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

X X f f Y Y\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=gfh=g\circ f and hh and gg are \dagger-closed, then so is ff.

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

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

Last revised on December 2, 2012 at 23:46:24. See the history of this page for a list of all contributions to it.