nLab posetal reflection

Contents

Contents

Idea

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

Construction

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

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

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

[a][b]iffab [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,)(A,)(A, \leq) \mapsto (A', \leq') is functorial from preorders to posets, and in fact exhibits Pos as a reflective subcategory of preorders (hence the name).

Last revised on November 26, 2024 at 11:46:46. See the history of this page for a list of all contributions to it.