Fractured topos | Higher Topos Theory notes

Definition 7.12.

A fractured topos is a topos $$E$$ equipped with a functor $j_!\colon E^{\corp} \to E$ satisfying the following properties:

The functor $$j_!$$ is faithful and induces a fully faithful functor on groupoid cores.
The category $E^{\corp}$ admits fiber products and $$j_!$$ preserves them.
The functor $$j_!$$ admits a right adjoint $$j^*$$ which is conservative and preserves colimits.
For every morphism $U \to V$ in $E^{\corp}$ , the commutative square

is a pullback square.

We refer to objects of $E^{\corp}$ , and to their images in $$E$$ , as corporeal objects. The functor $$j_!$$ is generally not fully faithful. The fractured topos is called complete if every retract in $$E$$ of a corporeal object is corporeal.