quotient type

**natural deduction** metalanguage, practical foundations

**type theory** (dependent, intensional, observational type theory, homotopy type theory)

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

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

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

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.

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

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 for. 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*, (arXiv:1305.3835)

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

The category theoretic (“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

Revised on July 28, 2016 12:31:30
by Urs Schreiber
(131.220.184.222)