nLab quotient type

Idea

In type theory the kind of type corresponding in categorical semantics to a quotient object / coequalizer.

Properties

From higher inductive types

Quotient type may be constructed as higher inductive types. See here.

History and effectiveness

This notion of quotient type predates the general notion of higher inductive type, however; it appears already in Martin Hofmann’s thesis.

Hofmann works in a theory that includes a type of propositions $Prop$ that may be assumed to satisfy proof irrelevance? — for instance, it might be the universe of h-propositions — and considers the effectiveness of quotient types only for quotients of Prop-valued equivalence relations. This corresponds to the correct categorical notion of effectiveness (see Barr-exact category) and (as Hofmann noted) is provable in the presence of propositional extensionality (which in turn follows from the univalence axiom).

However, before the introduction of homotopy type theory and the realization of the importance of homotopy levels such as h-propositions, some authors considered also quotients of arbitrary $Type$-valued “equivalence relations” (i.e. setoids). Maietti shows, by adapting Diaconescu’s argument, that “effectiveness” for quotients of this sort of “equivalence relation” implies the law of excluded middle.

From univalence

Quotient types are implied by adding univalence to Martin-Löf type theory. We indicate how this works:

A sufficient set of hypothesis that imply the existence of quotient types is this:

• ambient dependent product types dependent sum types and identity types;
• availability of propositional truncation;
• existence of a univalent family $U$;
• every h-prop is equivalent to some type in $U$.

In terms of categorical semantics in fibration categories, this comes about as follows:

In fibration categories where every map is a fibration, “hprop” means “subobject”, propositional truncation becomes a stable image factorisation, and from a univalent universe $U$ containing all hprops, we can construct a subobject classifier as

$$\underset{x \colon U}{\sum} isProp(x) \,.$$

(For the subobject classification axioms: existence comes from “every hprop is equivalent to some type in $U$”, and uniqueness comes from univalence of $U$.)

So in this setting, the HoTT argument just becomes the usual construction (due to Paré 74, see for instance MacLane-Moerdijk 92, IV.5) of exact quotients from a subobject classifier in a regular locally cartesian closed category. For more on this see at quotient object – in toposes.

One thing to be careful about, in the fibration category setting, is that “equivalences” as defined in type theory correspond not generally to the original equivalences, but to the right homotopy equivalences, which of course may be different. This is why e.g. Mike Shulman’s def of “type-theoretic fibration category” doesn’t include a class of equivalences as separate data, but takes them to be the right homot equivs. So for the type-theoretic arguments to apply, “univalent fibration” should be defined using right homot equivs.

Related concepts

• setoid/Bishop set

• quotient

• quotient space

• quotient stack

References

On the convoluted history of the notion of quotient types, from setoids/Bishop sets to higher inductive types and in univalent foundations of mathematics:

• Nuo Li, Quotient Types in Type Theory, Nottingham (2014) [pdf, oai:eprints.nottingham.ac.uk:28941, pdf]

• Zachary Murray, §4.3.2 in: Constructive Analysis in the Agda Proof Assistant [arXiv:2205.08354, github]

As a higher inductive type

As higher inductive types:

• Martin Hofmann, p. 111 in: Extensional concepts in intensional type theory, Ph.D. thesis, University of Edinburgh (1995), Distinguished Dissertations, Springer (1997) [ECS-LFCS-95-327, pdf, doi:10.1007/978-1-4471-0963-1]

• Maria Emilia Maietti, About effective quotients in constructive type theory, in “Types for proofs and programs” (1999)

• Thorsten Altenkirch, Paolo Capriotti, Gabe Dijkstra, Nicolai Kraus, Fredrik Forsberg, Quotient inductive-inductive types, Foundations of Software Science and Computation Structures (FoSSaCS 2018), Lecture Notes in Computer Science 10803 (2018) [doi:10.1007/978-3-319-89366-2_16, arXiv:1612.02346]

From univalence

The argument that adding univalence (plus universe resizing) to Martin-Löf type theory implies the existence of quotient types is attributed to Voevodsky in places like here: pdf, slides 41 & 61.

Voevodsky’s own account of this seems to be entirely in his Coq-code repository here:

• Vladimir Voevodsky, Equivalence classes with respect to a given relation,

  here, Set quotients of types, here

A more readable version of this is in section 6.10 of

• Homotopy Type Theory – Univalent Foundations of Mathematics

Def. 6.10.5 there is the definition of the quotients. This uses truncation, for which something like univalence and resizing is necessary and univalence and truncation is sufficient. Then theorem 6.10.6 right below that definition checks that these quotient do indeed behave like quotients should, and this is where univalence proper comes in.

This discussion of quotients from univalence in the HoTT book originally comes from

• Egbert Rijke, Bas Spitters, Sets in homotopy type theory, Mathematical Structures in Computer Science 25 5 “From type theory and homotopy theory to Univalent Foundations of Mathematics” (2015) 1172-1202

  [doi:10.1017/S0960129514000553, arXiv:1305.3835]

The relevant section there is 2.4 “Voevodsky’s impredicative quotients”, see def. 2.24 and the lines right below.

The categorical semantics-version of the argument for colimits from a subobject classifier in an elementary topos is due to

• Robert Paré, Colimits in Topoi, Bull. AMS 80 (1974) pp.556-661. (pdf)

reviewed for instance in

• Saunders MacLane, Ieke Moerdijk, section IV.5 of Sheaves in Geometry and Logic – A first introduction to topos theory, Springer Verlag, 1992

A more elementary discussion is in

• Todd Trimble, An elementary approach to elementary topos theory