Examples of Poset-Functor

(draft … references to be added) … (up)

Introduction

Ideally, coalgebraic logic over posets should follow the same lines as coalgebraic logic over sets. Just replace the category

$\mathsf{S}\mathsf{e}\mathsf{t}$ everywhere by the category

$\mathsf{O}\mathsf{r}\mathsf{d}$ of ordered sets (posets or preorders) and order-preserving (aka monotone) maps.

Really many new phenomena arise. One general interesting question is how much knowledge we can transfer from set-coalgebras to poset-coalgebras in a systematic way.

One approach to answer this question is to study relationships between set and poset-functors systematically. So let us start by recalling the notion of an order-extension (posetification, preordification)

$T^{\prime }$ of a set functor

$T:\mathsf{S}\mathsf{e}\mathsf{t}\to \mathsf{S}\mathsf{e}\mathsf{t}$.

Functors on Posets

Powerset: Let

$T=\mathcal{P}$ be the powerset functor. There are the following three extensions to

$Pos$-enriched functors.

• $\mathcal{D}$ with
  $\mathcal{D}X=[X^o,2]$ on objects. For
  $f:X\to Y$, one defines
  $\mathcal{D}f:[X^o,2]\to [Y^o,2]$ as the left Kan extension of
  $Yo{n}_Y\circ f$ along
  $Yo{n}_X$.
• $\mathcal{U}X=(\mathcal{D}{X}^o{)}^o$.
• $\mathcal{P}$ is the convex powerset.

Double Powerset:

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