Zhen Lin
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 Apr13 comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$? But there are non-projective finitely presented modules! Apr13 comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$? Compact objects in the category of modules over a ring (commutative or otherwise), in the sense of Definition 1 on the linked page, are precisely finitely presented modules. Compact objects in the sense of additive categories (esp. triangulated categories) are an even weaker notion. You need to impose projectivity separately. Apr13 comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$? The definition of "compact object" you link to actually refers to finitely presentable objects, not dualisable objects. Apr13 comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$? I would be inclined to say that the category of vector bundles is too small for the notion of monomorphism/epimorphism to behave well. It is better to look at the category of $\mathscr{O}_X$-modules, say. Apr13 comment Lifting adjunctions I don't know what you mean by Yoneda-style proof. Defining the unit and counit here is much easier. Apr13 comment Natural bijection between $\mathbb{N}$ and algebraic numbers? Actually, I just noticed that lexicographic ordering does not give a well-ordering of polynomials. Instead we should order it first by (degree) + (magnitude of largest coefficient), then lexicographically. (Then we can also skip the step of choosing a well-ordering of the integers and use the standard linear order instead.) Apr13 comment Natural bijection between $\mathbb{N}$ and algebraic numbers? I am not an expert in numerical analysis, but my understanding is that it is possible to find roots numerically to any given precision, so we just have to compute the set of roots sufficiently precisely. Infinite precision is not required because each polynomial has finitely many roots. Apr13 comment Can We Say 'There Exists an Element' Without saying 'This One'? In your second paragraph, you seem to be saying that we have the existence property. Surely you don't mean that. Apr13 comment What does it mean for a category to be “tensored over” another category? If $\mathcal{V}$ is symmetric monoidal closed then $\mathcal{V}$ is $\mathcal{V}$-tensored. But I don't see how you're going to get "associativity" for free like that for a general $\mathcal{V}$-category. So there's a tradeoff: either you impose "associativity" by hand in the definition of "tensor", or you require an enriched adjunction. Apr12 comment What does it mean for a category to be “tensored over” another category? Actually, there's a subtlety in the enriched setting: the adjunction itself must be enriched. Otherwise the action is not "associative". Apr12 comment Functorizing a choice of sections There is no reason to believe that a functorial choice of splitting always exists. Apr12 comment Inverse limit of small categories Limits in $\mathbf{Cat}$ are constructed in the obvious way. This is because the functors $\operatorname{ob}, \operatorname{mor} : \mathbf{Cat} \to \mathbf{Set}$ are representable. Apr12 comment Natural bijection between $\mathbb{N}$ and algebraic numbers? The procedure is simple. Fix a well-ordering of polynomials with integer coefficients – to do that, you could take a well-ordering of the integers and use the lexicographic ordering. Then throw away all the ones that are not irreducible. This can all be done computably, maybe even primitive-recursively. Then if you really wanted to, you can numerically approximate the complex roots and then order them lexicographically. Apr12 comment Natural bijection between $\mathbb{N}$ and algebraic numbers? In that case, would you consider the enumeration of polynomials that I suggest to be "natural"? Apr12 comment N-Tuples or N-functions in category theory You can either use products (which is the usual way) or exponential objects (which corresponds to curried functions). Apr12 comment Natural bijection between $\mathbb{N}$ and algebraic numbers? @ThomasAndrews It's not so bad though – you could write a program (for an ideal computer with infinite memory) that enumerates all of them. Apr11 comment Base-free proof: Set of generators is Zariski-open I don't think you can really say you are "writing down equations" if you don't explicitly define $f$... Apr11 comment Understanding the isomorphism of Picard group with the first cohomology group I echo John's comments. Actually, I would say it more strongly: Čech cohomology is the best way of understanding this isomorphism. Apr10 comment Base-free proof: Set of generators is Zariski-open How do you propose to write down equations if you don't have a basis? Apr10 comment What is an example of a proof that explicitly relies on the law of excluded middle? This use of the law of excluded middle can be eliminated: the argument in the "not" branch works just as well.