Terminology regarding injectivity of objects

["Martín Hötzel Escardó" <m.escardo@cs.bham.ac.uk>]

(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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