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Aug
29
comment intuition of mass function of random variable
@Did What's in a "size?" Obviously I mean the measure. But Ian's answer is much better.
Aug
28
comment Is there a way to visualize a group?
Good question on the first one, but the question of how to visualize a group seems to be totally separate from the question of why we can characterize injective group homomorphisms as those with trivial kernel. Maybe you want to separate that out as a second question.
Aug
27
answered intuition of mass function of random variable
Aug
23
answered Isomorphism between colimits.
Aug
23
comment Vector bundle morphism as section of a bundle?
Yep, that's right.
Aug
19
comment Mathematical formalism for the “dot product” of three vectors
You can certainly compute this, but the dot product has an actual meaning-it's used to compute the angle between two vectors.
Aug
18
comment What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?
Yeah, this definitely isn't the currying isomorphism from sets.
Aug
18
comment What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?
Oh, yeah, you're right.
Aug
18
comment What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?
This is basically the first example of an adjunction, so I'm not sure what your last sentence means. The homs can certainly be taken over $C$, if you replace $A\times B$ with $A\otimes B$.
Aug
18
comment What explicitly is the “adjunction” isomorphism $Hom_C(A,C)\to Hom_B(A,Hom_C(B,C))$?
There's no reason for this to be an isomorphism-perhaps you've misread something? Consider $A=B=C^2$, where the left-hand side is $C^2$ and the right-hand, $C^4$.
Aug
17
comment Why did Serre choose coherent sheaves?
There are several proofs of the basic theorem. A good place to start looking for a more specific reference might be Akhil Matthew's blog post here: amathew.wordpress.com/2010/11/15/the-cohomology-of-affine-space/…
Aug
17
comment Vector bundles in Ravi Vakil's notes on quasicoherent sheaves
Basically the same reason as locally free sheaves do, or as any additive category does. You want to have kernels and exact sequences and cohomology. Most treatments of vector bundles will be in topology, not algebraic geometry. Hatcher has a book on vector bundles and K-theory, but it's hard to know what to recommend if you haven't had much topology. Usually one picks it up on the way to knowing either algebraic or geometric topology.
Aug
17
answered Why did Serre choose coherent sheaves?
Aug
17
comment Vector bundles in Ravi Vakil's notes on quasicoherent sheaves
To your second question, the whole category of vector bundles tempts one to do homological algebra! But perhaps I've misunderstood. As to the first, yeah, I think people like vector bundle because they're easier to visualize than sheaves and because most people learn topology before algebraic geometry.
Aug
17
comment Is $O(2)$ really not isomorphic to $SO(2)\times \{-1,1\}$?
@YilongZhang yep.
Aug
3
comment Natural operators in differential geometry?
You might look at Kolar's book, "Naturality in Differential Geometry," which has a free PDF under Google. Naturality means vaguely that an operator is defined in a way that doesn't use properties of the object it's defined on, and has a precise definition in category theory. All in all, this question admits only a very long answer, if any.
Aug
2
comment Extending a Field Monomorphism
Well, I would look into (A3.4).
Jul
29
comment Dense Domain: Preimage
But I still think if you take $Y=\ell^1,W$ the sequences with finite support, $X=\mathbb{R}, A(1)=(1/n^2 e_n)$ that you'll get a counterexample, for then $A^{-1}W=\{0\}$.
Jul
29
comment Dense Domain: Preimage
Oh, silly, I wasn't thinking of $W$ as a linear subspace.
Jul
29
comment Which directed graphs correspond to “algebraic” diagrams?
@Batominovski but of course you just require all starting points to satisfy the condition.