Re: Terminology regarding injectivity of objects

[Martin Escardo <m.escardo@cs.bham.ac.uk>]

Thank you, Peter.

I am in a similar situation, where weak and strong injectivity agree for
all objects, and some distinguished objects are injective in the four
senses.

Martin

On 09/02/2019 21:43, ptj@maths.cam.ac.uk wrote:
> 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, 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
>>
>>
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>>

-- 
Martin Escardo
http://www.cs.bham.ac.uk/~mhe


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