@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?
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.
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).
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.
@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
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