Contents

category theory

# Contents

## Idea

In a cartesian closed category, given an object $S$, the selection monad, also known as the select monad, is the endofunctor $J_S(X) \mapsto [[X, S], X]$. It is a strong monad.

There is a monad homomorphism from the selection monad to the continuation monad for $S$, $K_S(X) = (X \to S) \to S$, which sends $\epsilon \in J_S(X)$ to $\bar{\epsilon} \in K_S(X)$, where $\bar{\epsilon}(p) = p(\epsilon(p))$.

If we understand the continuation monad as mapping an object to the generalized quantifiers over it, with $S$ a generalized truth value, a selection function for a generalized quantifier is an element of its preimage under the monad morphism.

For instance, a selection functional for the supremum functional $sup: (X \to S) \to S$, when it exists, applied to a function, $p: X \to S$, gives a point in $X$ at which $p$ attains its maximum value.

Due to the resemblance of an algebra, $J_S(A) \to A$, to Peirce's law in logic, $((p \Rightarrow q) \Rightarrow p) \Rightarrow p$, $J_S$ is also called the Peirce monad in (Escardó-Oliva 2012).

## Properties

There is a distributive law $T J_S \Rightarrow J_S T$ for every strong monad $T$ (Fiore 2019).