(Beginner's question) Uses of HITs beyond homotopy theory
Timothy Carstens
Niels van der Weide
Steve Awodey
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Michael Shulman
"non-recursive" variety). This includes both homotopy colimits and
ordinary colimits of sets (obtained by 0-truncating homotopy
colimits). Having colimits of sets is fairly essential for nearly all
ordinary set-based mathematics, even for people who don't care about
homotopy theory or higher category theory in the slightest. There
aren't really papers specifically about this, because it's so vast,
and because there's not much to say other than the observation that
colimits exist, since at that point you can just appeal to the
long-known fact that once the category of sets satisfies certain basic
properties (Lawvere's "Elementary Theory of the Category of Sets") it
suffices as a basis on which to develop a large amount of mathematics.
The verification of these axioms in HoTT with HITs can be found in
section 10.1 of the HoTT Book. (Before HITs, people formalizing
set-based mathematics in type theory used "setoids" to mimic quotients
and other colimits.)
Beyond this, in set-based mathematics HITs are used to construct free
algebraic structures, as Niels said. Some free algebraic structures
(free monoids, free groups, free rings, etc.) can be constructed based
only on the axioms of ETCS, but for fancier (and in particular,
infinitary) algebraic structures one needs more. In fact there are
algebraic theories for which free algebraic structures cannot be
constructed in ZF (at least, under a large cardinal assumption): the
idea is to use a theory to encode the existence of large regular
cardinals, which cannot be constructed in ZF (see Blass's paper
"Words, free algebras, and coequalizers"). But HITs suffice to
construct even free infinitary algebras of this sort; see e.g. section
9 of my paper with Peter Lumsdaine, "Semantics of higher inductive
types". Thus, HITs can be useful for doing (universal) algebra
constructively, where here "constructively" can even mean "with
classical logic but without the axiom of choice".
Timothy Carstens
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Thorsten Altenkirch
I think it is an excellent question. However, looking at the examples it may seem that we only need QITs, that is set-truncated HITs. However, this is not true when you are dealing with higher structures that arise naturally like the type of sets. For example when you define the integers as a quotient, or nicer as a QIT, you can only eliminate into types that are sets, for example you cannot define a function from the Integers into Set. However, this can be addressed by replacing set-truncation with a coherence law, in this case you basically say that integers have 0 and suc and suc is an equivalence. You can prove that the HIT constructed by these principles is a Set (this is actually harder than it seems) – Luis Soccola and I have recently written a paper about this (need to put it on arxiv). Another example is the intrinsic presentation of type theory as the initial Category with Families which was already mentioned. Again the problem is that you need to set-truncate but then you cannot even define the set-interpretation. This can be again addressed by adding some coherence laws (need to check the details) and you get a coherent version of CWFs which enable us to eliminate into any 1-type, including the universe of sets.
Thorsten
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