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Stream: learning: questions

Topic: notation for delooping

Ralph Sarkis (Jul 06 2021 at 20:59):

Where does the notation $\mathbf{B}$ for delooping objects come from?

Nathanael Arkor (Jul 06 2021 at 21:55):

This question was recently answered on Math.SE.

Ralph Sarkis (Jul 06 2021 at 22:12):

Thanks! It seems my google-fu needs tuning.

John Baez (Jul 07 2021 at 16:37):

For those too lazy to click: BG is the base space of the universal G-bundle.

Mike Shulman (Jul 07 2021 at 17:43):

And EG is the total space of the universal G-bundle. (In French, the word for "space" starts with "e".) But since everything in sight is a space, wouldn't it have made more sense to write TG for the total space of the bundle?

John Baez (Jul 07 2021 at 17:45):

When I think of the total space of a bundle the letter E instantly comes to mind, unless it's a principal bundle in which case P is also fine. All this is just habit based on what I've read.

Reid Barton (Jul 07 2021 at 17:48):

I think EG does stand for the "total space", it's just that in French the adjective comes after the noun.

Kenji Maillard (Jul 07 2021 at 17:51):

Couldn't the E letter come also from the étale space (espace étale) ? It is an instance of a total space and the notation could have contaminated other areas over time. (This is purely speculative but the MO answer does not provide any source reference either).

Mike Shulman (Jul 07 2021 at 18:32):

Reid Barton said:

I think EG does stand for the "total space", it's just that in French the adjective comes after the noun.

Certainly the adjective comes after the noun in French, but if there are a lot of spaces around it doesn't make sense to abbreviate only one of them with the first letter of "space". If "espace total" is abbreviated by E, shouldn't "espace de base" also be abbreviated by E? And conversely, if "espace de base" is abbreviated by B, shouldn't "espace total" be abbreviated by T?

Joe Moeller (Jul 07 2021 at 21:02):

Just a guess, maybe T for tangent bundle specifically came first?

Tim Hosgood (Jul 07 2021 at 22:19):

i think i've heard "base" more often than "espace de base" from french speakers, so maybe this is of some relevance to the choice of notation

Tim Hosgood (Jul 07 2021 at 22:20):

i would definitely expect to hear "fibre, espace total, et base"

Tim Hosgood (Jul 07 2021 at 22:21):

Joe Moeller said:

Just a guess, maybe T for tangent bundle specifically came first?

but also this, i think