Thank you for the excellent replies! It looks like I was struggling with a
lack of imagination while the answer was staring me right in the face.
On Thu, Aug 8, 2019 at 2:49 PM Michael Shulman wrote:
> More generally, all colimits other than coproducts are HITs (of the
> "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".
>
> On Thu, Aug 8, 2019 at 1:18 PM Steve Awodey wrote:
> >
> > quotients by equivalence relations.
> > see HoTT Book 6.10
> >
> > On Aug 8, 2019, at 2:32 PM, Timothy Carstens
> wrote:
> >
> > Sorry for the broad & naive question. I'm a geometer by training but
> have been working in compsci for most of my career (with lots of time 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 HITs? I'm
> embarrassed to admit it, but I don't know any applications outside of
> synthetic 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
> semantics; but I am extremely ignorant, so please correct me!)
> >
> > All my best and apologies in advance if this is off-topic for this list,
> > -t
> >
> >
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