nLab
h-set

Idea

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

Definition

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

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

(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

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

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

A provably equivalent definition is

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

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

Examples

• 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 1 below.

Properties

Equivalent characterizations

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

Theorem

Suppose that $A$ 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

$$\array{Paths_A + (0\to A\times A)^{(Paths_A\to A\times A)}\\ \downarrow\\ A\times A}$$

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

Then $A$ is a h-set.

Proof

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

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

Let $r$ be the image of $(1_x,p) \in Paths_{A\times A}((x,x),(x,x)$ under the section $d$. This is a path in the total space $Paths_A$ lying over the path $(1_x,p)$ in $A$. Equivalently, it is a path in the fiber over $x$ from $(1_x,p)_*(d(x,x))$ to $d(x,x)$, where $(1_x,p)_*$ denotes transport in the fibration $Paths_A \to A\times A$ along the path $(1_x,p)$. However, we have defined $d(x,x) = q$, and transport in a path-space is just composition, so $r$ may be regarded as a path from $q p$ to $q$. Canceling $q$, we obtain a path from $p$ to $1_x$.

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

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

• More generally, $A$ is an h-set if and only if there is some $R:A\to A\to Prop$ which is reflexive (i.e. $\prod_{(x:A)} R(x,x)$) and such that $\prod_{(x,y:A)} R(x,y) \to (x=y)$. 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 $(\infty,1)$-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”.

Related concepts

homotopy level	n-truncation	homotopy theory	higher category theory	higher topos theory	homotopy type theory
h-level 0	(-2)-truncated	contractible space	(-2)-groupoid		true/unit type/contractible type
h-level 1	(-1)-truncated		(-1)-groupoid/truth value	(0,1)-sheaf	mere proposition, h-proposition
h-level 2	0-truncated	homotopy 0-type	0-groupoid/set	sheaf	h-set
h-level 3	1-truncated	homotopy 1-type	1-groupoid/groupoid	(2,1)-sheaf/stack	h-groupoid
h-level 4	2-truncated	homotopy 2-type	2-groupoid	(3,1)-sheaf	h-2-groupoid
h-level 5	3-truncated	homotopy 3-type	3-groupoid	(4,1)-sheaf	h-3-groupoid
h-level $n+2$	$n$-truncated	homotopy n-type	n-groupoid	(n+1,1)-sheaf	h-$n$-groupoid
h-level $\infty$	untruncated	homotopy type	∞-groupoid	(∞,1)-sheaf/∞-stack	h-$\infty$-groupoid

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

logic	category theory	type theory
true	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	initial object	empty type
proposition	(-1)-truncated object	h-proposition, mere proposition
proof	generalized element	program
cut rule	composition of classifying morphisms / pullback of display maps	substitution
cut elimination for implication	counit for hom-tensor adjunction	beta reduction
introduction rule for implication	unit for hom-tensor adjunction	eta conversion
logical conjunction	product	product type
disjunction	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	internal hom	function type
negation	internal hom into initial object	function type into empty type
universal quantification	dependent product	dependent product type
existential quantification	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
equivalence	path space object	identity type
equivalence class	quotient	quotient type
induction	colimit	inductive type, W-type, M-type
higher induction	higher colimit	higher inductive type
completely presented set	discrete object/0-truncated object	h-level 2-type/preset/h-set
set	internal 0-groupoid	Bishop set/setoid
universe	object classifier	type of types
modality	closure operator, (idemponent) 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

References

• 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 (arXiv:1305.3835)