Re: [HoTT] Geometry in Modal HoTT now on Youtube

[Egbert Rijke <e.m.rijke@gmail.com>]

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Jonas found a flaw...

Thanks Jonas!

(and forget what I said above)

With kind regards,
Egbert

On Mon, Mar 18, 2019 at 11:40 PM Egbert Rijke <e.m.rijke@gmail.com> wrote:

> So I guess you're right then... and I made a mistake. Thanks so much for
> pointing it out!
>
> On Mon, Mar 18, 2019 at 11:33 PM Egbert Rijke <e.m.rijke@gmail.com> wrote:
>
>> Wait a minute... in the proof that it is Sigma-closed I do have the
>> diagrams the wrong way around. Yikes!
>>
>> On Mon, Mar 18, 2019 at 11:27 PM Egbert Rijke <e.m.rijke@gmail.com>
>> wrote:
>>
>>> Hi Jonas,
>>>
>>> The last question is easiest: I do everything internally, so maybe it is
>>> different for external factorization systems. That I don't know.
>>>
>>> The first thing to show is that from a reflective factorization system
>>> you get a modality. The modal types are the types for which the terminal
>>> projection is in the right class. Then by factoring the terminal projection
>>> you get the modal operator. Let's call it L. This gives a reflective
>>> subuniverse (I believe you have no problem with this fact, but the proof is
>>> easy, by orthogonality). Then you need to show that this reflective
>>> subuniverse is closed under Sigma. If you have a family of modal types over
>>> a modal type, then the Sigma should also be modal. Since the right class is
>>> closed under composition, it suffices to show that the projection map is a
>>> right map. This can be done by showing that it is right orthogonal to any
>>> map in the left class. So indeed I claim that any reflective factorization
>>> system gives rise to a modality in this way.
>>>
>>> Ah, and since this is now a private message I can attach the proof.
>>> Hopefully it is not false...
>>>
>>> Best,
>>> Egbert
>>>
>>>
>>>
>>> On Mon, Mar 18, 2019 at 11:15 PM Jonas Frey <jonas743@gmail.com> wrote:
>>>
>>>> Hi Egbert (answering privately),
>>>>
>>>> I think that's weird since at least in the 1-categorical setting, every
>>>> reflection on say a locally presentable category determines a reflective
>>>> factorization system whose left class consists of those maps that are
>>>> inverted by the reflection (that's in the cassidy-hebert-kelly paper). I
>>>> suppose that's also true for ∞-presentable categories so e.g. there's a
>>>> reflective facsys whose left class consists of those maps that are inverted
>>>> by localization at the degree two map of the circle.
>>>>
>>>> Are you saying that this rfsys is also the "formally etale facsys" of
>>>> some modality?
>>>>
>>>> More generally, are you saying your claim also holds up for other
>>>> infty-toposes than spaces? In this case, do you want the notion of
>>>> factorization system to be understood internally?
>>>>
>>>> Jonas
>>>>
>>>>
>>>> On Tue, 19 Mar 2019 at 00:01, Egbert Rijke <e.m.rijke@gmail.com> wrote:
>>>>
>>>>> Dear all,
>>>>>
>>>>> During my lecture on modal descent someone in the audience (my
>>>>> apologies, I forgot who it was) asked the question
>>>>> <https://www.youtube.com/watch?v=InaRkqKdyp4&t=900s> whether
>>>>> reflective factorization systems (i.e. orthogonal factorization systems for
>>>>> which the left class satisfies the 3-for-2 property) also characterize
>>>>> modalities. I thought this was a very nice question, because we have a
>>>>> theorem like that for stable factorization systems (i.e. orthogonal
>>>>> factorization systems for which the left class is stable under pullback).
>>>>>
>>>>> During the lecture I didn't know the answer, but now I do: It is
>>>>> indeed the case that modalities are equivalently described as reflective
>>>>> factorization systems. I wanted to attach hand-written notes containing the
>>>>> proof, but the files are too large to be contained in a message to this
>>>>> google group. I'll write it up properly soon.
>>>>>
>>>>> In particular it follows that the poset of stable factorization
>>>>> systems is the same as the poset of reflective factorization systems, even
>>>>> though there are subtle differences between the stable factorization system
>>>>> of a modality, and the reflective factorization system of a modality (i.e.
>>>>> under this equivalence the left and right classes have to change slightly).
>>>>>
>>>>> My lecture is available via the link Felix just shared.
>>>>>
>>>>> With kind regards,
>>>>> Egbert
>>>>>
>>>>> On Mon, Mar 18, 2019 at 11:53 AM Felix Wellen <felix.wellen@gmail.com>
>>>>> wrote:
>>>>>
>>>>>> Dear all,
>>>>>>
>>>>>> I just finished uploading the recordings of last week's workshop
>>>>>>
>>>>>> Geometry in Modal HoTT
>>>>>> <http://www.andrew.cmu.edu/user/fwellen/modal-workshop.html#schedule>
>>>>>>
>>>>>> to youtube <https://www.youtube.com/channel/UCfQdw845tAduRwW6FnWC9KQ>.
>>>>>> I'm afraid the audio is not good, but with enough volume it should be
>>>>>> possible to understand everything.
>>>>>> You can get an overview of the talks on the abstracts-page:
>>>>>>
>>>>>> http://www.andrew.cmu.edu/user/fwellen/abstracts.html
>>>>>>
>>>>>> All the best,
>>>>>> Felix
>>>>>>
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>>>>>
>>>>>
>>>>> --
>>>>> egbertrijke.com
>>>>>
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>>>>
>>>
>>>
>>
>> --
>> egbertrijke.com
>>
>
>
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> egbertrijke.com
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