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• Has anybody attacked this problem with modern technology?
• Do you think that the above is a valuable question?
• I believe it is, because
  1. Every category that breaks Isbell condition (let's call it a "non-Isbell category" for the sake of brevity) seems quite nasty. And yet its homotopy theory can be well-understood. Isbell condition itself is stated in terms of the set theory of $\cal A$ and it is (unsurprisingly) linked to $\cal A$ having "nice" factorization systems ("nice" here means proper+something; did somebody explicitly prove this, maybe even Freyd?). So one can "foresee" if $\cal M$ will have a non-concrete localization proving that there is no homotopy-nice factorization system on $\cal M$ (a factorization system on $\cal M$ is homotopy-nice if it is an homotopy FS in the sense of Bousfield, and the FS induced by it on $\textsf{Ho}(\cal M)$ is nice).
  2. All these categories seem to exist (but I will be happy to see you disproving me, especially in the nontrivial cases):
     • a concrete category whose localization is concrete
     • a non-concrete category whose localization is concrete
     • a non-concrete category whose localization is non-concrete
     • a concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
     • a non-concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is concrete
     • a non-concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
     • a concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
  3. There's an interesting problem: every category is a quotient of a concrete category (link). Is every category a localization of a concrete one?
• If $\textsf{Ho}(\cal M)$ is not concrete, there should be a property $P$ of $\cal M$ preventing $\textsf{Ho}(\cal M)$ to be Isbell. It seems obvious that every category $\cal N$ which is Quillen equivalent to $\cal M$ has $P$, and yet Freyd's technology seems to be rather context-specific, to the point that it's hard to believe that $P$ can be transported along the adjunction of a Quillen equivalence. Or maybe it is, under suitable assumptions?

• Has anybody attacked this problem with modern technology?
• Do you think that the above is a valuable question?
• I believe it is, because
  1. Every category that breaks Isbell condition (let's call it a "non-Isbell category" for the sake of brevity) seems quite nasty. And yet its homotopy theory can be well-understood. Isbell condition itself is stated in terms of the set theory of $\cal A$ and it is (unsurprisingly) linked to $\cal A$ having "nice" factorization systems ("nice" here means proper+something; did somebody explicitly prove this, maybe even Freyd?). So one can "foresee" if $\cal M$ will have a non-concrete localization proving that there is no homotopy-nice factorization system on $\cal M$ (a factorization system on $\cal M$ is homotopy-nice if it is an homotopy FS in the sense of Bousfield, and the FS induced by it on $\textsf{Ho}(\cal M)$ is nice).
  2. All these categories seem to exist (but I will be happy to see you disproving me, especially in the nontrivial cases):
     • a concrete category whose localization is concrete
     • a non-concrete category whose localization is concrete
     • a non-concrete category whose localization is non-concrete
     • a concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
     • a non-concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is concrete
     • a non-concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
     • a concrete category $\cal M$ whose allocalization ${\cal M}[\text{All}^{-1}]$ is non-concrete
  3. There's an interesting problem: every category is a quotient of a concrete category (Kučera, JPAA 1971, link). Is every category a localization of a concrete one?
• If $\textsf{Ho}(\cal M)$ is not concrete, there should be a property $P$ of $\cal M$ preventing $\textsf{Ho}(\cal M)$ to be Isbell. It seems obvious that every category $\cal N$ which is Quillen equivalent to $\cal M$ has $P$, and yet Freyd's technology seems to be rather context-specific, to the point that it's hard to believe that $P$ can be transported along the adjunction of a Quillen equivalence. Or maybe it is, under suitable assumptions?

• Let $\mho$ be a universe. If $\cal M \in \mho\text{-}{\bf Cat}$ has a model structure, how often is the localization $\textsf{Ho}(\cal M)$ a ($\mho\text{-}\bf Set$-)concrete category?

• Let $\mho$ be a universe. If $\cal M \in {\bf Cat}$ is locally $\mho$-small and has a model structure, how often is the localization $\textsf{Ho}(\cal M)$ a ($\mho\text{-}\bf Set$-)concrete category?

Let ${\cal A}/A$ the class of arrows with codomain $A$ in a category $\cal A$, and for $f\in {\cal A}/A$ let $C(f,B)$ be the class of pairs $u,v \in A/{\cal A}/B$ such that $uf=vf$ as morphisms $\text{src}(f) \to A \to B$.

Define an equivalence relation $\asymp$ on ${\cal A}/A$ that says $f\asymp g$ iff $C(f,B)=C(g,B)$ for every $B\in\cal A$, and let $\textsf{S}({\cal A}/A)$ the quotient $\left({\cal A}/A\right)_{/\asymp}$.

For $f\in {\cal A}/A$ an object of the slice category, let $C(f,B)$ be the class of pairs $u,v : A \to B$ such that $uf=vf$ as morphisms $\text{src}(f) \to A \to B$.

Define an equivalence relation $\asymp$ on $({\cal A}/A)_0$ that says $f\asymp g$ iff $C(f,B)=C(g,B)$ for every $B\in\cal A$, and let $\textsf{S}({\cal A}/A)$ the quotient $\left({\cal A}/A\right)_{0,/\asymp}$.