Thank you for the excellent replies! It looks like I = was struggling with a lack of imagination while the answer was staring me r= ight in the face.

On Thu, Aug 8, 201= 9 at 2:49 PM Michael Shulman <sh= ulman@sandiego.edu> wrote:
More generally, all colimits other than coproducts are HI= Ts (of the
"non-recursive" variety).=C2=A0 This includes both homotopy colim= its and
ordinary colimits of sets (obtained by 0-truncating homotopy
colimits).=C2=A0 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.=C2=A0 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&q= uot;) 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.=C2=A0 (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.=C2=A0 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.=C2=A0 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").=C2=A0 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".=C2=A0 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 traini= ng but have been working in compsci for most of my career (with lots of tim= e spent 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 HIT= s? I'm embarrassed to admit it, but I don't know any applications o= utside of synthetic homotopy theory and higher categories.
>
> Perhaps categorical semantics? But even still I'm not personally a= ware of any applied results from that domain (contrast with operational sem= antics; 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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