the most general, but nearly every useful computational example can be expressed more compactly with partial orders. For instance, graphs and abstract simplicial complexes are both special cases of *hypergraphs*, which are merely subsets of the power set of some set $V$. A hypergraph $H$ in which each element is a finite subset of $V$ can be thought of as locally finite partial order with the partial order coming from inclusion: for $A,B \in H$, $A \le B$ precisely when $A \subseteq B$.

We first build a rather specialized description of sheaves on partial orders, since they are the primary mathematical tool in this chapter. We then generalize to sheaves on general topological spaces, since they describe the spaces of functions we will discretize in later sections.

Sheaves on partial orders
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We will mostly deal with locally finite posets, since these avoid computational difficulties.

A *partial order* on a set $P$ is a relation $\le$ on that set that is

1.  Reflexive: $x \le x$ for all $x \in P$,

2.  Antisymmetric: if $x \le y$ and $y \le x$, then $x = y$, and

3.  Transitive: if $x \le y$ and $y \le z$ then $x \le z$.

