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8h
awarded  group-theory
18h
comment Is the category of monoids cartesian closed? Why?
Did you read the links you were given at MO?
1d
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.
2d
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.
2d
comment Does $\operatorname{Spec}$ preserve pushouts?
The same example works.
2d
revised Does $\operatorname{Spec}$ preserve pushouts?
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2d
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.
2d
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.
2d
answered Does $\operatorname{Spec}$ preserve pushouts?
2d
revised Is there a reasonable Grothendieck topology on the category of modules over a ring?
edited tags
2d
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
asked Are weakly étale ring homomorphisms of finite presentation étale?
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
revised Tensor of cocomplete categories
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