@HarryGindi When $A$ has a contractible classifying space, then the localisation on the $\infty$-categorical level definitely exists. Is there any result of the following form: Let $\mathbf{M}$ be a sufficiently nice model category, then any coreflexive subcategory of its $\infty$-categorical localisation is presented by a right Bousfield localisation of $\mathbf{M}$?

Let $C$ be an ∞-category and $C^\amalg$ its finite coproduct completion. Then the maximal subgroupoid $i(C^\amalg)$ is the free $E_\infty$-space on the maximal subgroupoid $iC$ of $C$. I have a proof of this fact, but is there a reference somewhere so I don't have to write it down?

@AdrianClough the localization can exist, but there are two different notions of localization here

Yes, I meant colocalisation.

three actually

a DK-localization at a small set always exists

Sure, but what I mean is the following:

then there's a left adjoint localization and a right adjoint colocalization

and the existence of the left adoint localization is a special property that's happening here

at an arbitrary small set of morphisms

I don't know of a theorem that guarantees the existence of a colocalization for presentable categories

I'd be interested to know though!

The inclusion of $\mathcal{S}$ into $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathcal{S})$ (as constant presheaves has both a left and right adjoint, corresponding to colimits and limits. The $\infty$-category $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathcal{S})$ is presented by $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathbf{SSet})$ either with the injective or projective model structure.

Yeah, but presheaf categories are not 'cofreely cogenerated'

and the proof that the left adjoint localization exists exploits the generating category

there's a kind of 'small object argument' that goes into the proof

I think it is actually literally the small object argument

If it's not exactly that, it's something along those lines

I see.

I could be wrong, but in this case I don't think I am

The problem to me seemed to rather be the following: Right Bousfield localisations on $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathbf{SSet})$ with the injective model structure don't seem hard to come by, as the model structure is cellular and right proper. The problem is finding a suitable generating set of local equivalences.

In contrast, if I understand correctly, you seem to view the problem more that sets of local equivalences may not generated right Bousfield localisations in the first place.

Oh I see now. The problem is then probably that even if a colocalisation of a presentable $\infty$-category exists, then it may not come from a set of local equivalences.

@AdrianClough Oops, I just checked: the projective model structure is cellular; no mention of the injective model structure (in Hirschhorn).

yeah

The problem is visible without looking at the model cats

the injective and projective model structures are equivalent infty cats

so I think that's a red herring

I'm trying to prove something about model categories rather than $\infty$-categories, and a step would be to find the above mentioned model structure on $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathbf{SSet})$, so I have to think about these things.

But yes, once I have replaced the injective model structure with the projective one, then finding generating sets on $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathbf{SSet})$ or its $\infty$-categorical localisation are equivalent.

@AdrianClough This here should be: The inclusion of $\mathcal{S}$ into $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathcal{S})$ (as constant presheaves has both a left and right adjoint, corresponding to colimits and limits. The $\infty$-category $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathcal{S})$ is presented by $\underline{\mathrm{Hom}}(A^{\mathrm{op}},\mathbf{SSet})$ either with the injective or projective model structure.

If you're writing super complicated TeX in chat, you should do it in the discord

@AdrianClough I can't understand what you've written, but Barwick wrote a paper a while back about RBL: arxiv.org/pdf/0708.3435.pdf

@HarryGindi Don't most people use the MathJax plugin? Also, what is the discord?

@HarryGindi Thank you, I am aware of this paper.

We made a homotopy theory discord for some events

Send me an e-mail because yours is hidden

I see. I vaguely remember other people complaining that it is difficult to get chatjax working on mobile devices.

@HarryGindi Ah, I see. I will use less latex in the future then :)

thanks!

Also, a comment on your TeX: You should use \operatorname for Hom

Ok

I think it kerns thinks slightly differently.

Afak, in this case it doesn't make a difference because it is followed by a "(". But I agree that bad keming should be avoided (-:

@HarryGindi Harry I'm not very comfortable with you pushing people away from here when they have to write TeX. I think it's ok to ask them to upload a picture instead if it is a problem for you and you're interested in the message, but telling people they should go somewhere else feels like crossing a line to me

Not everyone is comfortable with using seven different accounts for various math conversations

Oh, no, I meant because he pinged me with TeX.

Yeah, please, anyone, this was not an exhortation to avoid using TeX, just that I have a hard time reading it on mobile.

I hope that clarifies matters.

Sorry everyone!

Also, mobile issues aside, the TeX support here (if you remember to turn it on, which I just did for the first time in ages) is really way better than on discord, where it just awkwardly generates a picture