Zhen Lin
Reputation
43,191
392/400 score
 May14 comment Do Boolean rings always have a unit element? My definition of ring includes a $1$, and my definition of boolean algebra also includes a $1$. May14 answered Commuting square of functors May12 comment Why does every countable limit ordinal have cofinality $\omega$? If there is a cofinal $\alpha$-sequence in $\beta$ and a cofinal $\beta$-sequence in $\gamma$, then there is a cofinal $\alpha$-sequence in $\gamma$. Therefore cofinalities are initial ordinals. May12 answered Viewing groups as objects of the concrete category $\mathsf{Grp}$ May11 answered Commutativity of a sheaf of groups from an epimorphism May11 answered Space modelled on ring May11 comment Space modelled on ring Functors are not the best way to think about schemes if you're looking for spaces modelled on rings. Rather, a scheme is a locally ringed space that is locally isomorphic to the spectrum of a ring. There is a functorial way of looking at schemes but I find it much more artificial. May11 comment Space modelled on ring Do you mean a scheme? May10 comment A question about partitioning the unit cube into simplexes There is a triangulation of the $n$-cube satisfying condition (1): it is what is obtained by considering the simplicial set $(\Delta^1)^n$. I haven't thought about (2). May9 comment Axiom of Choice-esque argument to show that a proof of a statement exists without actually giving a proof Any such category or topological space must be connected: everything proves $\top$, and $\bot$ proves everything. May9 comment Is $\mathbb{N}$ impossible to pin down? If you believe that there is a "real" $\mathbb{N}$, then it sounds like you've pinned it down already! May8 comment $C(X) \cup C(Y) = C(X \cup Y)$ if $C$ is an algebraic closure and $X, Y$ are finite? The smallest closed subset containing $C(X)$ and $C(Y)$ is $C(X \cup Y)$. But in general that is not $C(X) \cup C(Y)$. May7 answered Existence of not locally small categories May7 comment Is a divisor in the hyperplane class necessarily a hyperplane divisor? Hmmm... yes, I suppose that is true. Do you have a better proof for the plane curve case, then? May7 comment Is a divisor in the hyperplane class necessarily a hyperplane divisor? Well, if if $h$ is a rational function, then there are some coprime $F$ and $G$ such that $h = F / G$. If they weren't of degree $1$ then that would contradict the condition on the degrees of the divisors. May7 comment Is a divisor in the hyperplane class necessarily a hyperplane divisor? If $F$ and $G$ are homogeneous of degree $m$ and coprime, then $F / G$ will be a rational function with $m \deg H$ zeros and $m \deg H$ poles, counted with multiplicity, by Bézout's theorem. May7 comment Can “proving mathematical theorems” as a 1 player puzzle be studied in combinatorial game theory? See Wikipedia about game semantics. May7 comment Can “proving mathematical theorems” as a 1 player puzzle be studied in combinatorial game theory? Actually, it's a two-player game between the verifier and the refuter. May6 asked Is a divisor in the hyperplane class necessarily a hyperplane divisor? May6 comment Existence of not locally small categories If those authors don't admit the existence of locally small categories, why would they talk about such things? To talk about such things would require them to admit there are categories that are not locally small! But one sometimes speaks of ‘metacategories’.