(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
A partial order on a set is a way of ordering its elements to say that some elements precede others, but allowing for the possibility that two elements may be incomparable without being the same. This is the fundamental notion in order theory.
By regarding the ordering $\leq$ as the existence of a unique morphism, preorders may be regarded as certain categories (namely, thin categories). This category is sometimes called the order category associated to a partial order; see below for details.
A poset may be understood as a set with extra structure.
Given a set $S$, a partial order on $S$ is a (binary) relation $\leq$ with the following properties:
reflexivity: $x \leq x$ always;
transitivity: if $x \leq y \leq z$, then $x \leq z$;
antisymmetry: if $x \leq y \leq x$, then $x = y$.
A poset is a set equipped with a partial order.
A poset is precisely a proset satisfying the extra condition that $x \leq y \leq x$ implies that $x = y$.
A poset may be understood as a category with extra property, sometimes called its order category.
A poset is a category such that:
for any pair of objects $x, y$, there is at most one morphism from $x$ to $y$
if there is a morphism from $x$ to $y$ and a morphism from $y$ to $x$ (which by the above implies that $x$ and $y$ are isomorphic), then $x = y$.
Equivalently, this says that a poset is a skeletal thin category, or equivalently a skeletal category enriched over the cartesian monoidal category of truth values or equivalently a skeletal (0,1)-category.
When we do this, we are soon led to contemplate a slight generalization of partial orders: namely preorders. The reason is that the antisymmetry law, saying that $x \le y$ and $y \le x$ imply $x = y$, violates the principle of equivalence in a certain sense. (On the other hand, it does not violate it if taken as a definition of equality.)
The morphisms of partially ordered sets are monotone functions; a monotone function $f$ from a poset $S$ to a poset $T$ is a function from $S$ to $T$ (seen as structured sets) that preserves $\leq$:
Equivalently, it is a functor from $S$ to $T$ (seen as certain categories).
In this way, posets form a category Pos.
A (closed bounded) interval in a poset $C$ is a set of the form
A poset is locally finite if every closed bounded interval is finite.
A poset with a top element and bottom element is called bounded. (But note that a subset of a poset may be bounded without being a bounded as a poset in its own right.) More generally, it is bounded above if it is has a top element and bounded below if it has a bottom element.
A poset with all meets and joins is called a lattice; if it has only one or the other, it is still a semilattice.
A poset in which every finite set has an upper bound (but perhaps not a least upper bound, that is a join) is a directed set.
As remarked above, a poset in which each interval $[x,y]$ is a finite set is called locally finite or a causet.
A poset with a bounding countable subset is called $\sigma$-bounded. That is, the poset is $\sigma$-bounded above if there exists a sequence $(x_n)_{n=1}^{N}$ (where $N$ is a natural number or infinity) such that for every $y$ in the poset there is an $x_m$ with $y \leq x_m$. (The poset is $\sigma$-bounded below if we have $x_m \leq y$ instead.) Note that every bounded poset is $\sigma$-bounded, but not conversely. Note that some authorities require $N = \infty$; this makes a difference only for the empty poset (we say it is $\sigma$-bounded, they say it is not).
A poset can be understood as a (0,1)-category. This suggests an obvious vertical categorification of the notion of poset to that of n-poset.
For $P$ a poset, write $Up(P)$ for the topological space whose underlying set is the underlying set of $P$ and whose open subsets are the upward closed subsets of $P$: those subsets $U \subset P$ with the property that
This is called the Alexandroff topology on $P$.
This construction naturally extends to a full and faithful functor.
For $P$ a poset, there is a natural equivalence
between the category of sheaves on the locale $Up(P)$ and the category of copresheaves on $P$.
For more see Alexandroff topology.
Every poset is a Cauchy complete category. Posets are the Cauchy completions of prosets. (Rosolini)
Cauchy completion of prosets and posets is discussed in
Here are some references on directed homotopy theory:
Marco Grandis, Directed homotopy theory, I. The fundamental category (arXiv)
Tim Porter, Enriched categories and models for spaces of evolving states, Theoretical Computer Science, 405, (2008), pp. 88–100.
Tim Porter, Enriched categories and models for spaces of dipaths. A discussion document and overview of some techniques (pdf)