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Stream: learning: questions

Topic: injective objects in a topos

James Deikun (Feb 14 2024 at 05:16):

Remark A4.4.7 in the Elephant claims that:

...by 2.2.6, the injective objects of $\bold{sh}_j(\mathcal{E})$ are exactly the retracts of power objects $\Omega_j^A$, $A \in \text{ob}\;\bold{sh}_j(\mathcal{E})$; ...

But A2.2.6 doesn't seem to prove any necessary condition for objects of a topos to be injective. Is this a straight up error or is there some other proof that injective objects in a topos are all of this form?

Morgan Rogers (he/him) (Feb 14 2024 at 09:57):

If you take an injective object, by A2.2.3(i) it injects into the its power-object, and hence is a retract of it. (I think he should have referred to 2.2.3 also for his justification to make sense).