## Presheaf

*presheaf*on a category $\mathcal{C}$ is a functor $$ \mathcal{F}: \mathcal{C}^{\mathrm{op}} \to \mathcal{S}, $$ namely, a contravariant functor. If $\mathcal{S}=\operatorname{Set}$ with morphism given by set inclusion and $\mathcal{C}$ is small, $\mathcal{F}$ is simply called a presheaf.

Let $\mathcal{C}=X$ be a topological space, seen as a subcategory of $\operatorname{Top}$ and let $S$ be the category of groups $\operatorname{Grp}$, then morphisms in $\mathcal{C}^\mathrm{op}$ are restrictions $r_{V,W}$ of open sets. There is a commutative diagram and applying functors doesn’t change the diagram structure, rewriting $\mathcal{F}(r)$ as $\rho$, we have Furthermore, functors yield $\mathsf{1}$ when applied to $\mathsf{1}$, hence $\mathcal{F}(r_{U,U}) = \rho_{U,U} = \mathsf{1}$.

It is helpful to use Brylinski’s notation, namely for a presheaf $\mathcal{F}:\mathcal{C}^\mathrm{op}\to \mathcal{S}$, denote it as $\underline{\mathcal{S}}_\mathcal{C}$.

### Stalk, section, germ

When $\mathcal{S}=\operatorname{Set}$ or is a subcategory of that, one can speak of the notion of *stalk*, *section* and *germ*. These notions are not fully captured by category theory since one needs to probe down to the elements of an object.

*sections*of $\mathcal{F}$ over $U$.

The notion of stalk is similar to that of *localization*: forming an equivalence class and thus reversing the arrows. When two functions behave similarly, after being “localized”, around a point, they are seen as equal (define a same germ), the stalk is the collection of equivalence classes thus defined.

*stalk*$\mathcal{F}_x$ of $\mathcal{F}:\mathcal{C}^\mathrm{op}\to \operatorname{Set}$ where $x$ is a point is the colimit associated to the cocone in which each $U_i$ contains $x$. It means that any morphism to the nadir $c$ from the objects in the directed diagram (order given by reverse inclusion of sets) factors through $\mathcal{F}_x$.

This is often written as $$ \mathcal{F}_x = \lim_{x\in U}\mathcal{F}(U). $$

Notice that the morphism $\mathcal{F}(U)\to\mathrm{P}_x$ takes a section $s\in\mathcal{F}(U)$ to its germ. More precisely, two sections in $\mathcal{F}(U)$ define the same germ if they are sent to the same element (*i.e.* their germ-equivalence class) in the stalk.

## Sheaf

Sheaf is a presheaf that is “glued together”. After *sheafification* a (good) presheaf becomes a sheaf.

Let $(\mathcal{C},J)$ be a site with $J$ its Grothendieck topology (recall that $J$ is an assignment of sieves on $\mathcal{C}$ that is maximal, transitive and stable under pullbacks).

A more abstract characterization can be given by the requirement that $$ \mathcal{F}(c) \xrightarrow{\{\mathcal{F}(f)\}_f}\prod_{f\in S}\mathcal{F}(\operatorname{dom}f)\overset{p}{\underset{a}{\rightrightarrows}}\prod_{f,g; f\in S; \operatorname{cod}g = \operatorname{dom}f}\mathcal{F}(\operatorname{dom}g) $$ where $p(\{x_f\}_{f\in S}) = \{x_{f\circ g}\}_{f,g}$ and $a(\{x_f\})_{f\in S} = \{\mathcal{F}(g)(x_f)\}_{f,g}$ is an equalizer of sets.

This means sections that agree, in objects coming from different branches (except for the trunks) of a sieve, comes from the restriction of a unique section in the root-image. A characterization that can be easier to understand is to take the domain of $\mathcal{F}$ as a topological space. The sheaf condition means

- Whenever sections $s,t\in\mathcal{F}(U)$ agree on a cover $\{U_i\}$ of $U$, $s=t$.
- Whenever sections $s_i\in\mathcal{F}(U_i)$ for $i\in I$ agree on any nonempty intersection $U_i\cap U_j$, there is a section $s$ s.t. $s|_{U_i} = s_i$.

The first condition is implicit in the definition: The maximal sieves are coverings, and the identity morphism of any object, which should be taken as $f$, is in its maximal sieve.

Some remarks,

- If the unique existence is replaced with “at most one”, the above definition becomes that of
*separated presheaf*. - After gluing process (sheafification) is done, a “presheaf of $K$ on $X$” often becomes a “sheaf of germs of $K$ on $X$”.

## Morphisms

Presheaves are first of all contravariant functors. Morphism between functors are natural transformations, which should be seen as family of morphisms of objects.

*morphism of presheaves*$\phi:\mathcal{F}_1\to\mathcal{F}_2$ between $\mathcal{C},\mathcal{S}$ is a family $\{\phi_c\in \operatorname{Mor}(\mathcal{S})\}$ indexed by $c\in\mathcal{C}$ s.t. where $(f:d\to c)\in\operatorname{Mor}(\mathcal{C})$.

Here $f:d\to c$ not $c\to d$ since $\mathcal{F}_i$ are contravariant. If $f\in\mathcal{C}^\mathrm{op}$, then it should be written as $f:c\to d$. For $\mathcal{S}=\operatorname{Set}$, $f:d\to c$ is the same thing as $\rho_{c,d}$, with $d\subset c$.

A good example will be the classical one: morphisms of presheaves of abelian groups. Let $\mathcal{F},\mathcal{G}$ be presheaves of groups over a space $X$ and $\mathcal{G}$ is that of abelian groups, then the set of morphisms $\operatorname{Hom}_X(\mathcal{F},\mathcal{G})$ can be endowed with the structure of a group. The group operation is given pointwise by $$ (\phi_U + \psi_U)(x) = \phi_U(x) + \psi_U(x) $$ where $x\in U$; note that $\psi_U$ and $\phi_U$ take values in the same group $\mathcal{G}(U)$. This is compatible with the restrictions $\rho_{U,V}$ since these are group homomorphisms, $$ \rho_{U,V}(g+h) = \rho_{U,V}(g) + \rho_{U,V}(h). $$