*[HoTT] Dependent path composition in ordinary higher category theory@ 2020-02-28 18:33 Noah Snyder2020-02-28 18:44 ` [HoTT] " Noah Snyder 0 siblings, 1 reply; 3+ messages in thread From: Noah Snyder @ 2020-02-28 18:33 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1: Type: text/plain, Size: 1698 bytes --] Section 2.3 of the book introduces "dependent paths" (which are paths in a fibration "lying over" a path in the base) and "dependent path composition" which composes such dependent paths when that makes sense. I'm working on a paper that's not about HoTT but where "dependent path composition" plays an important role. The problem I'm running into is that I don't know what dependent path composition is called in "standard" mathematics. Does anyone know if this has another name in higher category theory? (Naturally, we'll include a remark mentioning the HoTT way of thinking about this (since it's how I think about it!), but I think that won't be illuminating to most of our target audience.) The simplest example of what I have in mind here is if C is a category and F is an endofunctor and c is an F-algebra (i.e. we have endowed c with a chosen map f: F(c) --> c) then c is also an F^n-algebra by taking the "nth power of f" which actually means f \circ F(f) \circ F^2(f) \circ ... \circ F^{n-1}(f). In particular, I think this example illustrates that you can talk about "dependent k-morphisms" and their compositions without requiring anything in sight to be a (higher) groupoid. My best guess is that the right setting might be "indexed (higher) categories"? Best, Noah -- 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/CAO0tDg7uSiQrU8%2B0tnwJQHB_AWvRKAxmAePF88GOvd_ra%3DrpwA%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 2226 bytes --] ^ permalink raw reply [flat|nested] 3+ messages in thread

*[HoTT] Re: Dependent path composition in ordinary higher category theory2020-02-28 18:33 [HoTT] Dependent path composition in ordinary higher category theory Noah Snyder@ 2020-02-28 18:44 ` Noah Snyder2020-03-02 7:03 ` Michael Shulman 0 siblings, 1 reply; 3+ messages in thread From: Noah Snyder @ 2020-02-28 18:44 UTC (permalink / raw) To: Homotopy Type Theory [-- Attachment #1: Type: text/plain, Size: 2211 bytes --] Mike pointed out that I didn't explain how my example is a special case of dependent path composition. A type family over S^1 is a higher groupoid together with an (invertible) endofunctor F. A path over loop is an algebra for that endofunctor (where the map is an iso). If you dependent path compose it with itself n times you get a path over loop^n, i.e. an algebra for F^n. Best, Noah On Fri, Feb 28, 2020 at 10:33 AM Noah Snyder <nsnyder@gmail.com> wrote: > Section 2.3 of the book introduces "dependent paths" (which are paths in a > fibration "lying over" a path in the base) and "dependent path composition" > which composes such dependent paths when that makes sense. I'm working on > a paper that's not about HoTT but where "dependent path composition" plays > an important role. The problem I'm running into is that I don't know what > dependent path composition is called in "standard" mathematics. Does > anyone know if this has another name in higher category theory? > (Naturally, we'll include a remark mentioning the HoTT way of thinking > about this (since it's how I think about it!), but I think that won't be > illuminating to most of our target audience.) > > The simplest example of what I have in mind here is if C is a category and > F is an endofunctor and c is an F-algebra (i.e. we have endowed c with a > chosen map f: F(c) --> c) then c is also an F^n-algebra by taking the "nth > power of f" which actually means f \circ F(f) \circ F^2(f) \circ ... \circ > F^{n-1}(f). In particular, I think this example illustrates that you can > talk about "dependent k-morphisms" and their compositions without requiring > anything in sight to be a (higher) groupoid. > > My best guess is that the right setting might be "indexed (higher) > categories"? > > Best, > > Noah > -- 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/CAO0tDg5mstbw3Ujjfu%2BkMfgW7_7y7raxbWV_yh-2phB2ynwb8g%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 3040 bytes --] ^ permalink raw reply [flat|nested] 3+ messages in thread

*Re: [HoTT] Re: Dependent path composition in ordinary higher category theory2020-02-28 18:44 ` [HoTT] " Noah Snyder@ 2020-03-02 7:03 ` Michael Shulman0 siblings, 0 replies; 3+ messages in thread From: Michael Shulman @ 2020-03-02 7:03 UTC (permalink / raw) To: Noah Snyder;+Cc:Homotopy Type Theory Yes, I think probably you want indexed categories, incarnated as fibrations. You can then remove the invertibility too: if C is the 1-object category corresponding to the monoid of natural numbers, then an opfibration over C is a category equipped with an endofunctor, an endomorphism over the generating morphism of C is an algebra for that endofunctor, and its n-fold composition with itself is the corresponding algebra for F^n. On Fri, Feb 28, 2020 at 10:44 AM Noah Snyder <nsnyder@gmail.com> wrote: > > Mike pointed out that I didn't explain how my example is a special case of dependent path composition. A type family over S^1 is a higher groupoid together with an (invertible) endofunctor F. A path over loop is an algebra for that endofunctor (where the map is an iso). If you dependent path compose it with itself n times you get a path over loop^n, i.e. an algebra for F^n. Best, > > Noah > > On Fri, Feb 28, 2020 at 10:33 AM Noah Snyder <nsnyder@gmail.com> wrote: >> >> Section 2.3 of the book introduces "dependent paths" (which are paths in a fibration "lying over" a path in the base) and "dependent path composition" which composes such dependent paths when that makes sense. I'm working on a paper that's not about HoTT but where "dependent path composition" plays an important role. The problem I'm running into is that I don't know what dependent path composition is called in "standard" mathematics. Does anyone know if this has another name in higher category theory? (Naturally, we'll include a remark mentioning the HoTT way of thinking about this (since it's how I think about it!), but I think that won't be illuminating to most of our target audience.) >> >> The simplest example of what I have in mind here is if C is a category and F is an endofunctor and c is an F-algebra (i.e. we have endowed c with a chosen map f: F(c) --> c) then c is also an F^n-algebra by taking the "nth power of f" which actually means f \circ F(f) \circ F^2(f) \circ ... \circ F^{n-1}(f). In particular, I think this example illustrates that you can talk about "dependent k-morphisms" and their compositions without requiring anything in sight to be a (higher) groupoid. >> >> My best guess is that the right setting might be "indexed (higher) categories"? >> >> Best, >> >> Noah > > -- > 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/CAO0tDg5mstbw3Ujjfu%2BkMfgW7_7y7raxbWV_yh-2phB2ynwb8g%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/CAOvivQzAoLQkXUNDkauM5so1rNu9-LsyWRHM4xO16d4O8Wt-0A%40mail.gmail.com. ^ permalink raw reply [flat|nested] 3+ messages in thread

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