course are defined recursively
• Not so important from the standpoint of theory
because we can construct them from what we have…

22. EXCEPT…

23. Π(x:A)B(x) Dependent Functions
• Type of output depends on element of input
not the type of input
hence, not just a “parameterized function”
• Normal function is a special case
resulting type family a constant type

24. Σ(x:A)B(x) Dependent pair type
• Second element depends on a dependently typed function on first
• Example:
1, 10
2, “I am a string”

25. Logical operations, represented in types
• True 1
• False 0
• A and B A x B
• A or B A + B
• If A then B A  B
• A if and only if B (A  B) x (B  A)
• Not A A  0

26. Predicate logic in type theory
• For all x
∀x corresponds to Π(x:A)P(x)
• There exists x
∃x corresponds to Σ(x:A)P(x)

27. Connection to computability
• Constructive logic confines itself to that which can be effectively
constructed by computation
• It does not include Law of Excluded Middle or Proof by Contradiction
• In general the tautologies of classical logic involving NOT cannot be
constructed

28. IdA(a,b) Indentity function
• Has special importance in type theory
IdA(a,b) type representing proposition of equality
p : a =A b
a = b (shorthand)
refl : Π(a:A)(a=Aa)reflexivity
• Homotopy Type Theory –
there is a path between equals
reflexivity is an infinitesimally short path from self to self
paths are types (inverse path is not the same type)

29. Fundamental “Mathy” Properties of Path
• Reflexivity
constant path, e.g. identity
• Symmetry
paths can be reversed (a =A b)  (b =A a)
• Transitivity
concatenation of paths

30. Operations on paths
• Identity (fundamental compositional element)
• Associativity
(p ∙ q) ∙ r = p ∙ (q ∙ r)
• Transitivity
p ∙ p-1 = Id(p)

31. …but equalities are proofs
• So paths are proofs p : x =A y
• The equivalence of 2 paths (proofs) is called a homotopy
• And equality of proofs can have proofs
p' : p =x=Ay q Paths between paths (i.e. equality of paths)
• And these are all types
• It’s turtles all the way down (equivalences of paths)

32. Exercise:
• Why do paths have direction?

33. Exercise:
• Why do paths have direction?
• A space X is a set of points with a topology
• x =A y can be seen as giving a point in X at each “moment in time”
• More intuitively, proofs are directional

34. The Fundamental Theorem of Type Theory
• Induction principal for identity types
• Similar to 2 step number theory induction (prove base case, prove a step)
• Except there is only one step to prove
• To prove a statement (or construct an object) that depends on path
identity,
p: x = A y
it suffices to prove (or construct) the special case where x and y are the
same. p is thus the reflexivity element.
• “by induction it suffices to assume…”

35. Axiom of function extensionality
• (f = g) ⋍ Π(x:A)(f(x) =B(x) g(x))
2 functions that are pointwise equal, would be equal
a path in a function space would be a continuous homotopy
…but basic type theory is insufficient to prove this
• Axiom: there is a certain function
happly : (f = g)  Π(x:A)(f(x) = B(x)g(x))

36. Exercise:
• Why is function extensionality an axiom?

37. Axiom of univalence
• Lemma: for types A,B : U, there is a certain function
idtoeqv : (A =U B)  (A ⋍ B)
• Axiom (univalence): for any A,B : U, the function idtoeqv is
an equivalence,
(A =U B) ⋍ (A ⋍ B)

38. More exercises (from the Hott Book):
• 2.4 – Define, by induction on n, a general notion of n-dimensional
path in a type A, simultaneously with the type of boundaries for such
paths.

39. More exercises (from the Hott Book):
• 1.15 – Show the indiscernibility of identicals follows from path
induction.

40. More exercises (from the Hott Book):
• 1.14 – Why do the induction principles for identity types not allow us
to construct a function f : Π(x:A) Π(p:x=x)(p= reflx)
with the defining equation f(x, reflx) :≡ reflreflx
?