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Jun
2
comment Decomposing Sheaves into a direct sum
This is somewhat misguided. Have you thought about the analogous questions for, say, modules over a ring?
Jun
2
comment Beck-Chevalley via coend-calculus
Anyway, the coend calculus is not a good way of verifying the Beck–Chevalley condition. For one thing, the condition is that a specific natural transformation is invertible, not that a natural isomorphism exists, and the coend calculus makes it hard to check that the natural isomorphism you get is the one you want. It should be doable in principle, but I never managed to do it.
Jun
2
comment Beck-Chevalley via coend-calculus
What? The Beck–Chevalley condition is not automatic.
Jun
2
comment etale morphism between sheaves
Your definition is not the usual one – it is too restrictive if your site is the category of affine schemes – but it does have the advantage of making it easy to prove the cancellation property you want.
Jun
1
comment Pullbacks and sections
There is no reason to expect it to be the identity. After all, there are usually lots of sections.
Jun
1
comment etale morphism between sheaves
What is your definition of étale morphism of sheaves?
Jun
1
comment Pullbacks in the Ind-completion
The claim implies that $[\mathcal{J}, \mathbf{Ind}(\mathcal{C})]$ is a finitely accessible category where the finitely presentable objects are the diagrams whose vertices are finitely presentable. You have to express every diagram as a filtered colimits of such diagrams – but in fact you just have to check that the canonical diagram whose vertices are all finitely presented diagrams works.
May
31
comment Pullbacks in the Ind-completion
Well, we don't really need that in full generality – it would be enough to know it for $\mathcal{J} = \emptyset$ and $\mathcal{J} = \{ \bullet \rightarrow \bullet \leftarrow \bullet \}$; the former is easy and the latter can be done by hand.
May
31
comment Fibrations over topoi
There's nothing special about $\mathcal{S}$ being a topos. Just look up indexed categories and fibred categories in general.
May
31
comment The contrapositive
@Jared You need to learn about intuitionistic logic.
May
31
comment How to read “realize the mapping $x \cdot -: T \rightarrow T$”
It's not a minus, it's a dash. It indicates a blank.
May
30
comment Monomorphism preservation by pullback
You need to check your understanding of the definition of monomorphism. To show that $f'$ is a monomorphism, you start by assuming $f' \circ n = f' \circ m$; what you need to prove is that $n = m$.
May
29
comment Description of free Lie algebra in Weibel's book
Perhaps you are supposed to use the PBW theorem to embed $\mathfrak{g}$ in its universal enveloping algebra.
May
28
comment Examples of base points of linear systems
The complete linear system of $D$ has no base points. But $\{ D \}$ is not the complete linear system of $D$.
May
28
comment Examples of base points of linear systems
There are very simple examples. For instance, take $D = P$; then the singleton $\{ D \}$ is a linear system, and $P$ is a base point of this linear system.
May
27
comment The coproduct of a family of objects of a Preorder (seen as a category)
Yes, it is the least upper bound. The proof is the same, except for some fiddling with equality.
May
25
comment is axiom of powers required?
Is your collection of singletons actually a set? You can only take unions over sets.
May
25
comment What is the relationship between the path-loop space fibration and path induction?
Well, based path induction is more or less the statement that the space $P X$ is contractible (plus some auxiliary facts about contractible spaces). But $\Omega X$ doesn't come into it.
May
22
comment Is it possible to develop differential geometry without points?
Locales are like topological spaces, not manifolds. Besides, every Hausdorff space is sober, so the category of Hausdorff spaces embeds as a full subcategory of the category of locales.
May
21
comment Is $\coprod \subseteq \prod$ true in any (complete cocomplete) Abelian category?
Well, no: if it were true then the dual result would force it to be an isomorphism, which is absurd.