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Jun
3
comment Beck-Chevalley via coend-calculus
Of course, one can construct the Beck–Chevalley transformation by pasting natural transformations together as well, which is what these manipulations are hiding. Indeed: $$h_! k^* \Rightarrow g^* g_! h_! k^* \Rightarrow g^* f_! k_! k^* \Rightarrow g^* f_!$$
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 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
answered What is the relationship between the path-loop space fibration and path induction?
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
25
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