nLab Introduction to Homotopy Type Theory

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Homotopy theory

This entry collects links related to the forthcoming book

Egbert Rijke:

Introduction to Homotopy Type Theory

Cambridge Studies in Advanced Mathematics,

Cambridge University Press

arXiv:2212.11082 (359 pages)

which introduces homotopy type theory in general and in particular Martin-Löf's dependent type theory, the Univalent Foundations for Mathematics and synthetic homotopy theory.

The book is based on a course taught by the author at Carnegie Mellon University in the spring semester of 2018:

course webpage

The original writeup of these course notes

pdf (2018 course notes), pdf (221 pages, but smaller margins)

contains considerably more material than retained in the above arXiv version.

Other versions:

pdf (2019 summer school notes) (134 pages)
GitHub repositories
pdf (2022 HoTTEST summer school) (478 pages)

Chapter I introduces the reader to Martin-Löf's dependent type theory. The fundamental concepts of type theory are explained without immediately jumping into the homotopy interpretation of type theory.

$\;\;\;1.$ Dependent Type Theory

$\;\;\;2.$ Dependent function types

$\;\;\;3.$ The natural numbers

$\;\;\;4.$ More inductive types

$\;\;\;5.$ Identity types

$\;\;\;6.$ Universes

$\;\;\;7.$ Modular arithmetic via the Curry-Howard isomorphism

$\;\;\;8.$ Decidability in elementary number theory

Chapter II is an exposition of the Univalent Foundations for Mathematics. This chapter gradually extends dependent type theory with function extensionality, propositional truncation, the univalence axiom, and the type theoretic replacement axiom.

$\;\;\;9.$ Equivalences

$\;\;\;10.$ Contractible types

$\;\;\;11.$ The fundamental theorem of identity types