nLab posetal reflection

Idea

The posetal reflection of a preorder is the poset obtained by enforcing antisymmetry by quotienting out isomorphisms.

Construction

Let $(A, \leq)$ be a preorder. Define the equivalence relation:

$$a \simeq b \quad\text{iff}\quad a \leq b \ \text{and}\ b \leq a.$$

The corresponding posetal reflection $A'$ of $A$ is the preorder on the quotient $A/\simeq$ given by

$$[a] \leq' [b] \quad\text{iff}\quad a \leq b$$

It’s easy to show both $\simeq$ and $\leq'$ are well-defined.

Abstractly, the correspondence $(A, \leq) \mapsto (A', \leq')$ is functorial from preorders to posets, and in fact exhibits posets as a reflective subcategory of preorders (hence the name).