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Oct
16
comment Yoneda implies $\text{Hom}(X,Z)\cong \text{Hom(}Y,Z)\Rightarrow X\cong Y$??
It also has to be a natural bijection, or at least, natural enough.
Oct
16
comment Variety of Connected Components
Isn't it clear that $\pi_0 (X)$ should be a disjoint union of $n$ copies of $\operatorname{Spec} k$, where $n$ is the number of connected components?
Oct
15
comment Let $\pi: E \to M$ a vector bundle. Is $E$ a direct summnad of $M\times\mathbb{R}^{d}$, for some $d$?
Note that the base is assumed to be compact Hausdorff.
Oct
15
comment Can a model (of a general theory) be viewed as a (less general) theory?
A group does not "contain" representations. Moreover, representations are not themselves groups but rather homomorphisms of groups. There are many other objections. Please read an introduction to first-order logic.
Oct
15
comment kan extension,(co)ends,natural bijection
Did you look at [CWM, Ch. X §4]?
Oct
15
comment Some Galois theory
Search for "simple extensions".
Oct
14
comment Singular homology of cofinite topology space
My impression is that these spaces are contractible.
Oct
13
comment How can I visualize principal bundles?
@NajibIdrissi It is the easiest non-trivial example of a principal $G$-bundle that can be embedded in $\mathbb{R}^3$.
Oct
13
comment How can I visualize principal bundles?
There is a unique non-trivial principal $O(1)$-bundle over $S^1$, namely the unique connected double cover of $S^1$. This is easy enough to visualise.
Oct
12
comment Confusion in basic defintion of sheaf cohomology
The meaning of exactness depends on the category.
Oct
11
comment (Are there) subtleties in the definition of 'biproduct'
Well, consider an infinite set $X$. Then for cardinality reasons, $X + X \cong X \times X$, yet we do not say that it is a biproduct.
Oct
10
comment Under which assumptions counit of the adjunction $f^* f_* \to 1$ is epimorphic?
The counit of an adjunction is a componentwise epimorphism if and only if the right adjoint is faithful. However, being an epimorphism is not the same as being surjective...
Oct
10
comment Hartshorne Chapter II exercise 5.7 on Invertible sheaves
That's a good approach. You might want to think about what happens when you tensor with the residue field. What does that imply about the number of generators you need?
Oct
10
comment does a commutative diagram implies pull-back?
No. The dimensions don't even have to match.
Oct
10
comment Show that C is a split exact chain complex if and only if the identity map on C is null homotopic.
Start by unfolding the definitions.
Oct
8
comment Seeking help to understand a simple Kripke model
You can prove in intuitionistic logic that LEM is equivalent to DNE. So LEM fails if and only if DNE fails.
Oct
8
comment How to show $\mathbb{A}_k^2 - \{ (x,y) \}$ is not an affine scheme
Well, it is a matter of fact that the inclusion induces an isomorphism of coordinate rings. So if both the domain and the codomain were affine then it would have to be an isomorphism, but it isn't even bijective on points.
Oct
8
comment Colimits in the 2-category of partial functions (which is locally posetal)
More precisely: if bi(co)limits or pseudo(co)limits exist, then they are ordinary (co)limits in particular. This is not true for a general bicategory.
Oct
8
comment Colimits in the 2-category of partial functions (which is locally posetal)
Bi(co)limits and pseudo(co)limits in locally posetal categories reduce to ordinary (co)limits (assuming existence). They beocme more interesting in locally preordered categories.
Oct
8
comment Does Extremal Mono imply Split Mono in (Epi, Regular Mono)-factorization categories?
Every regular monomorphism is in particular extremal. So you are claiming that every regular mono splits, which is of course false.