Questions tagged [presheaves]

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Is extension of a presheaf on a base to a presheaf on the whole space a left adjoint?

Let $\mathcal{B}$ be a base for a topological space $X$. We denote by $\mathbf{PSh}_\mathcal{B}$ the category of presheaves on the base $\mathcal{B}$, and by $\mathbf{PSh}$ the category of presheaves ...
Marc-André Brochu's user avatar
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How does one define the observable presheaf?

My professor had limited to talk about matrices, then since the functional calculus for self-adjoint matrices $M \in \mathbb{M}_{n\times n}$ is such that $\sigma(f(M))=f(\sigma(M))$ he then defined: ...
Felipe Dilho's user avatar
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Can the functor to empty sets be defined?

Can these functors from C to Sets be defined? Functor $F$ from one object $A$ and its identity arrow to a empty set $F(A)$ = (empty set) $F(id_A)$ = (function from empty set to empty set) Functor G ...
qS2's user avatar
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Does a morphism of presheaves extend to a morphism of the corresponding sheaves of discontinuous sections uniquely?

The textbook George Kempf: Algebraic Varieties defines a presheaf as a contravariant functor from the category of the topology of a space $X$ with ordering by inclusion to Set. For a presheaf $F$, its ...
Muhammad Mursaleen's user avatar
2 votes
1 answer
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(Pre)Sheaves of abelian groups

Let $X$ be a topological space, $A$ an abelian group, $p\in X$. Define the presheaf $i_p(A)$ to be $$i_p(A)(U):=\begin{cases} \text{A}, & \text{if } p\in U \\ 0, & \text{otherwise} \end{cases}$...
Mario's user avatar
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1 answer
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Natural transformations and canonical maps

I’ve had a lingering question that I haven’t been able to fully resolve. I’ve often noticed that many demonstrations don’t detail the proof of natural transformation, instead simply stating that the ...
Antonio Hernando's user avatar
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1 answer
71 views

Hom of sheafification [closed]

Let $\mathcal{F},\mathcal{G}$ be presheaf of modules. Can we say that there exist a bijection between $\hom(\mathcal{F},\mathcal{G})$ and $\hom_{\mathrm{sheaf}}(\mathcal{F}^{\sharp},\mathcal{G}^{\...
Hamed Khalilian's user avatar
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Is the quotient of separated presheaves separated? [closed]

Let $X$ be a topological space. If $A \leq B$ are Abelian group-valued sheaves on $X$, then the presheaf quotient $U \mapsto B(U)/A(U)$ is a separated presheaf. Does this still hold if we require $A$ ...
Adelhart's user avatar
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2 votes
0 answers
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Questions about étale space construction of sheafification

I'm trying to wrap my head around sheafification via the étale space construction from, say, Hartshorne's or Liu's books on algebraic geometry (I guess this is a construction of Bourbaki?). In these ...
SBRJCT's user avatar
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What is the difference between a presheaf and a contravariant Hom functor

I'm currently delving into category theory and came across two concepts that appear similar: presheaves and contravariant Hom-functors. Both of these constructs are associated with the mapping from $\...
whymgang's user avatar
3 votes
2 answers
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A short exact sequence of sheaves of vector spaces that always splits

Even though a short exact sequence of sheaves of vector spaces does not split in general, I want to prove that there is one particular short exact sequence of sheaves of vector spaces that always ...
Flavius Aetius's user avatar
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0 answers
38 views

Show the existence of a sheafification for every presheaf $\mathcal{O} '$

I need to show that the existence of a sheafification for every presheaf $\mathcal{O}'$. I know that it's enough to show that the covariant functor $$Hom(\mathcal{O}' , ·) :\textbf{Sh}_X\...
Mousa Hamieh's user avatar
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0 answers
47 views

Existence of a sheafification for every presheaf $ \mathcal{O} '$

I need to show that the existence of a sheafification for every presheaf $\mathcal{O}'$. I know that it's enough to show that the covariant functor $$Hom(\mathcal{O}' , ·) :\textbf{Sh}_X\...
Mousa Hamieh's user avatar
1 vote
1 answer
111 views

For the sheaf $\mathscr{F}^+$ associated to the presheaf $\mathscr{F}$, how to prove $\mathscr{F}^+_P=\mathscr{F}_P$?

In Hartshorne's book algebraic geometry, he introduced the concept the sheaf $\mathscr{F}^+$ associated to the presheaf $\mathscr{F}$ in proposition 1.2, chapter II. Then he claim that for each point $...
Frank's user avatar
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1 answer
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Higher direct images of a sheaf in Kempf's Algebraic Varieties

Let $f:X\longrightarrow Y$ be a continuous mapping of topological spaces. Let $F$ be an abelian sheaf on $X$. We define $R^i f_{\star} F$ as the $i$-homology sheaf of the complex $f_{\star}(D^{\star}(...
WindUpBird's user avatar
1 vote
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Inverse image - direct image adjunction

I've been struggling to prove the adjunction between the inverse image and direct image functors for sheaves, and am looking to find either a reference/book that explains it, suggestions for an ...
subrosar's user avatar
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1 vote
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What is a germ of a presheaf?

I’m trying to figure out the notions of germs and stalks of a presheaf. I had understand the definition is there an example though? Also I was thinking why there was a reason to construct these two ...
Mathematician's user avatar
1 vote
0 answers
57 views

Representable presheaves on the slice category

$\def\sfC{\mathsf{C}} \def\op{\mathrm{op}} \def\set{\mathsf{Set}} \def\psh{\operatorname{PSh}} \def\ob{\operatorname{Ob}} \def\hom{\operatorname{Hom}}$Let $\sfC$ be a category. A presheaf over $\sfC$ ...
Elías Guisado Villalgordo's user avatar
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1 answer
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If the presheaf Hom$_\mathcal{C}(- \times A, B) : \mathcal{C}^\text{op} \to \textbf{Set}$ is representable, then $\mathcal{C}$ is ccc

We consider a small category $\mathcal{C}$ with binary products, and we consider, for any objects $A, B$ of $\mathcal{C}$, the assignments \begin{eqnarray*} F :\,\, &\mathcal{C}^\text{op} &\...
Jos van Nieuwman's user avatar
1 vote
1 answer
97 views

A particular presheaf on a small category. What if it's representable?

We consider a small category $\mathcal{C}$ with binary products, and we consider, for any objects $A, B$ of $\mathcal{C}$, the assignments \begin{eqnarray*} F :\,\, &\mathcal{C}^\text{op} &\...
Jos van Nieuwman's user avatar