Type theory

Constructivism, Realizability, Computability



In homotopy type theory an h-set is a type XX – hence a homotopy type – with the special property that any two of its terms x,y:Xx,y : X are equal (equivalent) in an at most essentially unique way, hence that the identity type (x=y):Type(x = y) : Type is an h-proposition.

The notion of h-set is an internalization of the notion of 0-truncated object into homotopy type theory, essentially an internalization of the notion of set (or possibly of preset). See below in Relation to internal sets for more on this.

h-Sets can also be regarded as a way of embedding extensional type theory into intensional type theory.


Let AA be a type in intensional type theory with dependent sums, dependent products, and identity types. We define a new type isSet(A)isSet(A) as follows:

isSet(A) x:A y:AisProp(x=y)isSet(A) \coloneqq \prod_{x\colon A} \prod_{y\colon A} isProp(x=y)

(using any equivalent definition of the predicate isProp for h-propositions; and where “\prod” denotes dependent product types and “==” denotes identity types).

In other words, the only relationship between two elements of an h-set is whether they are equal; there is no room for more than one path between them. By beta-reducing this definition, we can express it as

isSet(A) x,y:A p,q:x=y(p=q)isSet(A) \coloneqq \prod_{x,y\colon A} \prod_{p,q\colon x=y} (p=q)

In other words, any two parallel paths in AA are equal.

A provably equivalent definition is

isSet(A) x:A p:x=x(p=id x)isSet(A) \coloneqq \prod_{x\colon A} \prod_{p\colon x=x} (p=id_x)

This says that a version of Streicher’s “axiom K” holds for h-sets. (See also at axiom UIP.)


  • Most (non-higher) inductive types are h-sets (assuming that all their parameters and indices are so). In particular, the type of natural numbers is an h-set. This can be proven from Theorem below.


Equivalent characterizations

One interesting consequence of this definition is the following, first proven in (Hedberg)


Suppose that AA is a type which has decidable equality in the propositions as types logic (which is not the logic of h-propositions usually used in HoTT). In other words, the projection

Paths A+(0A×A) (Paths AA×A) A×A\array{Paths_A + (0\to A\times A)^{(Paths_A\to A\times A)}\\ \downarrow\\ A\times A}

(where Paths APaths_A is the path type of AA, “+” forms the sum type, and on the right we have the A×AA \times A-dependent function type into the empty type), has a section.

Then AA is a h-set.


Let dd be the given section. Thus, for any x,y:Ax,y\colon A, d(x,y)d(x,y) is either a path from xx to yy or a function from Paths(x,y)Paths(x,y) to the empty type (implying that Paths(x,y)Paths(x,y) is also empty).

It suffices to exhibit an operation connecting any endo-path pPaths(x,x)p \in Paths(x,x) to the identity path 1 x1_x. Given such a path, define q=d(x,x)q = d(x,x). If d(x,x)d(x,x) lies in the second case, then Paths(x,x)Paths(x,x) is empty, a contradiction since we know it contains 1 x1_x; hence we may assume qPaths(x,x)q\in Paths(x,x) as well.

Let rr be the image of (1 x,p)Paths A×A((x,x),(x,x))(1_x,p) \in Paths_{A\times A}((x,x),(x,x)) under the section dd. This is a path in the total space Paths APaths_A lying over the path (1 x,p)(1_x,p) in A×AA\times A. Equivalently, it is a path in the fiber over xx from (1 x,p) *(d(x,x))(1_x,p)_*(d(x,x)) to d(x,x)d(x,x), where (1 x,p) *(1_x,p)_* denotes transport in the fibration Paths AA×APaths_A \to A\times A along the path (1 x,p)(1_x,p). However, we have defined d(x,x)=qd(x,x) = q, and transport in a path-space is just composition, so rr may be regarded as a path from qpq p to qq. Canceling qq, we obtain a path from pp to 1 x1_x.

Not every h-set has decidable equality (unless the law of excluded middle hold), but there are some other related equivalent characterizations.

  • A type AA is an h-set if and only if all its identity types x= Ayx=_A y have split support, i.e. (x,y:A)x=y(x=y)\prod_{(x,y:A)} \Vert x=y\Vert \to (x=y). This is proven in (KECA).

  • More generally, AA is an h-set if and only if there is some R:AAPropR:A\to A\to Prop which is reflexive (i.e. (x:A)R(x,x)\prod_{(x:A)} R(x,x)) and such that (x,y:A)R(x,y)(x=y)\prod_{(x,y:A)} R(x,y) \to (x=y). This is Theorem 7.2.2 in the HoTT Book.

Relation to internal sets

When using homotopy type theory as the ambient foundations, h-sets generally play the role of the sets. When homotopy type theory is the internal logic of some (∞,1)-category, then the h-sets are the “internal sets” in this internal logic. (Not to be confused with the other meaning of internal set.)

Note, though, that this notion of “internal set” is of a different sort from the usual notions of internal category or internal groupoid. If an internal set is an h-set, then an “internal groupoid” should mean a 1-truncated type, whereas an internal groupoid usually means some kind of groupoid object in an (∞,1)-category. Conversely, the usual meaning of “internal groupoid” suggests that the meaning of “internal set” should be something more like a setoid, with the h-sets being more like presets. This latter meaning is how “sets” are more often defined by constructive type theorists.

The point is that to be worthy of the name “set”, a notion ought to come with “quotients of equivalence relations”. If we start with a notion which does not have quotients, such as the types in ordinary Martin-Löf dependent type theory, then in order to get a good notion of “set” we need to “freely add quotients”, which semantically means passing to the exact completion whose objects are setoids. But if we start with a notion that does have quotients, then this is unnecessary. In homotopy type theory, h-sets do have quotients, which can be constructed using higher inductive types; thus it makes sense to call them “sets” rather than “presets”.

A good way to reconcile these seemingly clashing terminologies is to talk about exact completions of unary sites or (∞,1)-sites. The presence of a Grothendieck topology allows us to “remember” to what extent our given notion has well-behaved quotients: if we have no quotients, then we use the trivial topology, whereas if we have quotients, we can use the regular topology. And the exact completion builds in quotients “freely” but preserving those which the topology asserts to already exist. In particular, if we start with quotients (an exact category or (,1)(\infty,1)-category), then the exact completion of the regular topology is idempotent, whereas if we start with a trivial topology, then the exact completion gives a category of setoids. Thus, in general, the good notion of “internal set” in a unary site is “object of the exact completion”.

Pretopos of hsets

The h-sets in HoTT form a ΠW-pretopos (Rijke-Spitters 13). See also at structural set theory.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/​unit type/​contractible type
h-level 1(-1)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid

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
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idemponent) 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


  • Michael Hedberg, A coherence theorem for Martin-Löf’s type theory, J. Functional Programming, (1998)

  • Nicolai Kraus, A direct proof of Hedberg’s theorem, blog post

  • Nicolai Kraus and Martin Escardo and Thierry Coquand and Thorsten Altenkirch, “Generalizations of Hedberg’s theorem”, M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. PDF

Formalization of set theory via h-sets in homotopy type theory is discussed in

  • Egbert Rijke, Bas Spitters, Sets in homotopy type theory, Mathematical Structures in Computer Science, Volume 25, Issue 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics) (arXiv:1305.3835)

which became one chapter in

Last revised on February 10, 2018 at 06:28:22. See the history of this page for a list of all contributions to it.