nLab associative H-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

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




In dependent type theory, an associative H-space (cf. BCFR23) or A 3A_3-space is a naive translation of the notion of monoid. If the underlying type is an h-set (such as if the type theory is extensional), then it yields a correct notion of monoid. Otherwise, it is only a “partly-coherent” notion of “homotopy monoid”: a type-theoretic version of the notion of A 3A_3-space from homotopy theory.


Similar to the definition of H-spaces, there are coherent and non-coherent versions of associative H-spaces:

Non-coherent associative H-spaces

A non-coherent associative H-space or non-coherent A 3A_3-space 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: x:Aμ(e,x)=x\lambda:\prod_{x:A} \mu(e,x)=x
  • A right unitor
    ρ: x:Aμ(x,e)=x\rho:\prod_{x:A} \mu(x,e)=x
  • An asssociator
    α: x:A y:A z:Aμ(μ(x,y),z)=μ(x,μ(y,z))\alpha:\prod_{x:A} \prod_{y:A} \prod_{z:A} \mu(\mu(x, y),z)=\mu(x,\mu(y,z))

Coherent associative H-spaces

A coherent associative H-space or coherent A 3A_3-space is a non-coherent associative H-space AA which additionally has the coherence condition

μ λρ:λ(e)=ρ(e)\mu_{\lambda \rho}:\lambda(e) = \rho(e)

since λ(e)\lambda(e) and ρ(e)\rho(e) are elements of the identity type μ(e,e)=e\mu(e, e) = e.

Homomorphisms of associative H-spaces

There are also two different notions of homomorphisms between associative H-spaces, depending on whether the H-space is coherent or not.

Homomorphisms of non-coherent associative H-spaces

A homomorphism of non-coherent associative H-spaces between two non-coherent associative H-spaces AA and BB is a function ϕ:AB\phi:A \to B such that

  • The basepoint is preserved, i.e. there is a specified identification

    ϕ e:ϕ(e A)=e B\phi_e:\phi(e_A) = e_B
  • The binary operation is preserved, i.e. there is a specified dependent function

    ϕ μ: x:A y:Aϕ(μ A(x,y))=μ B(ϕ(x),ϕ(y))\phi_\mu:\prod_{x:A} \prod_{y:A} \phi(\mu_A(x, y)) = \mu_B(\phi(x),\phi(y))
  • The left unitor is preserved, i.e. there is a specified homotopy identifying the concatenation of identifications

    ϕ μ(e A,x):ϕ(μ A(e A,x))=μ B(ϕ(e A),ϕ(x))\phi_\mu(e_A, x):\phi(\mu_A(e_A,x)) = \mu_B(\phi(e_A),\phi(x))
    ap λy:B.μ B(y,ϕ(x))(ϕ e):μ B(ϕ(e A),ϕ(x))=μ B(e B,ϕ(x))\mathrm{ap}_{\lambda y:B.\mu_B(y,\phi(x))}(\phi_e):\mu_B(\phi(e_A),\phi(x)) = \mu_B(e_B,\phi(x))
    λ B(ϕ(x)):μ B(e B,ϕ(x))=ϕ(x)\lambda_B(\phi(x)):\mu_B(e_B,\phi(x)) = \phi(x)

    with the image ap ϕ(λ A(x)):ϕ(μ A(e A,x))=ϕ(x)\mathrm{ap}_{\phi}(\lambda_A(x)):\phi(\mu_A(e_A,x)) = \phi(x) of the left unitor of AA.

    ϕ λ: x:Aϕ μ(e A,x)ap λy:B.μ B(y,ϕ(x))(ϕ e)λ B(ϕ(x))=ap ϕ(λ A(x))\phi_\lambda:\prod_{x:A} \phi_\mu(e_A, x) \bullet \mathrm{ap}_{\lambda y:B.\mu_B(y,\phi(x))}(\phi_e) \bullet \lambda_B(\phi(x)) = \mathrm{ap}_{\phi}(\lambda_A(x))
  • Similarly, the right unitor is preserved, i.e. there is a specified homotopy identifying the concatenation of identifications

    ϕ μ(x,e A):ϕ(μ A(x,e A))=μ B(ϕ(x),ϕ(e A))\phi_\mu(x, e_A):\phi(\mu_A(x,e_A)) = \mu_B(\phi(x),\phi(e_A))
    ap λy:B.μ B(ϕ(x),y)(ϕ e):μ B(ϕ(x),ϕ(e A))=μ B(ϕ(x),e B)\mathrm{ap}_{\lambda y:B.\mu_B(\phi(x),y)}(\phi_e):\mu_B(\phi(x),\phi(e_A)) = \mu_B(\phi(x),e_B)
    ρ B(ϕ(x)):μ B(ϕ(x),e B)=ϕ(x)\rho_B(\phi(x)):\mu_B(\phi(x),e_B) = \phi(x)

    with the image ap ϕ(ρ A(x)):ϕ(μ A(x,e A))=ϕ(x)\mathrm{ap}_{\phi}(\rho_A(x)):\phi(\mu_A(x,e_A)) = \phi(x) of the right unitor of AA.

    ϕ ρ: x:Aϕ μ(x,e A)ap λy:B.μ B(ϕ(x),y)(ϕ e)ρ B(ϕ(x))=ap ϕ(ρ A(x))\phi_\rho:\prod_{x:A} \phi_\mu(x, e_A) \bullet \mathrm{ap}_{\lambda y:B.\mu_B(\phi(x),y)}(\phi_e) \bullet \rho_B(\phi(x)) = \mathrm{ap}_{\phi}(\rho_A(x))
  • The associator is preserved in an analogous way.

Homomorphisms of coherent associative H-spaces

A homomorphism of coherent associative H-spaces between two coherent associative H-spaces AA and BB is a homomorphism of non-coherent associative H-spaces ϕ:AB\phi:A \to B in which the coherence condition is preserved.


  • The integers are an associative H-space under addition, as are the natural numbers. More generally, as noted above, any h-set monoid is an associative H-space, and is coherent.

  • Every loop space type Ω x(X)(x= Xx)\Omega_x(X) \equiv (x=_X x) is naturally an associative H-space, with path concatenation as the operation. In this case, the operation is in fact coherent, and indeed every loop space is a \infty-group (although this is difficult to express in type theory).

  • The type of endofunctions AAA \to A has the structure of an associative H-space, with basepoint id Aid_A, operation function composition. Once again the operation is coherent, so this is an \infty-monoid (inverses need not exist).

  • The type of coherent H-space structures on a central type is a coherent associative H-space.


Last revised on December 20, 2023 at 15:08:07. See the history of this page for a list of all contributions to it.