Segal's "classifying spaces and spectral sequences" paper, proposition 5.1

Wow has anyone else heard about this work of Crockett and Crutwell on "tangent categories?" arxiv.org/pdf/1610.08774v1.pdf

They've written a few papers on it, and their work seems very interesting.

Grothendieck topologies: If a site is subcanonical, it quickly follows the composite of sheafification on Yoneda embedding is fully faithful. Exercise 3.7.2 of Borceux, vol III suggests proving the converse with some effective epi characterization. Is there a more direct proof of the converse?

Let L be sheafification and h a representable. Then Hom(h', h) ---> Hom(Lh', Lh)=Hom(h', Lh) is an isomorphism for every representable h'. We can write every presheaf as a colimit of representables, so we conclude that Hom(F, h) --> Hom(F, Lh) is an equivalence for every presheaf F and the result follows by Yoneda.

@RuneHaugseng I like Tamaki-Kono's "Generalized Cohomology" book, section 5.5

the definition of localization of a category given in the nLab ncatlab.org/nlab/show/localization#general is 2-categorical, i.e. it doesn't ask that the universal property for the localization functor hold on the nose, but up to a natural isomorphism. That's not what I've seen in other, more traditional sources. is there a conceptual reason to prefer the 2-categorical one? (I'm aware that the homotopy category of a model category satisfies the stronger 1-categorical version)

@BrunoStonek Asking equality for functors is evil. I like when my definitions do not change if I happen to replace a category with an equivalent one (since it is something I do quite a lot!)

aha! that's definitely a good reason. I guess classical sources are evil then :)