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May
24
comment Questions about epimorphisms and projectives in functor categories
I meant each $@_i$ has a left adjoint. Of course, this is because each $@_i$ is representable, as you observed.
May
23
comment Questions about epimorphisms and projectives in functor categories
Each representable presheaf $\mathcal{I}(i, -)$ is free because it occurs as the image of $1$ under the left adjoint of the evaluation-at-$i$ functor. (This, in some sense, is the content of the Yoneda lemma.) That they are projective is easily shown to be a consequence of this, exactly like how free modules are projective. For the canonical projective covering of a presheaf $P$, just take the collection of all morphisms from any representable presheaf to $P$, and then take the amalgamation of all those.
May
23
answered Analogy between prime numbers and singleton sets?
May
23
revised Questions about epimorphisms and projectives in functor categories
added 234 characters in body
May
23
answered Questions about epimorphisms and projectives in functor categories
May
23
comment a group is not the union of two proper subgroups - how to internalize this into other categories?
My previous comments about toposes with enough points were incorrect; please disregard.
May
23
revised a group is not the union of two proper subgroups - how to internalize this into other categories?
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May
23
comment uniformization theorem - squares and circles
The closed square is not a Riemann surface – it isn't even a manifold without boundary!
May
23
revised Grothendieck topology on pre/sheaves
added 161 characters in body
May
23
comment Grothendieck topology on pre/sheaves
That defines, at best, a pretopology. However the sieve generated by any such family is a covering sieve in the canonical topology.
May
23
comment What are some examples of subtle logical pitfalls?
Actually, the construction of JDH indicated here shows that Fact I is enough to construct, within any given model of ZFC, a transitive set that externally is a model of ZFC!
May
23
answered Grothendieck topology on pre/sheaves
May
23
revised a group is not the union of two proper subgroups - how to internalize this into other categories?
added 669 characters in body
May
23
revised a group is not the union of two proper subgroups - how to internalize this into other categories?
added 10 characters in body
May
23
revised a group is not the union of two proper subgroups - how to internalize this into other categories?
added 506 characters in body
May
23
revised a group is not the union of two proper subgroups - how to internalize this into other categories?
added 2327 characters in body
May
22
comment a group is not the union of two proper subgroups - how to internalize this into other categories?
Actually, there's an easy counterexample in $\mathbf{Grp}$ as well, since internal groups there are just abelian groups.
May
22
answered a group is not the union of two proper subgroups - how to internalize this into other categories?
May
22
comment When is the pullback functor on sheaves faithful?
It's true at the level of toposes and at the level of sheaves of abelian groups as soon as the map is surjective, but I think it is also true at level of quasicoherent sheaves if the morphism is faithfully flat. So what kind of sheaves are you asking about?
May
21
comment A question regarding etale morphisms of affine varieties
Well, the closed subvarieties are either the whole variety or finite sets of points, so those won't do either. And I don't think you really want to consider subvarieties that are neither open nor closed...