On 09/02/2019 21:43, ptj@maths.cam.ac.uk wrote:
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.
The paper John Bourke, 2017, Equipping weak equivalences with algebraic structure. https://arxiv.org/abs/1712.02523. has a terminology for this that is appealing: an algebraic injective object, with respect to a class J of arrows, is an object D equipped with extensions c(j,f) : Y -> D for each j:X->Y in J and f : X -> D. (Then you can consider the obvious morphisms of algebraic injective objects that commute with the designated extensions c(j,f).) Thanks to Mike Shulman for this reference. Martin
Peter Johnstone
On Feb 9 2019, Mart??n H??tzel Escard?? 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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
-- Martin Escardo http://www.cs.bham.ac.uk/~mhe [For admin and other information see: http://www.mta.ca/~cat-dist/ ]