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Aug
3
comment ind-completion and functors which are full with respect to isomorphisms
Q1. This is sometimes called "full with respect to isomorphisms". See Definition 4.9 here.
Aug
3
comment Right Kan extension along a diagonal functor.
Why go to the trouble of using the general formula? You just have to check that the given formula does actually define a right adjoint, and that is quite straightforward. More generally, you might like to show that for any functor $f : \mathcal{C} \to \mathcal{D}$, $f_* : [\mathcal{C}^\mathrm{op}, \mathbf{Set}] \to [\mathcal{D}^\mathrm{op}, \mathbf{Set}]$ can be defined by $f_* X (d) = [\mathcal{C}^\mathrm{op}, \mathbf{Set}](\mathcal{D}(f -, d), X)$.
Aug
3
comment Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1 $
Your description of $\Omega^1$ is not correct: $\Omega^0 \ncong \Omega^1$, after all. The transition should be given by $\sum_{n \in \mathbb{Z}} a_n x^n \, \mathrm{d} x \mapsto - \sum_{n \in \mathbb{Z}} a_n x^{n+2} \, \mathrm{d} x^{-1}$.
Aug
2
comment Two questions on the definition of $\mathcal{O}_X(U)$ for an affine scheme $X$.
Yes, that's right.
Aug
2
comment Can a rational map $X\leadsto Y$ be defined as a scheme morphism $Z\to Y$ for some $Z$?
Take $X = \operatorname{Spec} \mathbb{Z}$, or if you prefer, $X = \mathbb{A}^1_\mathbb{Q}$. Then there is no morphism to the generic point, by observing that the function field is not finitely generated as a ring (resp. as a $\mathbb{Q}$-algebra).
Aug
2
comment Can a rational map $X\leadsto Y$ be defined as a scheme morphism $Z\to Y$ for some $Z$?
Indeed. Nonetheless, the point (hah) is that there is a morphism $\operatorname{Spec} (\mathscr{O}_{X, \eta}) \to \operatorname{Spec} (\mathscr{O}_{X, \eta})$ but in general no rational map $X \leadsto \operatorname{Spec} (\mathscr{O}_{X, \eta})$.
Aug
1
comment Natural Transformations Without Objects
I like to think that is because the definition of homotopy is wrong! But unfortunately exponential objects in $\mathbf{Top}$ are a bit complicated to describe (assuming they even exist).
Aug
1
comment Can a rational map $X\leadsto Y$ be defined as a scheme morphism $Z\to Y$ for some $Z$?
Er, $\mathscr{O}_{X, \eta}$ is a field, so its spectrum is just a point...
Aug
1
comment Natural Transformations Without Objects
Use the category $\mathbb{3} = \{ 0 \to 1 \to 2 \}$. (Exercise.)
Aug
1
comment Size of Hom-Sets in A Functor Category
So, you are asking: why is the union of a small family of small sets small? This follows by the replacement axiom of universes.
Aug
1
comment About the functor between varieties over $k$ and $k$-schemes
Well, complex conjugation defines a morphism of schemes $\mathbb{A}^1_\mathbb{C} \to \mathbb{A}^1_\mathbb{C}$ that is not a morphism of $\mathbb{C}$-schemes. Does this example help?
Aug
1
comment What are differences between affine space and vector space?
First, do you understand the definition of affine space that the authors have given? If so, can you distinguish between the notion of a vector space and the notion of an affine space?
Jul
31
comment how would you define the term “elementary” in the context of categories and sets?
Elementary here is meant in the sense of logic, i.e. something related finitary first order logic (as opposed to higher order logic and infinitary logic).
Jul
30
comment For an inductive limit $X = \bigcup X_n$ of vector spaces, show that $X$ is complete if $X_n$ is complete for all $n$
Oh, I suppose I was thinking about Banach spaces rather than topological vector spaces in general. Still, the direct limit $\varinjlim_n \mathbb{R}^n$ has the topology induced by the obvious metric (because the inclusions are all isometries), and that is not complete.
Jul
30
comment The semidirect product as a deformation of the direct product
Well, to be more precise: the semidirect product $G \rtimes_\phi H$ is the product $G \times H$ "deformed" by the parameter $\phi : H \to \mathrm{Aut}(G)$, and if you set $\phi$ to be the constant function with value $\mathrm{id}$, then you get back the ordinary direct product. How is this not already analogous to deformation quantisation?
Jul
30
comment The semidirect product as a deformation of the direct product
What's wrong with the simple fact that the direct product is a special case of the semidirect product?
Jul
30
comment Generalizing a statement about direct limits in the category of $A$-modules to other categories
The embedding is guaranteed to be fully faithful and preserve finite limits and finite colimits (in particular, kernels, cokernels, and finite direct sums), which is enough to establish many of the basic lemmas of homological algebra.
Jul
30
comment Self-duality in a lattice
I think there is a counterexample where $X$ is a (finite) poset rather than a lattice.
Jul
30
comment For an inductive limit $X = \bigcup X_n$ of vector spaces, show that $X$ is complete if $X_n$ is complete for all $n$
The claim seems to be false. Take, for instance, $X_n = \mathbb{R}^n$. Then the inductive limit is a real vector space of countably infinite dimension, and these are never complete (regardless of the topology).
Jul
30
comment Generalizing a statement about direct limits in the category of $A$-modules to other categories
@dorebell No. The embedding does not preserve direct limits in general, so you can't use the embedding theorem to say anything about direct limits.