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Sep
11
comment Direct and inverse limits of sheaves
Since no one has mentioned it, I'll just quickly say that sheafification is a left exact left adjoint, so it preserves all colimits and all finite limits. In particular it is an exact functor.
Sep
10
comment Why can a map be factored out if $\ker(f) ⊆ \ker(h)$
It is an equivalence relation, if you think of binary relations on a set $X$ as being subsets of $X \times X$. The proof of this is easy and amounts to the fact that $=$ is an equivalence relation on the codomain. The fact that the quotient by this equivalence relation has a natural bijection with the image is more-or-less automatic once you realise the equivalence relation is "$x \sim y$ if and only if $f(x) = f(y)$".
Sep
10
comment Why can a map be factored out if $\ker(f) ⊆ \ker(h)$
This website is not suitable for discussions – it is a question-and-answer site.
Sep
10
comment Why can a map be factored out if $\ker(f) ⊆ \ker(h)$
This is called the kernel pair of $f$. It is worth noting that it defines an equivalence relation on $\operatorname{dom} f$, and the quotient of $\operatorname{dom} f$ by this equivalence relation has a natural bijection with $\operatorname{im} f$. It is in fact very interesting.
Sep
10
comment Rings' second isomorphism theorem
Of course, we could also say $S / (I \cap S)$ instead, and the two are isomorphic.
Sep
10
comment What do those different separators of arguments mean?
It's a way of avoiding confusion when you have a family of functions that can be parametrised in different ways.
Sep
10
comment Notation to work with vector-valued differential forms
Different alphabets. Note that indices related to the coordinate basis are written in Latin while indices related to the bundle frame are written in Greek.
Sep
10
comment Notation to work with vector-valued differential forms
It all transforms as you expect, so you can pretend it's abstract index notation if you like.
Sep
10
comment Why use ZF over NFU?
Feferman [2011] proposes a system that extends both ZFC and NFU, so apparently you can have (a bit of) cake and eat it too!
Sep
10
comment Does $\mathbb{Z}_5(\sqrt[3]{3})$ make sense? Or, can we always extend a field by a root of a reducible polynomial?
You can talk about the ring $\mathbb{F}_5 [x]/(x^3 - 3)$, but it is not a field.
Sep
9
comment Why use ZF over NFU?
NFU is known to be of weaker consistency strength than PA. (!) Holmes has a very readable survey of alternative set theories. Models of NFU can be constructed from models of Zermelo equipped with an automorphism that moves a rank. Forster conjectures that plain NF is also quite weak.
Sep
9
comment Understanding the trivialisation of a normal bundle
The normal bundle is the orthogonal complement of the tangent bundle of a (Riemannian) manifold embedded in another. It is a generalisation of the normal vector field of a surface embedded in $\mathbb{R}^3$.
Sep
9
comment Completeness and Topological Equivalence
Being compact is a topological property, and a metric space is compact if and only if it is complete and totally bounded. I suppose the trick is to find a way of replacing any metric with a topologically equivalent totally bounded one...
Sep
9
comment Question on the definition of ample vector bundles
Isn't this just the relative form of the tautological isomorphism $\mathscr{O}_{\operatorname{Proj} S}(1) \cong S_1$?
Sep
9
comment Tablet for reading textbooks and writing math by hand?
The iPad is not designed for stylus input. PC tablets (a dying breed!) are better, but I still find it hard to get used to...
Sep
8
comment Is the dualizing functor $\mathcal{Hom}( \cdot, \mathcal{O}_{X})$ exact?
I'm sure the point Matt is making is that the next term in the sequence is $\mathscr{E}xt^1(\mathscr{O}_X, \mathscr{O}_X)$, which is zero by the observation.
Sep
8
comment Pointfree generalization of uniform spaces?
Yes: uniform locales. Isbell discusses them in his 1972 paper, though I'm not sure if the definition on nLab is the same as his.
Sep
8
comment Is the dualizing functor $\mathcal{Hom}( \cdot, \mathcal{O}_{X})$ exact?
@KeenanKidwell A good point. But if I'm not mistaken, when $A$ has a projective resolution and $B$ has an injective resolution, then you can compute $\textrm{Ext}(A, B)$ using either and get the same answer. (If you have enough projectives and enough injectives, at some point you will have to show that the derived functors of $\textrm{Hom}(A, -)$ and $\textrm{Hom}(-, B)$ give the same answer!) Weibel gives a proof of this in the category of $R$-modules using the total complex of a double complex [Thm 2.7.6], but I think everything carries over without change to $\mathscr{O}_X$-modules.
Sep
8
comment Cokernels - how to explain or get a good intuition of what they are or might be
It's worth mentioning that there are semi-abelian categories in which $\ker f = 0$ is the same thing as $f$ being monic, but $\operatorname{coker} f = 0$ does not necessarily imply $f$ is epic.
Sep
8
comment The category of adjoint functors
The one in the main text (with "conjugate" pairs of natural transformations) is standard and can be made into part of a double category: see the examples here. It's not clear to me why you want to make $\textrm{Adj}(-, -)$ into a functor though.