Zhen Lin
Reputation
96/100 score
 Jan 23 comment Proper maps in terms of projection from pullback What's your definition of proper map if not this? Jan 23 revised Homotopy colimit,weighted colimit, homotopy theory edited tags Jan 23 comment Homotopy colimit,weighted colimit, homotopy theory Take $m = \operatorname{hocolim} F$ on the LHS. Then you have $\mathrm{id}$. Follow it across the bijection to get the desired universal simplicial natural transformation. Jan 19 comment Scheme morphism properties that aren't stable under taking triangles? It has very little to do with the empty set. It's just an extreme case – you can easily conjure up examples with non-empty domain. What's more interesting is to find ones where $g$ is surjective... 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.