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Stream: practice: terminology & notation

Topic: Name for class of maps induced by universal property

David Michael Roberts (Jul 28 2024 at 23:30):

What do people call the map from a (co)cone to a limit (co)cone? Is there a generally accepted terminology? Or one that appears in a standard reference of your choice?

(I suspect I know it, but I don't want to bias the answers. And I want to know if there are other terms people use)

Nathanael Arkor (Jul 29 2024 at 07:55):

I would say "mediating morphism". I see from a quick look on Google Scholar that this terminology appears to be more common in computer science publications. The earliest reference I can find to use it is Plotkin's 1976 "A powerdomain construction" (though I didn't look for very long).

Josh Chen (Jul 29 2024 at 18:28):

For pullbacks/pushouts these are sometimes called the "gap" and "cogap" maps I believe?...

David Michael Roberts (Jul 29 2024 at 23:32):

@Josh Chen so for instance the map $A \to X\times_Y Z$ induced by the commuting square involving $A,X,Z,Y$ is called the "gap" map?

David Michael Roberts (Jul 29 2024 at 23:33):

@Nathanael Arkor and same question to you, but substituting 'gap' for 'mediating'

Josh Chen (Jul 29 2024 at 23:47):

David Michael Roberts said:

Josh Chen so for instance the map $A \to X\times_Y Z$ induced by the commuting square involving $A,X,Z,Y$ is called the "gap" map?

That's right

David Michael Roberts (Jul 30 2024 at 00:42):

Interesting, thanks.

Nathanael Arkor (Jul 30 2024 at 06:42):

David Michael Roberts said:

Nathanael Arkor and same question to you, but substituting 'gap' for 'mediating'

Yes.

Morgan Rogers (he/him) (Aug 02 2024 at 05:55):

I have also seen the adjective "universal" used to refer to such maps, as in "the universal map to the pullback [induced by/corresponding to the pair $(f,g)$ ]"

David Michael Roberts (Aug 02 2024 at 11:41):

Thanks!

Matteo Capucci (he/him) (Aug 04 2024 at 09:57):

I call them comparison maps since they compare solutions to a problem (find a co/cone), am I delusional? :S

Graham Manuell (Aug 04 2024 at 10:51):

The 'universal' terminology seems like it could be a bit confusing, since it's actually the projections that are universal according to the usual definition of universal, no?

David Michael Roberts (Aug 04 2024 at 11:11):

Graham Manuell said:

The 'universal' terminology seems like it could be a bit confusing, since it's actually the projections that are universal according to the usual definition of universal, no?

That was my thought. The actual quotient map is the universal one arising from (say) an equivalence relation, in my mind. And this extends to the maps in a (co)cone more generally, I guess.

Joe Moeller (Aug 04 2024 at 20:03):

I like comparison map, but I’m afraid my tendency is to say something like “the map given by the universal property of pullbacks” or similar. This has obvious positives and negatives.