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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
2
revised Adjunction counit for sheaves is isomorphism
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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
answered Natural Transformations Without Objects
Aug
1
revised Size of Hom-Sets in A Functor Category
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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
answered What are differences between affine space and vector space?
Aug
1
revised What are differences between affine space and vector space?
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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
31
awarded  Nice Question
Jul
31
awarded  Popular Question
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.