I think it is an excellent question. However, lookin= g 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 structure= s 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 a= ddressed 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 b= y these principles is a Set (this is actually harder than it seems) =E2=80= =93 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 F= amilies 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 coheren= ce 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 set= s.

Thorsten

From: <homotopytypethe= ory@googlegroups.com> on behalf of Timothy Carstens <intoverflow@gmai= l.com>
Date: Thursday, 8 August 2019 at 23:09
Subject: Re: [HoTT] (Beginner's question) Uses of HITs beyond homoto= py theory

Thank you for the excel= lent replies! It looks like I was struggling with a lack of imagination whi= le the answer was staring me right in the face.

On Thu, Aug 8, 2019 at = 2:49 PM Michael Shulman <shulman= @sandiego.edu> wrote:

More generally, all col= imits other than coproducts are HITs (of the
"non-recursive" variety).  This includes both homotopy colim= its 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
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 quot= ients
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 "w= ith
classical logic but without the axiom of choice".

On Thu, Aug 8, 2019 at 1:18 PM Steve Awodey <steveawodey@gmail.com> wrote:
>
> quotients by equivalence relations.
> see HoTT Book 6.10
>
> On Aug 8, 2019, at 2:32 PM, Timothy Carstens <intoverflow@gmail.com> wrote:<= br> >
> Sorry for the broad & naive question. I'm a geometer by training b= ut have been working in compsci for most of my career (with lots of time sp= ent in Coq verifying programs).
>
> I've got a naive question that I hope isn't too inappropriate for this= list: can anyone suggest some papers that show applications of HITs? I'm e= mbarrassed to admit it, but I don't know any applications outside of synthe= tic homotopy theory and higher categories.
>
> Perhaps categorical semantics? But even still I'm not personally aware= of any applied results from that domain (contrast with operational semanti= cs; but I am extremely ignorant, so please correct me!)
>
> All my best and apologies in advance if this is off-topic for this lis= t,
> -t
>
>
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