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 > > > > > > -- > > You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAJGt_zG%2B04Rfbs_py%3DPYkubbwzeYb0TRhhfek-RT663uVUo%3D-A%40mail.gmail.com > . > > > > > > -- > > You received this message because you are subscribed to the Google > Groups "Homotopy Type Theory" group. > > To unsubscribe from this group and stop receiving emails from it, send > an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/728FA1EA-014C-4242-8B34-33A17D7B9208%40gmail.com > . > > -- > You received this message because you are subscribed to the Google Groups > "Homotopy Type Theory" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to HomotopyTypeTheory+unsubscribe@googlegroups.com. > To view this discussion on the web visit > https://groups.google.com/d/msgid/HomotopyTypeTheory/CAOvivQyYyPzpT0Y04vi27gdg6Un147RkJ4tyPcCRC_Tsed5PMA%40mail.gmail.com > . > -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAJGt_zH1qWjaQYy-jOGfuxm--cLw3AJ6y6WPheqHmA8Dr1B%2Bww%40mail.gmail.com.