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Stream: learning: questions

Topic: Epi pullback

Sean Gloumeau (Oct 06 2021 at 12:04):

image.png

Hi, I'm trying to understand precisely what an epi pullback is (it's a term mentioned in Sheaves in Geometry and Logic). Does it mean that both p and q in the commutative diagram are epi?

Mike Shulman (Oct 06 2021 at 12:11):

What is the exact phrase in which it's mentioned?

Dylan Braithwaite (Oct 06 2021 at 12:11):

Do you mean in contexts where it says something like "pullbacks of epis are epi"?

It's referring to the terminology where you would say "$p$ is the pullback of $g$ along $f$". So the pullback in this case is a single morphism rather than the whole square.

Ie it means that (in some categories) if $g$ is epi, then $p$ is epi. And if $f$ is epi, then $q$ is epi.

Sean Gloumeau (Oct 06 2021 at 12:13):

Mike Shulman said:

What is the exact phrase in which it's mentioned?

image.png
Here's the exact phrasing, it's in one of the exercises for Chapter 1

Mike Shulman (Oct 06 2021 at 12:14):

That doesn't use the phrase "epi pullback" at all!

Mike Shulman (Oct 06 2021 at 12:14):

It means what Dylan said.

Sean Gloumeau (Oct 06 2021 at 12:14):

Yeah, I believe Dylan got it. Thanks Mike & Dylan :)!

Mike Shulman (Oct 06 2021 at 12:14):

More generally, in a diagram like the one you pictured, we say that p is a "pullback of g" and that q is a "pullback of f".

Matteo Capucci (he/him) (Oct 06 2021 at 15:47):

lol though what @Sean Gloumeau understood is grammatically plausible

Matteo Capucci (he/him) (Oct 06 2021 at 15:48):

Indeed pullbacks are described in one way (universal spans over a cospan) and then often used in another ('pull back' a morphism along another one)

John Baez (Oct 06 2021 at 16:03):

Beginners often get confused about the relation between the verb "to pull back" and the noun "pullback", especially because lots of mathematicians use the former who have no understanding of the latter.

For example, as Matteo hinted, a lot of mathematicians who know little category theory will talk about pulling back a function $f : Y \to A$ along a function $g: X \to Y$ to get the function $f \circ g: X \to A$. And then when they meet the categorical concept of "pullback", they will wonder whether their "pulling back" is an example of a "pullback".

John Baez (Oct 06 2021 at 16:03):

And the answer is: yes, but you have to think a little about how.

Matteo Capucci (he/him) (Oct 07 2021 at 05:45):

Mmh it's the first time I realize the verb pull back is used in two ways: to pullback $f : Y \to A$ 'along' $g : X \to Y$ and (most commonly in ct, in my experience) to pullback $f$ along $h : B \to A$

Matteo Capucci (he/him) (Oct 07 2021 at 05:47):

John Baez said:

And the answer is: yes, but you have to think a little about how.

Now I have a puzzle to solve :laughing:

Fawzi Hreiki (Oct 07 2021 at 07:44):

It seems like ‘precompose’ is more appropriate than pullback for the first one

Zhen Lin Low (Oct 07 2021 at 09:08):

I think the precomposition meaning is older.

Sean Gloumeau (Oct 07 2021 at 10:07):

Precompose is certainly clearer, its name implies its meaning

Sean Gloumeau (Oct 07 2021 at 10:09):

In any case, I now feel more confident I can distinguish which pullback is meant now that I understand the different contexts for its use. Thank you @John Baez !

Reid Barton (Oct 07 2021 at 10:13):

The classical example of "pulling back a function" would be something along the lines of: $X$ and $Y$ are smooth manifolds, say, $g : X \to Y$ is a smooth map, and $f : Y \to \mathbb{R}$ is a smooth real-valued function. We think of $f$ as assigning a real number to each point of $Y$, and $f \circ g$ "pulls back" this assignment to points of $X$.