Dear Martin, I encountered this situation when I considered injectivity in Top: see my paper in SLNM 871, and also pages 738-9 in the Elephant. I used the terms `weakly injective' and `strongly injective' (not very imaginative, but they did the job), and also `completely injective' for the case where the `extension along j' operation can be taken to be right adjoint to restriction along j (you could of course use `cocompletely injective' for the case where it's left adjoint). Fortunately, in Top the notions of weak injective, strong injective and complete injective coincide. Peter Johnstone On Feb 9 2019, Martín Hötzel Escardó wrote:

(1) An object D is called injective over an arrow j:X->Y if the "restriction map"

hom(Y,D) -> hom(X,D) g |-> g o j

is a surjection. This is fairly standard terminology (where does it come from, by the way).

(2) I am working with the situation where the restriction map is a *split* surjection.

I though of the terminology "D is split injective over j", but perhaps this is awkward. Is there a standard terminology for this notion. Or, failing that, a terminology that at least one person has already used in the literature or in the folklore. Or, failing that too, a good suggestion by any of you?

Thanks, Martin

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