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Apr
19
comment Is it useful to view magmas as diagrams?
The data is a diagram, and the axiom is that the diagram commutes!
Apr
18
comment When $\operatorname{Hom}_{R}(M,N)$ is finitely generated as $\mathbb Z$-module or $R$-module?
If $M$ is finitely-generated and $N$ is noetherian, then $\textrm{Hom}_R (M, N)$ is also finitely-generated (indeed, noetherian).
Apr
18
comment Why is a variety etale locally like affine space?
I doubt it. Take $Y = \operatorname{Spec} k$ and $X = \mathbb{A}^2_k \setminus \{ (0, 0) \}$; then we must have $V = Y$ and $d = 2$. It's not clear to me where one would get a étale surjection $U \to \mathbb{A}^2_k$ where $U$ is an affine open subscheme of $X$.
Apr
18
comment Intuition behind the Axiom of Choice
@AsafKaragila I do think AC is a kind of local-to-global principle of the same kind as induction, however: it says, if I can make a choice locally (i.e. for each member of a set), then I can make a choice globally (i.e. for the whole set at once).
Apr
17
comment explicitly represent a representable functor
Read the penultimate paragraph.
Apr
17
comment What analysis is needed for AG?
If you haven't done some differential geometry, then the notion of tangent space or smooth variety will probably be somewhat mysterious. But you don't need to know differential geometry, because the definitions in algebraic geometry are purely algebraic!
Apr
17
comment What analysis is needed for AG?
The main text of Hartshorne does not need any analysis whatsoever... but it may be difficult to appreciate what is going on, as Georges Elencwajg said.
Apr
15
comment Open subset of irreducible affine curve is an affine curve
Regard the open subset as a topological space by using the subspace topology, then apply the usual definition of dimension.
Apr
14
comment Worst category with first isomorphism?
The first isomorphism theorem says (essentially) that every morphism factors as an effective epimorphism followed by a monomorphism, which is precisely what happens in a regular category.
Apr
14
comment Connected groupoids and action groupoids
How about the group of automorphisms of the action groupoid itself?
Apr
14
comment Quotient space and Retractions
Question 1: You could say that it is an effective epimorphism, or even an extremal epimorphism.
Apr
14
comment Elephant: how do I prove Lemma 2.1.7, section C2.1?
@user72134 I know of at least one other mistake in this section, so it is not inconceivable. My claim about the existence of $\mathfrak{Y}$ is axiom (C) for coverages, and the sheaf condition for $\mathfrak{Y}$ says, among other things, that equality of elements of $\mathscr{F}(Y)$ can be detected by restricting along all the morphisms in $\mathfrak{Y}$.
Apr
14
comment Can intuitionistic proofs be laid out in a format that parallels reductio ad absurdum?
I think the short answer is no, but I don't see how to justify it. It is probably related to the reason why intuitionistic sequent calculus only allows one formula on the right of the turnstile.
Apr
14
comment Understanding the Yoneda lemma
A natural transformation is not a map $\textrm{Hom}(x, a) \to F x$, it is a family of such maps satisfying certain equations.
Apr
13
comment What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?
Nope, I don't understand that either. (It would be extremely surprising if it does turn out to be decided by the universe axiom, however!) I guess what I want to understand is how it is that the existence of "big" sets can affect "small" sets.
Apr
13
comment What are the “ordinary” (e.g. arithmetic) consequences of the universe axiom?
I don't understand what $0^\sharp$ is, so that surely can't be down-to-earth!
Apr
13
comment What is the point of extremal epimorphisms in category theory? Why not just use strong epis instead?
On the other hand, if you are trying to prove the uniqueness of epi–mono factorisations, you would need a lifting condition like that of a strong epimorphism.
Apr
13
comment Do Groebner bases give the smallest generating set for Ideals?
Is the ideal $(x z - y^2, x^2 y - z^2, x^3 - y z)$ a counterexample?
Apr
12
comment Why is 'isomorphism' defined more generally in Category theory than in Abstract Algebra?
Except for what they teach first-year undergraduates (bijective homomorphism)...
Apr
12
comment Are there versions of the axiom of choice that restrict the size of the factors?
You need AC to choose a bijection for all $s$ at once. To use Conway's terminology, products of counted sets are inhabited, but products of countable sets could be anything.