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3h
comment In an abelian category, what are the projectives of the chain complex category?
No, there is no need to add anything to my last assertion. There is a notion of projective object in any category, and coproducts of projective objects – whenever they exist – are automatically projective.
7h
comment In an abelian category, what are the projectives of the chain complex category?
@TobiasKildetoft The embedding theorem does not guarantee the preservation of infinite direct sums. If it did, then every abelian category with infinite direct sums would satisfy AB4.
1d
comment Is the category of monoids cartesian closed? Why?
Did you read the links you were given at MO?
2d
comment Does $\operatorname{Spec}$ preserve pushouts?
It does not in general, but it does in this case. Compute explicitly.
2d
comment Correct definition of model category
I'm not aware of any results that genuinely require functorial factorisation – however, doing without is sometimes a lot harder. See also here and here.
Feb
6
comment Is there a reasonable Grothendieck topology on the category of modules over a ring?
Incidentally, in the case of vector spaces over a field, the regular topology coincides with the topology in which only the maximal sieves cover.
Feb
6
comment Does $\operatorname{Spec}$ preserve pushouts?
The same example works.
Feb
6
comment Is there a reasonable Grothendieck topology on the category of modules over a ring?
There are at least two: one where only maximal sieves are covering, and another where every sieve is covering.
Feb
6
comment Does $\operatorname{Spec}$ preserve pushouts?
This answers the OP's question: the corresponding cospan in $\mathbf{CRing}$ has a pullback, and $\operatorname{Spec}$ does not send it to a pushout.
Feb
6
comment Is there a reasonable Grothendieck topology on the category of modules over a ring?
Much as in point set topology, we have minimal and maximal Grothendieck topologies. So it all depends on what you mean by "reasonable".
Feb
6
comment motivation for the direct limit
The claim about algebraic closures needs to be made more precise. While it is true that $\overline{\mathbb{F}_p}$ is (isomorphic to) the direct limit of its finite subfields, that diagram cannot be constructed without first having an algebraic closure. The issue is, of course, the existence of automorphisms of field extensions – merely taking the category of finite fields of characteristic $p$ will not give you the right diagram.
Feb
5
comment Name for categories with a certain property on coproducts
Er, any category of modules is an example...
Feb
5
comment Name for categories with a certain property on coproducts
The category is pointed, so you get morphisms by sending all the other summands to zero.
Feb
5
comment cokernel in the pointed set category $Set.$
What kind of examples are you looking for? For instance, it is a fact that the cokernel of an identity morphism is a zero morphism, but perhaps that is not very interesting...
Feb
5
comment Quasicoherent sheaf on the functor of points is the same as on the scheme itself
Yes, you can glue sheaves together: see Exercise 1.22 in [Hartshorne, Ch. II]. Quasicoherence is more or less automatic, since it is a local property.
Feb
5
comment Gluing along infinitely many trivial cofibrations
The description in the first paragraph doesn't fit well with the actual situation – I would have expected the cells to be cofibrations, not the attaching maps.
Feb
5
comment Tensor of cocomplete categories
I think it is just true in general, though you might have to give up some things like local smallness. Do it by generators and relations like any other tensor product.
Feb
4
comment Exactness of a right adjoint functor
Actually, that direction is not true, but fortunately we do not need it.
Feb
4
comment “Coforgetful” functors?
If you don't send the morphisms somewhere then you don't have a functor. So you have to send the morphisms somewhere. I wouldn't say $F S$ has no morphisms – identity morphisms are still morphisms...
Feb
4
comment “Coforgetful” functors?
It is not correct. A functor from a category to a set is just a mapping of objects, but the morphisms still have to go somewhere...