Joyal's CatLab Introduction to sheaves on a topological space

Introduction

Here we will set up some basics of sheaves of sets, and perhaps in the future sheaves with values in an abelian category. Here our sheaves will always be on a topological space, the reader should be aware that there are generalizations of the concept of sheaves. One may generalize from either end, that is one may construct a theory of sheaves on suitable categories, or we can allow the ‘value category’ to change.

Preliminary definitions

Let $X$ be a topological space, and let $T$ be the category of open sets of $X$, whose morphisms are the inclusion morphisms.

Presheaves, Sheaves

A presheaf of sets on $X$ is a contravariant functor from $T^{\mathrm{op}}$ to $\mathrm{Sets}$.

A sheaf of sets on $X$ is a presheaf, we will denote it by $\mathcal{F}: T^{\mathrm{op}} \rightarrow \mathrm{Sets}$, which satisfies that for every open set $U$ and every open covering $\{ U_i\}_{i \in I}$of $U$,

Is an equalizer diagram, where the maps are being induced via $U_i \cap U_j \hookrightarrow U_i \hookrightarrow U$ and $U_k \cap U_i \hookrightarrow U_i \hookrightarrow U$. For an even more categorical flavor, note that this the same as $U_i \times_U U_j \rightarrow U_i \rightarrow U$ and $U_k \times_U U_i \rightarrow U_i \rightarrow U$, so we can rewrite this as

being an equalizer diagram for every open $U$ and open cover $\{ U_i \}_{i \in I}$.

Another method to define a sheaf uses two axioms:

1. if we have an open cover $\{ U_i \}_{i \in I}$ of $U$, and a $f_i \in \mathcal{F}(U_i)$ with $\mathcal{F}(i_{U_i\cap U_j,U_i})(f_i)=\mathcal{F}(i_{U_i \cap U_j ,U_j})(f_j)$ for each $i,j \in I$ then there is a ‘gluing’ $f \in \mathcal{F}(U)$ where $f_i = \mathcal{F}(i_{U_i,U})(f)$

2. any such gluing is unique.

Note there is a notion of a monopresheaf which does not satisfy the above diagram being an equalizer diagram, but the left-most map is injective. Equivalently it satisfies axiom 2 above.

Morphisms of presheaves and sheaves

Since presheaves are functors, we may simply the category $\mathrm{PSh}$ of presheaves of sets on $X$ to be $\mathrm{PSh}(X) := \mathrm{Sets}^{T^{\mathrm{op}}}$ the functor category from $T^{\mathrm{op}}$. The category $\mathrm{Sh}(X)$ of sheaves of sets on $X$ is defined to be the full subcategory of presheaves containing the sheaves as objects. Since sheaves form a full subcategory of presheaves, we have a natural forgetful functor $\mathrm{I}: \mathrm{Psh}(X) \rightarrow \mathrm{Sh}$

To decategorify: a morphism of presheaves $\phi: \mathcal{F} \rightarrow \mathcal{G}$ is a natural transformation from the functor $\mathcal{F}$ to the functor $\mathcal{G}$, that is a set of morphisms $\phi_U: \mathcal{F}(U) \rightarrow \mathcal{G}(U)$ for each open set $U$ such that for any $i: V \rightarrow U$, $\phi_V \circ \mathcal{F}(i) = \mathcal{G}(i) \circ \phi_U$.

Stalks

Let $x \in X$ and $\mathscr{F}$ a presheaf of sets on $X$. Define the stalk of $\mathcal{F}$ at $x$ to be $\mathrm{colim}_{U \ni x} \mathcal{F}(U)$.

Sheafification

Sheafification via Étalé Spaces

Sheafification as a left adjoint to the forgetful functor