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[2a00:1450:4864:20::12d]) by gmr-mx.google.com with ESMTPS id s30si2196159eda.4.2019.08.08.15.09.34 for (version=TLS1_3 cipher=AEAD-AES128-GCM-SHA256 bits=128/128); Thu, 08 Aug 2019 15:09:34 -0700 (PDT) Received-SPF: pass (google.com: domain of intoverflow@gmail.com designates 2a00:1450:4864:20::12d as permitted sender) client-ip=2a00:1450:4864:20::12d; Received: by mail-lf1-x12d.google.com with SMTP id z15so63724191lfh.13 for ; Thu, 08 Aug 2019 15:09:34 -0700 (PDT) X-Received: by 2002:a19:4f42:: with SMTP id a2mr10462818lfk.23.1565302173489; Thu, 08 Aug 2019 15:09:33 -0700 (PDT) MIME-Version: 1.0 References: <728FA1EA-014C-4242-8B34-33A17D7B9208@gmail.com> In-Reply-To: From: Timothy Carstens Date: Thu, 8 Aug 2019 15:09:21 -0700 Message-ID: Subject: Re: [HoTT] (Beginner's question) Uses of HITs beyond homotopy theory Cc: Homotopy Type Theory Content-Type: multipart/alternative; boundary="000000000000045e09058fa250f9" X-Original-Sender: intoverflow@gmail.com X-Original-Authentication-Results: gmr-mx.google.com; dkim=pass header.i=@gmail.com header.s=20161025 header.b="tHrCjo/1"; spf=pass (google.com: domain of intoverflow@gmail.com designates 2a00:1450:4864:20::12d as permitted sender) smtp.mailfrom=intoverflow@gmail.com; dmarc=pass (p=NONE sp=QUARANTINE dis=NONE) header.from=gmail.com Precedence: list Mailing-list: list HomotopyTypeTheory@googlegroups.com; contact HomotopyTypeTheory+owners@googlegroups.com List-ID: X-Google-Group-Id: 1041266174716 List-Post: , List-Help: , List-Archive: , --000000000000045e09058fa250f9 Content-Type: text/plain; charset="UTF-8" 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. --000000000000045e09058fa250f9 Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable
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
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&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
>
>
> --
> You received this message because you are subscribed to the Google Gro= ups "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 ht= tps://groups.google.com/d/msgid/HomotopyTypeTheory/CAJGt_zG%2B04Rfbs_py%3DP= YkubbwzeYb0TRhhfek-RT663uVUo%3D-A%40mail.gmail.com.
>
>
> --
> You received this message because you are subscribed to the Google Gro= ups "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 &= quot;Homotopy Type Theory" group.
To unsubscribe from this group and stop receiving emails from it, send an e= mail to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://group= s.google.com/d/msgid/HomotopyTypeTheory/CAOvivQyYyPzpT0Y04vi27gdg6Un147RkJ4= tyPcCRC_Tsed5PMA%40mail.gmail.com.

--
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