Discussion of Homotopy Type Theory and Univalent Foundations
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From: Michael Shulman <shulman@sandiego.edu>
To: Timothy Carstens <intoverflow@gmail.com>
Cc: Homotopy Type Theory <HomotopyTypeTheory@googlegroups.com>
Subject: Re: [HoTT] (Beginner's question) Uses of HITs beyond homotopy theory
Date: Thu, 8 Aug 2019 14:49:28 -0700
Message-ID: <CAOvivQyYyPzpT0Y04vi27gdg6Un147RkJ4tyPcCRC_Tsed5PMA@mail.gmail.com> (raw)
In-Reply-To: <728FA1EA-014C-4242-8B34-33A17D7B9208@gmail.com>

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 <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:
>
> 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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  reply index

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2019-08-08 18:32 Timothy Carstens
2019-08-08 18:52 ` [HoTT] " Niels van der Weide
2019-08-08 20:18 ` [HoTT] " Steve Awodey
2019-08-08 21:49   ` Michael Shulman [this message]
2019-08-08 22:09     ` Timothy Carstens
2019-08-09 10:36       ` Thorsten Altenkirch

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Discussion of Homotopy Type Theory and Univalent Foundations

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