Terminology regarding injectivity of objects

Terminology regarding injectivity of objects

[Martín Hötzel Escardó]

(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/ ]

Re: Terminology regarding injectivity of objects

[Martin Escardo]

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
>>
>>
>> [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/ ]

Re: Terminology regarding injectivity of objects

[Martin Escardo]

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/ ]