This is just a quick note on the paper Brown representability does not come for free by Casacuberta and Neeman. This is going to be completely bare bones as you can read more details in the paper.
We are going to define a “big” abelian category A as follows. An object of A consists of a pair (M, α, s_β) where M is an abelian group and α is an ordinal and s_β : M —> M is a commuting family of homomorphisms parametrized by β ∈ α. A morphism (M, α s_β) —> (N, γ, t_δ) is given by a homomorphism of abelian groups f : M —> N such that f(s_β(m)) = t_β(f(m)) for any ordinal β where the rule is that we set s_β equal to zero if β is not in α and similarly we set t_β equal to zero if β is not an element of γ.
A special object is Z = (Z, 0, ∅), i.e., all the operators are zero. The observation is that computed in A the “group” Ext^1_A(Z, Z) is a proper class and not a set. Namely, for each ordinal β we can find an extension M of Z by Z whose underlying group is M = Z ⊕ Z and where s_β acts by a nonzero operator s_β, e.g. via the matrix (0, 1; 0, 0). This clearly produces a proper class of isomorphism classes of extensions.
In my world forming the category D(A) doesn’t make sense because the Hom’s aren’t sets. Another conclusion is that in K(A) the subcategory of acyclic complexes does not give rise to a Bousfield localization or colocalization.
Scarier than Halloween?