nLab A3-space


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


Higher algebra

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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
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




Sometimes we can equip a type with a certain structure, called an A 3A_3-algebra structure, allowing us to derive some nice properties about the type and 0-truncate it to form monoids.


In classical mathematics

An A 3A_3-space is a homotopy associative H-space (but no coherence is required of the associator).

An HH-monoid is a monoid internal to the classical homotopy category of topological spaces Ho(Top), or in the homotopy category Ho(Top) *Ho(Top)_* of pointed topological spaces, which has a unit up to homotopy.

In homotopy type theory

Both notions coincide in homotopy type theory. An A 3A_3-space or H-monoid consists of

  • A type AA,
  • A basepoint e:Ae:A
  • A binary operation μ:AAA\mu : A \to A \to A
  • A left unitor
    λ: (a:A)μ(e,a)=a\lambda:\prod_{(a:A)} \mu(e,a)=a
  • A right unitor
    ρ: (a:A)μ(a,e)=a\rho:\prod_{(a:A)} \mu(a,e)=a
  • An asssociator
    α: (a:A) (b:A) (c:A)μ(μ(a,b),c)=μ(a,μ(b,c))\alpha:\prod_{(a:A)} \prod_{(b:A)} \prod_{(c:A)} \mu(\mu(a, b),c)=\mu(a,\mu(b,c))

Homomorphisms of A 3A_3-spaces

A homomorphism of A 3A_3-spaces between two A 3A_3-spaces AA and BB consists of

  • A function ϕ:AB\phi:A \to B such that

    • The basepoint is preserved
      ϕ(e A)=e B\phi(e_A) = e_B
    • The binary operation is preserved
      (a:A) (b:A)ϕ(μ A(a,b))=μ B(ϕ(a),ϕ(b))\prod_{(a:A)} \prod_{(b:A)} \phi(\mu_A(a, b)) = \mu_B(\phi(a),\phi(b))
  • A function

ϕ λ:( (a:A)μ(e A,a)=a)( (b:B)μ(e B,b)=b)\phi_\lambda:\left(\prod_{(a:A)} \mu(e_A,a)=a\right) \to \left(\prod_{(b:B)} \mu(e_B,b)=b\right)

such that the left unitor is preserved:

ϕ λ(λ A)=λ B\phi_\lambda(\lambda_A) = \lambda_B
  • A function
ϕ ρ:( (a:A)μ(a,e A)=a)( (b:B)μ(b,e B)=b)\phi_\rho:\left(\prod_{(a:A)} \mu(a, e_A)=a\right) \to \left(\prod_{(b:B)} \mu(b, e_B)=b\right)

such that the right unitor is preserved:

ϕ ρ(ρ A)=ρ B\phi_\rho(\rho_A) = \rho_B
  • A function
ϕ α:( (a 1:A) (a 2:A) (a 3:A)μ(μ(a 1,a 2),a 3)=μ(a 1,μ(a 2,a 3)))( (b 1:B) (b 2:B) (b 3:B)μ(μ(b 1,b 2),b 3)=μ(b 1,μ(b 2,b 3)))\phi_\alpha:\left(\prod_{(a_1:A)} \prod_{(a_2:A)} \prod_{(a_3:A)} \mu(\mu(a_1, a_2),a_3)=\mu(a_1,\mu(a_2,a_3))\right) \to \left(\prod_{(b_1:B)} \prod_{(b_2:B)} \prod_{(b_3:B)} \mu(\mu(b_1, b_2),b_3)=\mu(b_1,\mu(b_2,b_3))\right)

such that the associator is preserved:

ϕ α(α A)=α B\phi_\alpha(\alpha_A) = \alpha_B


  • The integers are an A 3A_3-space.

  • Every loop space is naturally an A 3A_3-space with path concatenation as the operation. In fact every loop space is a \infty-group.

  • The type of endofunctions AAA \to A has the structure of an A 3A_3-space, with basepoint id Aid_A, operation function composition.

  • A monoid is a 0-truncated A 3A_3-space.

See also

Created on June 9, 2022 at 10:21:02. See the history of this page for a list of all contributions to it.