Why sheaves are important and why do we care about them? [closed]

Presheaves are contravariant functors from a category $C$ to the category $Set$, that is functors $P$:
$$P:C^{op}to Set.$$

For every topology $J$ on $C$ we can generate a reflexive subcategory
$$Sh(C,J)subseteq Fun(C^{op},Set).$$

Part of the beauty/usefulness of this procedure is that the resulting objects are topoi and working in them is “easy”.

The first question is: why do we prefer $Set$ above all else? Why $Set$ seems to be the center of this construction?
Usually when consider functions/morphisms/functors targeting some structure, the set/category of functions/functors inherits this structure. (“Usually” our structure comes from products and limits so “usually” it works. I never worked with Hopf algebras to not use the usually.)
My guess is that it seems we want to “pullback” $Set$’s structure, that is a topos. Is this all there is to it?

The second question is why we do it for other categories as well? We consider sheaves of groups or rings. We will never get a topos, but we are nevertheless interested. And it seems that some people are interested even in sheaves over other less concrete categories. (I don’t have an explicit example.)

So the main question is: why do sheaves always seem to pop out? Why it seems that sheaves contain interesting information?

[EDIT]

It seems my question is ambiguous and noons gets it…
A group us not a sheaf, is a category. A ring is not a sheaf, is a category. A metric space is not a sheaf is a category. A posed is not a sheaf is a category ( in this case both enriched and not).
When we consider many concepts in math close to these concepts it seems that sheaves comes out pretty often. I do not know about stochastic processes, but I wouldn’t except them to be far away just because today none treats them as sheaves.
Most base theories are categories, because we need indexes. Most interesting construction on these theories are categories of sheaves. The question was why it is so.

Specialization of étale sheaves

Let $(R,mathfrak{m})$ be a henselian local ring with separably closed residue field and fraction field $K$.

Let:

  • $overline{s}=text{Spec}(R/mathfrak{m})totext{Spec}(R)$ be the closed point of $text{Spec}(R)$
  • $overline{eta}=text{Spec}(K)to text{Spec}(R)$ be the generic point of $text{Spec}(R)$.

Let $f: Xtotext{Spec}(R)$ be a smooth $R$-scheme and $nin R^{times}$ an integer invertible in $R$.

Is the specialization map between stalks of étale sheaves:

$$Gamma(text{Spec}(R), R^jf_*(mathbf{Z}/n)) = R^jf_*(mathbf{Z}/n)_{overline{s}}to R^jf_*(mathbf{Z}/n)_{overline{eta}}$$

an isomorphism?

Smoothness should play a crucial role, probably through the fact that smooth morphisms are universally locally acyclic.

Segre embedding and Hilbert polynomial of coherent sheaves

Let $X subset mathbb{P}^n$ and $Y subset mathbb{P}^m$ be smooth, projective subvarieties, $F$ and $G$ coherent, torsion-free, sheaves on $X$ and $Y$ with Hilbert polynomials $P_{F}$ and $P_G$, respectively (under the fixed embedding). Consider now the Segre embedding $X times Y subset mathbb{P}^N$. Is the Hilbert polynomial of $mbox{pr}_1^*F otimes mbox{pr}_2^*G$ under the Segre embedding just $P_F.P_G$?

Any idea/reference will be most welcome.

Cokernel of map of ètale sheaves

This is probably a very dumb question. Let $p:mathbb{G}_mto operatorname{Spec} k$ be the structure map, and let $T$ be an algebraic $k$-torus viewed as an étale sheaf over $k$. Why is the cokernel of the canonical map $Tto p_*p^*T$ canonically isomorphic to the cocharacter lattice $L$ of $T$?

If $operatorname{Spec}A$ is affine scheme, it seems to me that $p^*T=Ttimes mathbb{G}_m$, so $$p_*p^*T(A)=p_*((Ttimes mathbb{G}_m))(A)=T(A[t^{pm 1}])times mathbb{G}_m(A[t^{pm 1}]).$$ Say that $A=K$ is a field. Then $$p_*p^*T(K)=Loplusmathbb{Z},$$ but why is the image of $T(K)$ equal to $mathbb{Z}$?

Serre duality for compactly supported sheaves

Given a smooth quasi-projective variety $X$ over $mathbb{C}$ and bounded complexes of vector bundles $(P,d)$ and $(P’,d’)$ with compactly supported cohomology. It is well-known that such complexes satisfy Serre-duality. The standard proof that I have heard is to complete $X$ to a projective variety $bar{X}$.

For me the intuition behind Serre duality is integration, and the intuition for the above result is that one can always integrate compactly supported differential forms. Unfortunately, I’ve never seen a place where this result is actually proven in this way.

Question: Is there a reference which proves Serre duality using compactly supported Dolbeault cohomology?

The proof I have in mind is the standard proof of Serre duality for projective varieties, but I remember in Serre’s original paper there are some tricky points of topology on Frechet spaces which are complicated. I haven’t actually seen the compactly supported Dolbeault theory used in any other papers since then, so I’m wondering whether there is any newer reference with the above statement and which covers its properties more systematically.

Composition of functions between sheaves

Let $(X, mathcal{O}_X)$ and be a topological space endowed with a sheaf $mathcal{O}_X(U)$ of regular functions for every open subset $U subset X$. Let $(Y, mathcal{O}_Y)$ be a structure of the same kind. Let $U$ be an open subset of $X$ and $f:Y rightarrow U$ a map. Finally let $i:U rightarrow X$ be the inclusion map of $U$ in $X$. Knowing that for all open sets $W subset X$, for every $f’ in mathcal{O}_{X}(W), space f’ circ j circ f$ is in $mathcal{O}_Y(f^{-1}(j^{-1}(W))$, can I conclude that for all open sets $V subset U$, for every $f’ in mathcal{O}_{X}(V),space f’ circ f$ is in $mathcal{O}_Y(f^{-1}(V))$?

Examples of degree zero, rank one reflexive sheaves without r-th roots

Let $X$ be a normal, projective surface (or more generally a variety) over $mathbb{C}$ (i.e., $X$ is irreducible). Fix a polarisation on $X$. I am looking for examples of rank one, degree zero (degree under the fixed polarisation) reflexive sheaf $F$ such that there exists an integer $r>0$ for which there is no rank one reflexive sheaf $G$ for which $G^{otimes r}=F$. Is there some general strategy to produce such examples? Moreover, what happens if $F$ is an invertible sheaf?

Semicontinuity of cohomology of torsion-free sheaves restricted to divisors

Let $X$ be a smooth projective variety, $mathcal{E}$ a torsion-free coherent sheaf on $X$ and $mathfrak{d}$ a linear system of divisors in $X$.

I would like to show (at least when $X$ is a surface) the upper semicontinuity of the map
$$ Dinmathfrak{d}longmapsto h^0(D,mathcal{E}{restriction_D}),.$$
I considered the incidence locus $Zsubset Xtimesmathfrak{d}$ and the projections $p:Zto X$ and $q:Ztomathfrak{d}$. If $mathcal{E}$ is locally free, then $p^*mathcal{E}$ is flat over $mathfrak{d}$, so by the semicontinuity theorem
$$ h^0(D,mathcal{E}{restriction_D})=h^0(q^{-1}(D),p^*mathcal{E}{restriction_{q^{-1}(D)}})$$
is indeed upper semicontinuous in $D$. But what if $mathcal{E}$ is only torsion-free? Is $p^*mathcal{E}$ still flat over $mathfrak{d}$?

isomorphism between category of sheaves and morphisms of abelian groups

I am working on theory of category and I found this exercise. I tried a lot but I didn’t know how I could do. Let $A$ a discrete valuation ring. Show that the category of sheaves of abelian groups on $Spec(A)$ is equivalent to the category which objects are defined as below:

${ f: S rightarrow L quad S,Lin Ab }$ and if we take two morphisms $f: S rightarrow L$ and $f’: S’ rightarrow L’$ then $gcirc f = f’ circ g’$ with $g: L rightarrow L’$ and $g’: S rightarrow L$.

I would really appreciate your answers. Thanks!