nLab A1-cohesive homotopy type theory


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

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

syntax object language

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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts





𝔸 1\mathbb{A}^1-cohesive homotopy type theory, affine-cohesive homotopy type theory, or algebraic-cohesive homotopy type theory is a version of cohesive homotopy type theory which has an affine line type 𝔸 1\mathbb{A}^1 and the axiom of 𝔸 1 \mathbb{A}^1 -cohesion. 𝔸 1\mathbb{A}^1-cohesive homotopy type theory provides a synthetic foundation for 𝔸 1 \mathbb{A}^1 -homotopy theory, where the affine line is the standard affine line in the Nisnevich site of smooth schemes of finite type over a Noetherian scheme.

Motivic homotopy type theory

Mitchell Riley‘s bunched linear homotopy type theory [Riley (2022)] is a synthetic foundations for parameterized stable homotopy theory. This means that there should be an 𝔸 1\mathbb{A}^1-cohesive bunched linear homotopy type theory which behaves as a synthetic foundations for motivic homotopy theory.


Similarly to real-cohesive homotopy type theory, we assume a spatial type theory presented with crisp term judgments a::Aa::A. In addition, we also assume the spatial type theory has an affine line type 𝔸 1\mathbb{A}^1, and 𝔸 1\mathbb{A}^1-localizations 𝔸 1()\mathcal{L}_{\mathbb{A}^1}(-).

Given a type AA, let us define const A,𝔸 1:A(𝔸 1A)\mathrm{const}_{A, \mathbb{A}^1}:A \to (\mathbb{A}^1 \to A) to be the type of all constant functions in the affine line 𝔸 1\mathbb{A}^1:

δ const A,𝔸 1(a,r):const A,𝔸 1(a)(r)= Aa\delta_{\mathrm{const}_{A, \mathbb{A}^1}}(a, r):\mathrm{const}_{A, \mathbb{A}^1}(a)(r) =_A a

There is an equivalence const A,1:A(1A)\mathrm{const}_{A, 1}:A \simeq (1 \to A) between the type AA and the type of functions from the unit type 11 to AA. Given types BB and CC and a function F:(BA)(CA)F:(B \to A) \to (C \to A), type AA is FF-local if the function F:(BA)(CA)F:(B \to A) \to (C \to A) is an equivalence of types.

A crisp type Ξ|()A\Xi \vert () \vdash A is discrete if the function () :AA(-)_\flat:\flat A \to A is an equivalence of types.

The axiom of 𝔸 1\mathbb{A}^1-cohesion states that for the crisp affine line Ξ|()𝔸 1type\Xi \vert () \vdash \mathbb{A}^1 \; \mathrm{type}, given any crisp type Ξ|()Atype\Xi \vert () \vdash A \; \mathrm{type}, AA is discrete if and only if AA is (const A,1 1const A,𝔸 1)(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})-local.

This allows us to define discreteness for non-crisp types: a type AA is discrete if AA is (const A,1 1const A,𝔸 1)(\mathrm{const}_{A, 1}^{-1} \circ \mathrm{const}_{A, \mathbb{A}^1})-local.

The shape modality in 𝔸 1\mathbb{A}^1-cohesive homotopy type theory is then defined as the 𝔸 1\mathbb{A}^1-localization ʃ(A) 𝔸 1(A)\esh(A) \coloneqq \mathcal{L}_{\mathbb{A}^1}(A), which ensures that the shape of 𝔸 1\mathbb{A}^1 itself is a contractible type.

See also


For the presentation of the underlying spatial type theory used for 𝔸 1\mathbb{A}^1-cohesive homotopy type theory, see:

Last revised on November 16, 2022 at 04:22:15. See the history of this page for a list of all contributions to it.