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Jan
19
comment Scheme morphism properties that aren't stable under taking triangles?
There are lots of examples. For instance, suppose every morphism $\emptyset \to X$ is in $\mathcal{P}$. Then the two-out-of-three condition would imply every morphism is in $\mathcal{P}$ – but this is plainly not true when e.g. $\mathcal{P}$ is the class of open immersions, étale morphisms, affine morphisms, morphisms of finite presentation...
Jan
19
comment Is Set “prime” with respect to the cartesian product?
The functor sending a category to its maximal subgroupoid is a right adjoint, so it preserves products in particular. It therefore suffices to show that the maximal subgroupoid of $\mathbf{Set}$ is "prime". If we assume GCH, there is an easy argument: supposing both factors are non-trivial, one has a connected component with non-trivial finite $\pi_1$ and the other has a connected component with $\pi_1$ of cardinality $\aleph_1$, so the product will have at least two connected components with $\pi_1$ of cardinality $\aleph_1$, but this is impossible.
Jan
18
comment Could there be an “$n$-th root” of the category $\mathsf{Set}$?
That's a more interesting question. It would be enough to show that the maximal subgroupoid has no interesting product decompositions – then it becomes a question of homotopy theory.
Jan
18
revised Could there be an “$n$-th root” of the category $\mathsf{Set}$?
added 226 characters in body
Jan
18
answered Could there be an “$n$-th root” of the category $\mathsf{Set}$?
Jan
18
comment What are the generating cofibrations of the canonical model structure on Cat?
"A model category for categories". It's available here.
Jan
18
answered Spec $\mathbb{Z}[X]$ in Mumford's Red Book
Jan
18
comment Compactness and directed systems of subspaces
I'm afraid I don't know about nets, so I can't follow your proof. I wonder if there might be a more elementary proof, along the lines of Proposition 2.4.2 in [Hovey, Model categories]?
Jan
18
comment (Co-)limit characterization of subobjects mapping to subobjects?
You need to use image factorisations. Look up regular categories.
Jan
18
answered What are the generating cofibrations of the canonical model structure on Cat?
Jan
18
awarded  Notable Question
Jan
17
revised Compactness and directed systems of subspaces
added 116 characters in body
Jan
17
comment Compactness and directed systems of subspaces
Huh, right. So I need more hypotheses on the system of closed subsets...
Jan
16
comment Example of a Zariski sheaf which is not representable?
Yes, sorry. I was thinking of the constant presheaf, which is different.
Jan
16
comment How can you actually do universal algebra with monads?
@StefanPerko It is always true for finitary algebraic theories, and with AC it is also true for infinitary algebraic theories.
Jan
15
comment How can you actually do universal algebra with monads?
The category of algebras for a monad on $\mathbf{Set}$ is always an exact category. That means quotients and subalgebras behave as you expect.
Jan
15
comment Has a natural transformation between functors with codomain $Cat$ that is an equivalence on each component a weak inverse?
In general no, but you do get a pseudonatural transformation.
Jan
15
comment Kan extensions and double application of Yoneda lemma
It's usually easier to simplify a complicated expression than to make a complicated expression from a simple one.
Jan
15
comment “Minimal upper bounds” in a categorical setting
There are notions like weak colimit or multicolimit but they're not quite the same.
Jan
15
comment Example of a Zariski sheaf which is not representable?
Even the constant sheaf with two elements would do.