# Questions tagged [presheaves]

The tag has no usage guidance.

20 questions
Filter by
Sorted by
Tagged with
1 vote
34 views

### 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 ...
69 views

### 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: ...
• 157
72 views

### 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 ...
• 21
56 views

### 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 ...
67 views

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\...
1 vote
111 views

• 111
1 vote
103 views

### 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 ...
• 4,624
1 vote
37 views

### 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 ...
1 vote
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 $\sfC$ ...
62 views

### 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} &\...
• 2,074
1 vote
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} &\...
• 2,074