is injective (or surjective).

Like sheaf morphisms, hybrid morphisms transform local (or global) sections of a dual sheaf to local (or global) sections of a sheaf.

Sheaves on topological spaces {#sec:topo_sheaves}
-----------------------------

This section explains the appropriate generalization of sheaves on posets with the Alexandroff topology to sheaves over arbitrary topological spaces. Topological spaces and partial orders are closely related, because every topology defines a unique partial order.

A *topology* on a set $X$ consists of a collection $\col{T}$ of subsets of $X$ that satisfy the following four axioms

1.  $\emptyset \in \col{T}$,

2.  $X \in \col{T}$,

3.  If $U, V \in \col{T}$, then $U \cap V \in \col{T}$, and

4.  If $\col{U} \subseteq \col{T}$, then $\cup \col{U} = \{x\in X: x \in U \text{ for some }U\in\col{U}\} \in \col{T}$.

We will call $(X,\col{T})$ a *topological space*.

\[df:open\] Every topological space $X=(X,\col{T})$ defines a poset ${\bf Open}(X,\col{T})=(\col{T},\subseteq)$ on the open sets, partially ordered by the subset relation. When the topology $\col{T}$ is clear from context, we shall usually write ${\bf Open}(X)={\bf Open}(X,\col{T})$.

The axioms for a topology ensure that in ${\bf Open}(X,\col{T})$, infima of finite sets exist, namely via $U_1 \wedge U_2 = U_1 \cap U_2$, and that suprema of 