# Category with hom-sets of order less than or equal to $1$

Does there exist a name for a category $$\mathcal{C}$$ with the following property: For any two objects $$X,Y \in \mathcal{C}$$, we have $$\mathrm{Order}|Hom(X,Y)| \leq 1 < \infty,$$ where Order means order of a set. Do this property have any immediately interesting consequences?

As another answer said, such categories exist and arise from sets with a pre-order. The most common kind of these are partial orders, which is a set $$S$$ equipped with a relation $$\leq$$ such that $$x \leq x$$, and if $$x \leq y$$ and $$y \leq x$$ then $$x = y$$, and if $$x \leq y$$ and $$y \leq z$$ then $$x \leq z$$. Some common examples of partials orders are the integers with divisibility or any collection of sets with $$\subseteq$$. We can make these orders into categories by saying the objects of the category are the elements of the set, and defining $$Hom(x, y)$$ to be $$\{(x, y)\}$$ if $$x \leq y$$ and empty otherwise. The identity morphisms are pairs $$(x, x)$$ (this is well defined because we assumed $$\leq$$ was reflexive) and the composition of $$(x, y)$$ and $$(y, z)$$ is $$(x, z)$$ (this is well defined because we assumed $$\leq$$ was transitive). Note that we did not use the assumption that $$\leq$$ was antisymmetric, and dropping this assumption gives us the definition of a pre ordered set.
Interpreting pre/partial orders as categories is not immediately useful, but it can simplify some things. For example, in algebraic geometry there's this important concept called a presheaf on a topological space. Given a space $$X$$, the collection of open sets of $$X$$ has a partial order on it given by "is a subset of" (see my example above) and so we can make it into a category. It turns out that the standard definition of "presheaf on $$X$$" is equivalent to "contravariant functor from the category of open sets of $$X$$ to the category of sets", and this shift in perspective can be very illuminating.