Zhen Lin
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 May6 comment Existence of not locally small categories The point is that there are people who take ‘category’ to mean locally small by definition. So for them a category that is not locally small does not exist – by definition! May6 comment Existence of not locally small categories Well, the collection of all sets is certainly well-defined, yet some people say such a thing does not exist. The question is of "legitimacy" more than whether we can define it at all. May6 comment Irreducible closed subsets of projective varieties Well, it is certainly the case that every closed irreducible subset of a projective variety is a projective variety. But if your definition of "projective variety" does not include irreducibility then even every closed subset of a projective variety is a projective variety. Therefore if $Y$ has a irreducible closed subset that is not a projective variety, then $Y$ cannot be projective. May6 comment Pole of differential That's not correct. Both $x$ and $y$ evaluate to $\infty$ at $[0 : 1 : 0]$. (Otherwise $x$ and $y$ would be functions that are regular everywhere on a projective variety, hence, constant – an absurdity.) May6 comment Irreducible closed subsets of projective varieties Some people include irreducibility in their definition of "variety". May6 comment Pole of differential $x / y$ is a rational function with a simple pole at infinity. Do you understand that calculation? May6 comment Examples of epimorphisms which are not split epimorphisms? You mean coequaliser. May6 comment What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom? Yes, I am aware of Solovay's result. That does not count as a statement in ordinary mathematics to me. May6 awarded Nice Question May5 answered Determine whether two primitive recursive functions are equal May5 comment Confusion about cofinality Yes. But if $[0, \kappa)$ is the set of all ordinals of cardinality less than $\kappa$, then the answer is the same, provided the well-ordering principle is available. May4 comment What's stronger: projective or locally free? flat or locally free? The question does not apply to general abelian categories, since there is no notion of "locally free" and no (obvious) notion of "flat". May4 comment The Axiom of Choice and definability The inductive step is far too vague. Here is one problem: in the set-forming axioms, one can use parameters, so the "constant" terms must have variables. But once you allow that, then one can derive a form of meta-AC: suppose we have a proof of $\forall x . x \in X \to \exists y . y \in x$; then we can construct a definable term $c (x)$ such that $\forall x . x \in X \to c (x) \in x$; then by replacement, $\exists f . (f : X \to \bigcup X) \land \forall x . x \in X \to f (x) \in x$. May4 comment The Axiom of Choice and definability My gut feeling is that your "Theorem" is false, because the same kind of reasoning can be used to "deduce" the axiom of choice. (See, for instance, the derivation of AC in Martin-Löf type theory.) May4 comment Calculating the Picard group of $\mathbb{C}P^1$ in an elementary way @ArthurStuart First of all, what is your definition of Picard group? If you are defining it as $H^1 (X, \mathscr{O}_X^{\times})$ then there's not going to be much chance of explaining it without sheaf theory! May4 answered Confusion about cofinality May4 comment Confusion about cofinality @AndresCaicedo In his notation, $[0, \aleph_1)$ denotes $\{ 0, 1, \ldots, \aleph_0 \}$, which is countable. May4 comment Confusion about cofinality The cofinality of an ordinal is always an initial ordinal, hence, may be regarded as a cardinal. The definition of the cofinality of a cardinal is, in your notation, $\operatorname{cf} \eta (\alpha)$. May3 comment How to define topology in terms of subobjects? There's an obvious way of categorifying the third axiom: say that the subcategory has all colimits. But this is still not enough. You should ask that finite meets distribute over all joins. In any case, one will not get close to the idea of a Grothendieck topology in this way... they are not really related to topologies at all. May3 comment What is the notion for this arrow in category theory? There is not enough context to determine what your notation means. It would be helpful if you could give a precise reference.