8.5.6 Modules Over Internal Monads

    Let $A$ be a set.

    Proposition 8.5.6.1.1 ▶ Modules Over Internal Monads in $\boldsymbol {\mathsf{Rel}}$

    Let $\preceq _{A}$ be a preorder on $A$, viewed also as an internal monad on $A$ via Proposition 8.5.4.1.1.

    1. 1.

      Left Modules. We have a natural identification

      \[ \left\{ \text{Left modules over $\mathord {\preceq _{A}}$}\right\} \cong \left\{ \begin{aligned} & \text{Relations $R\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ such that,}\\ & \text{for each $b\in B$, the set $R(b)$ is}\\ & \text{upward-closed in $A$}\end{aligned} \right\} . \]
  • 2.

    Right Modules. We have a natural identification

    \[ \left\{ \text{Right modules over $\mathord {\preceq _{A}}$}\right\} \cong \left\{ \begin{aligned} & \text{Relations $R\colon A\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}B$ such that,}\\ & \text{for each $b\in B$, the set $R^{-1}(b)$ is}\\ & \text{downward-closed in $A$}\end{aligned} \right\} . \]
  • 3.

    Bimodules. We have a natural identification

    \[ \left\{ \text{Bimodules over $\mathord {\preceq _{A}}$}\right\} \cong \left\{ \begin{gathered} \text{Quadruples $(B,C,R,S)$ such that:}\\ \begin{aligned} & \text{1. For each $b\in B$, the set $R(b)$ is}\\ & \phantom{\text{1. }}\text{upward-closed in $A$.}\\ & \text{2. For each $c\in C$, the set $S^{-1}(c)$ is}\\ & \phantom{\text{2. }}\text{downward-closed in $A$.}\end{aligned} \end{gathered} \right\} . \]

    • Proof of Proposition 8.5.6.1.1.

    Item 1: Left Modules
    A left module over $\mathord {\preceq _{A}}$ in $\boldsymbol {\mathsf{Rel}}$ consists of a relation $R\colon B\mathrel {\rightarrow \kern -9.5pt\mathrlap {|}\kern 6pt}A$ together with an inclusion

    \[ \alpha _{B}\colon \mathord {\preceq _{A}}\mathbin {\diamond }R\subset R \]

    making appropriate diagrams commute. Since $\boldsymbol {\mathsf{Rel}}$ is locally posetal, however, the commutativity of the diagrams in question is automatic (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2), and hence all that is left is the data of the inclusion $\alpha _{B}$. This corresponds to the following condition:

    • (★)
      • For each $a,a'\in A$, if there exists some $b\in B$ such that $b\sim _{R}a$ and $a\preceq _{a}a'$, then $b\sim _{R}a'$.

    This condition is equivalent to $R(b)$ being downward-closed for all $b\in B$.

    Item 2: Right Modules
    The proof is dual to Item 1, and is therefore omitted.

    Item 3: Bimodules
    Since $\boldsymbol {\mathsf{Rel}}$ is locally posetal, the diagram encoding the compatibility conditions for a bimodule commutes automatically (Chapter 11: Categories, Item 4 of Proposition 11.2.7.1.2), and hence a bimodule is just a left module along with a right module.

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