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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?

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Such a category is called a thin category, or preorder. If it is also skeletal, then it is a poset.

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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.

The most important result about these categories (ones where every hom set has at most one element) is the that every diagram you can draw in them commutes. Put in less fancy language, any two morphisms with the same domain and target are equal. But this is just another way to state the fact that there is always at most one morphisms between two objects!

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