All Questions
Tagged with limits-colimits sheaf-theory
37
questions
2
votes
0
answers
138
views
Cech cohomology on infinite open cover commutes with colimit on Noetherian space? (Exercise 5.2.6 in Qing Liu's book)
This is Exercise 5.2.6 in Qing Liu's book Algebraic Geometry and Arithmetic Curve. In part (b), I can show (b) if the open covering has only finitely many open subsets, since the the colimit of the ...
- 1,785
3
votes
1
answer
180
views
On kernels and stalks of sheaves.
Suppose we're given sheaves $F,G$ on a space $X$ and a morphism of sheaves (of abelian groups) $\phi:F\to G$.
I want to prove two things :
the presheaf $\ker \phi$, defined by $(\ker\phi)(U):=\ker(\...
- 1,048
1
vote
0
answers
59
views
Sheaves valued in a $k$-category
Let $\mathcal{C}$ be a $k$-category which we regard as an $\infty$-category whose objects all happens to be $k$-truncated.
A sheaf $F$ valued in $\mathcal{C}$, in the $\infty$-categorical sense, is a ...
- 1,613
0
votes
1
answer
70
views
Does invertibility of a section $f$ (of a sheaf of rings) in every open set containing $x$ imply invertibility of $f$ in the stalk at $x$?
Let $\mathcal{O}$ be a sheaf of commutative rings on a topological space $X$.
Let a point $x \in X$ and a global section $f \in \mathcal{O}(X)$ be given.
Suppose that for every open $U \subset X$ with ...
- 2,461
4
votes
1
answer
141
views
Why should the target category of a sheaf be complete?
The categorical definition of sheaf given on Wikipedia (https://en.wikipedia.org/wiki/Sheaf_(mathematics)#Complements) requires that the target category $\mathcal{C}$ of the functor $\mathcal{F}:\...
- 4,794
4
votes
2
answers
347
views
Is the limit of a family of sheaves a sheaf?
So, I can prove that the kernel of a morphism of sheaves or a product of sheaves is a sheaf, but I do not know how to prove in general that $lim F_{i}$ is a sheaf for $F_{i}$ sheaves. I know that if ...
- 697
2
votes
1
answer
118
views
Does the (pseudo)functor that assigns a commutative monoid $M$ to the topos of $M\text{-Sets}$ preserve limits? [closed]
Let $\mathrm{CMon}$ be the category of commutative monoids, and $\mathrm{Topos}$ be the bicategory of (Grothendieck) toposes with geometric morphisms. Consider the (pseudo)functor $\mathrm{CMon}\to\...
- 1,278
0
votes
2
answers
456
views
The stalk as a colimit
From Vakil's notes:
The notion of colimit is defined for diagrams $D:I\to\textbf{Set}$. The colimit of $D:I\to\textbf{Set}$ is the limit of $D^{op}:I^{op}\to \mathbf{Set}^{op}$. What exactly is the ...
- 12k
4
votes
0
answers
375
views
Sheaf Axioms and Limits - Intuition
The question is based on the following problem from Vakil's notes in Algebraic Geometry:
2.2.C. The identity and gluability axioms (of sheaves) may be interpreted as saying that $\mathcal{F}(\cup_i ...
- 990
2
votes
2
answers
131
views
$\ker \varphi_p \subset (\ker \varphi)_p$ where $(\cdot)_p$ is taking the stalk of sheaves at the point $p$ (Diagram inside!).
I've already proven that $(\ker \varphi)_p \subset \ker \varphi_p$ using a commutative diagram and the definition $F_p = \lim\limits_{\longrightarrow \\ U \ni p} F(U) = \bigsqcup\limits_{U \ni p} F(U)/...
0
votes
0
answers
369
views
Injective map and surjective map between inductive limits
I'm learning Hartshorne's Algebraic Geometry and trying to solve the following exercise: Exercise II.1.4
Let $\phi: \mathcal{F} \rightarrow \mathcal{G}$ be a morphism of presheaves such that $\phi(U):...
- 2,177
1
vote
1
answer
158
views
Cosheaf on a base
There is the well-known construction of sheaves on a base, i.e. rather than specifying a sheaf $S$ on all open sets of a topological space $M$, specify its data only on a topological base $\mathcal{B}$...
- 1,663
1
vote
1
answer
340
views
Show that $\Gamma$, $\Lambda$, and the associated sheaf functor are all left exact.
This is Exercise II.6 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to the first few pages of this Approach0 search, it is new to MSE.
The Details:
The ...
- 45.7k
1
vote
1
answer
247
views
A presheaf $P$ on $X$ is a sheaf iff for every covering sieve $S$ on an open set $U$ of $X$ one has $PU=\varprojlim_{V\in S}PV.$
This is Exercise II.2 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]".
The Details:
Adapted from Adámek et al.'s, "Abstract and Concrete Categories: The Joy of Cats&...
- 45.7k
1
vote
1
answer
151
views
Inverse image functor and restriction of colimits
Let $f:X\longrightarrow Y$ be a morphism of topological spaces. I want to prove, that the inverse-image functor of (set-valued) presheaves
\begin{equation}
f^{-1} : \mathrm{PSh}(Y)\longrightarrow\...
- 329