A_2$ and $\bigvee A = A_1 \vee A_2$.

Figure \[fig:lattice\_sheaves\] shows a poset $P$ with four elements at left and its dual poset $P^{op}$ at center. The diagram is to be read that $d \le a \le c$ in $P$. In $P$, the infimum of $A=\{a,b\}$ is $d$, while the supremum is $c$.

The definition of a *sheaf* is rather crisply stated in terms of the diagram of a poset, where the vertices represent elements and arrows point from lesser elements to greater ones. Merely replace each vertex by a set or a space and each arrow by a function so that the composition of functions in the diagram is path independent. If all of the functions’ inputs are at the tails of each arrow, then the diagram is that of a *sheaf on the Alexandroff topology* for that poset. (We generalize to arbitrary topologies in Section \[sec:topo\_sheaves\].) If all of the functions’ inputs are at the heads of each arrow, then the diagram defines a sheaf over the dual poset. When discussing a particular poset, we will emphasize this difference by setting sheaves over that poset in script type, and by setting sheaves over the dual poset in fraktur 