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Apr
15
answered Is there a more general obstruction to the existence of moduli spaces than the existence of automorphisms?
Apr
15
comment Questions about the function fields of complex algebraic surfaces
Well, for a start, you should take $f : X \to Y$ to be dominant. And then you should probably also impose a finiteness condition on $f$.
Apr
15
comment Why do counits go that way?
I think the easiest answer is that it would break self-duality.
Apr
15
comment Why is the internal hom of a Kan complex also a Kan complex?
There is an argument of that type in [Goerss and Jardine, Ch. I, §4].
Apr
15
comment Why is the internal hom of a Kan complex also a Kan complex?
No. It only works for sufficiently nice model categories, e.g. cartesian model categories in which all objects are cofibrant and cofibrations = monomorphisms.
Apr
14
comment When is $\mathbb{Q}(x)$ a finite extension of $\mathbb{Q}$?
@RobertLewis Use Zorn's lemma to construct transcendence bases of $\mathbb{C}$ and $\overline{\mathbb{Q}}$, and then use Zorn's lemma again to construct the appropriate automorphisms by extending automorphisms of $\overline{\mathbb{Q}}$.
Apr
14
answered Why is the internal hom of a Kan complex also a Kan complex?
Apr
14
comment Functorizing a choice of sections
I think $\mathcal{T}'$ is supposed to be the category obtained from $\mathcal{T}$ by freely adjoining a section for every morphism in $\mathcal{T}$ subject to the constraint that the sections compose.
Apr
14
comment Subobjects(A) $\cong \operatorname{Hom}(A,\Omega)$ in a topos is a natural transformation?
Well, it is a correct observation, but I don't think it is especially interesting. You can get rid of $\Omega$ and turn it into a statement about pullbacks: the pullback of $n \circ m$ along $n$ is $m$ again, which is what you expect.
Apr
14
revised Subobjects(A) $\cong \operatorname{Hom}(A,\Omega)$ in a topos is a natural transformation?
deleted 31 characters in body
Apr
14
comment Subobjects(A) $\cong \operatorname{Hom}(A,\Omega)$ in a topos is a natural transformation?
The directions of your morphisms are mixed up. Think through everything carefully and re-ask your question.
Apr
14
answered Representability criterion with universal element
Apr
14
comment First axiom of sheaves: in noetherian topological spaces the direct limit presheaf is a sheaf.
The point is indeed that every cover of every open subset of a noetherian topological space has a finite subcover. So everything that needs to be amalgamated can be amalgamated by going "far enough" in the direct system.
Apr
14
comment Representability criterion with universal element
You are correct: a universal element for $X$ is precisely an initial object in $(1 \downarrow X)$.
Apr
14
comment Is GRP a subcategory of SET, or not?
See also this question.
Apr
14
comment Is GRP a subcategory of SET, or not?
Well, is a group a set, or not? Think very carefully.
Apr
13
comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$?
But there are non-projective finitely presented modules!
Apr
13
comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$?
Compact objects in the category of modules over a ring (commutative or otherwise), in the sense of Definition 1 on the linked page, are precisely finitely presented modules. Compact objects in the sense of additive categories (esp. triangulated categories) are an even weaker notion. You need to impose projectivity separately.
Apr
13
comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$?
The definition of "compact object" you link to actually refers to finitely presentable objects, not dualisable objects.
Apr
13
comment What are monomorphisms in the category of real vector bundles over a fixed base space $X$?
I would be inclined to say that the category of vector bundles is too small for the notion of monomorphism/epimorphism to behave well. It is better to look at the category of $\mathscr{O}_X$-modules, say.