Spahn Pi-closure

Definition

Let $\dagger$ be a monad on a topos $H$. A morphism $f:X\to Y$ is called $\dagger$-closed if

$$\array{ X&\to &\dagger X \\ \downarrow^f &&\downarrow^{\dagger f} \\ Y&\to& \dagger Y }$$

is a pullback square.

Closure properties

The class of $\dagger$-closed morphisms satisfies the following closure properties:

(1) Every equivalence is $\dagger$-closed.

(2) The composite of two $\dagger$-closed morphisms is $\dagger$-closed.

(3) The left cancellation property is satisfied: If $h=g\circ f$ and $h$ and $g$ are $\dagger$-closed, then so is $f$.

(4) Any retract of a $\dagger$-closed morphism is $\dagger$-closed.

(5) The class is closed under pullbacks which are preserved by $\dagger$.