type.

\[df:sheaf\_poset\] Suppose that $P=(P,\le)$ is a poset. A *sheaf $\shf{S}$ of sets on $P$ with the Alexandroff topology* (briefly, a *sheaf $\shf{S}$ on $P$*) consists of the following specification:

1.  For each $p \in P$, a set $\shf{S}(p)$, called the *stalk at* $p$,

2.  For each pair $p \le q \in P$, there is a function $\shf{S}(p \le q):\shf{S}(p)\to\shf{S}(q)$, called a *restriction function* (or just a *restriction*), such that

3.  For each triple $p \le q \le r \in P$, $\shf{S}(p \le r) = \shf{S}(q \le r) \circ \shf{S}(p \le q)$.

When the stalks themselves have structure (they are vector spaces or topological spaces, for instance) one obtains a sheaf *of* that type of object when the restrictions or extensions preserve that structure. For example, a *sheaf of vector spaces* has linear functions for each restriction, while a *sheaf of topological spaces* has continuous functions for each restriction.

Similarly, a *sheaf $\dshf{C}$ of sets on the dual poset $P^{op}$ with the Alexandroff topology* (briefly, a *dual sheaf $\dshf{C}$ on $P$*) consists of the same kind of thing, just backwards. Namely,

1.  For each $p \in P$, a set $\dshf{D}(p)$, called the *stalk at* $p$,

2.  For each pair $p 