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Nov
1
comment Showing that representations are isomorphic if the “forgetful representations” are isomorphic.
If you do know that it is the direct limit (in the appropriate category), then sure.
Nov
1
comment Logic and geometry
The connection is nothing so superficial. On the one hand, to every logical theory of a certain type is associated a "classifying topos", which is a geometric object whose "points" correspond to models of the theory, and many properties of the classifying topos correspond to properties of the theory. This is at the heart of the "bridges" technique expounded by Olivia Caramello. On the other hand every topos is a mathematical universe unto itself and gives rise to new interpretations of intuitionistic logic, the so-called "Joyal–Kripke semantics".
Oct
31
comment Given, the cartesian product of two non-empty sets A and B (subsets of a metric space M) is sequentially compact, show that A and B are compact
Do you mean compact metric spaces? In general sequential compactness is not the same as compactness. Shawn's suggestion is the easiest proof for general compact Hausdorff spaces. The easiest proof for general compact topological spaces is to use projections and the fact that the image of a compact space is again compact.
Oct
31
revised A question about Killing vector and Riemann curvature tensor
edited body
Oct
31
comment For what algebraic curves do rational points form a group?
The set of rational points always has a group structure because any non-empty set has a group structure. But some group structures are more natural than others – for example, it is well-known that the rational points of an elliptic curve form a group...
Oct
31
comment Left adjoint and right adjoint/ Nakayama isomorphism
There is a Nakayama lemma in ordinary commutative algebra, but I don't see how this is related. As for the double description – when you have a pullback $f^*$ defined by $f^* G (y) = F(f(y))$, then, assuming you have enough limits and colimits, $f^*$ has left and right adjoints $f_!$ (a.k.a. left Kan extension) and $f_*$ (a.k.a. right Kan extension). But usually these are very different kinds of functors.
Oct
30
comment Almost A Vector Bundle
The naïve (i.e. fibrewise) kernel of a bundle homomorphism can fail to be a vector bundle: just a take a bundle homomorphism whose rank jumps.
Oct
30
comment Is there any way to define morphisms between filters in order to get a category, one which its opposit category would be the category of ideals?
That's not the sense in which filters and ideals are dual. Rather, a filter is a certain kind of subcategory of a poset (thought of as a category), and if you take the opposite of the poset, then filters become ideals.
Oct
30
comment Differentiable manifolds as locally ringed spaces
Yes. The point is that once you have a morphism of ringed spaces then you know that the map has an expression in local coordinates that is smooth/analytic/algebraic etc. as according to the nature of your structure sheaf. Brian Conrad has notes on the locally ringed space approach to differential geometry, if I recall correctly.
Oct
30
comment Why $C^{\infty}(M)$ module of sections of a vector bundle $E\rightarrow M$ is a reflexive module?
Surely this just boils down to the fact that it's true fibrewise?
Oct
30
comment What is a complete lattice?
That's just bad notation. What he means is, $\bigcap S$ is smaller than anything in $S$, and $\bigcup S$ is bigger than anything in $S$.
Oct
29
comment When are kernels (resp. cokernels) finite limits (resp. finite colimits)?
Assuming you have the right definition of kernel and cokernel, of course, to say nothing of the assumptions needed so that such things even make sense...
Oct
29
accepted Prime ideals of height less than the dimension
Oct
29
asked Prime ideals of height less than the dimension
Oct
29
comment What is the intuition behind the Lie derivative of a vector field.
It's just a long laborious calculation, I'm afraid.
Oct
29
comment What is the intuition behind the Lie derivative of a vector field.
It measures the "difference" between two flows. If you take local coordinates and expand $\Phi^X_{-t} \circ \Phi^Y_{-t} \circ \Phi^X_t \circ \Phi^Y_t$ as a power series in $t$, you should find that the $t$ term vanishes and the next term is $[X, Y] t^2$. (Equivalently, take the second derivative.)
Oct
28
comment Non-trivial homomorphism between multiplicative group of rationals and integers
Asking about homomorphisms $f : \mathbb{Z} \to G$ for any group $G$ is boring: any such $f$ is freely and uniquely determined by $f (1)$.
Oct
28
comment Calculating the Fundamental group of $\Bbb R P^2$
It's neither. The fundamental group of $\mathbb{R P}^2$ is $\mathbb{Z} / 2 \mathbb{Z}$.
Oct
28
comment Does $a = 0$ iff $a \equiv 0 \ (\operatorname{mod} p)$ for every prime $p$?
By martini's observation, we see that not every sequence of residues comes from an integer. So what is the nature of the ring $\mathbb{F}_2 \times \mathbb{F}_3 \times \mathbb{F}_5 \times \cdots$? Hmmm...
Oct
28
comment A property of a sheaf in an arbitrary category
You also need to assume $\mathbf{C}$ has, say, filtered colimits in order to make sense of stalks at all. I suspect if you could make filtered colimits in $\mathbf{C}$ behave sufficiently badly, then you could construct counterexamples. But why on earth would you want to do that? The reason why things are easy in the case when $\mathbf{C}$ is a category of "algebras" is because then it can be embedded nicely in the topos of sheaves of sets, and then properties like "having enough points" are just inherited from the topos.