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Stream: learning: questions

Topic: How can we study a collection of functions with the same ...

Julius Hamilton (Sep 20 2024 at 16:41):

For a fixed set $I$, we can consider the set $F$ of functions $f : X \to Y$ for all $X, Y$ such that $Im(f) = I$. How should we define that set, and how can we study it?

For example, it can be defined $F = \{ f : X \to Y | Im(f) = I \}$, where $Im(f) = \{ y \in Y | \exists x \in X [f(x) = y] \}$.

John Baez (Sep 20 2024 at 17:27):

Are you talking about functions $F$ with a fixed domain $X$ or all possible domains $X$? Your question didn't make clear which of these two very different cases you mean, though "all function" suggests the latter.

Julius Hamilton (Sep 20 2024 at 17:27):

The latter.

John Baez (Sep 20 2024 at 17:48):

Your question was "ungrammatical" because you started using the sets $X$ and $Y$ without introducing them. In introducing you'd typically say whether you're fixing them or quantifying over them.

John Baez (Sep 20 2024 at 17:50):

Julius Hamilton said:

The latter.

Okay. Then your "set" is not a set: it's a proper class.

Kevin Carlson (Sep 20 2024 at 17:52):

One natural way to think about this not-quite-set is as the category of all surjections $f:X\to I,$ where morphisms $f\to g$ for $g:Y\to I$ are commutative $h:X\to Y$ with $g\circ h=f.$ And maybe you want $h$ to be surjective too.

David Egolf (Sep 20 2024 at 18:22):

I'm curious why you are interested in some particular image $I$! Do you have some specific $I$ in mind?

In any case, you might consider trying to create a category where objects are functions, but we don't require the image of each function to be $I$. One could still consider functions with image $I$ in the context of this larger category.

Julius Hamilton (Sep 20 2024 at 20:58):

I guess this is a good opportunity for me to study NBG class theory, then, to axiomatize this.

John Baez (Sep 20 2024 at 21:18):

I'd instead recommend thinking about the issue some better way! I would treat the fact that you're getting a proper class here as a red flag sent down from the math gods, warning you that you are not doing things the best way.

For example instead of working with the class of all functions whose image is some set - a huge and poorly organized thing - it could be better to work with a representable presheaf on the category of surjections. And learning what this means is likely to be more generally rewarding than learning NBG set theory, unless you have a special hankering to learn nonstandard approaches to set theory.

There's too much math to learn all of it, so it's good to be strategic about what you study.

Julius Hamilton (Sep 23 2024 at 18:30):

Cool. I believe the idea is to epi-mono factorize functions $\{f : X \to Y | Im(f) = I\}$, into $(\iota : I \to Y) \circ (f^* : X \to I)$. $\iota$ should be the inclusion map $i(x) = x$.

John Baez (Sep 23 2024 at 18:34):

That's fine, but now I hope you're talking about the set of morphisms from a fixed $X$ into a fixed $Y$, rather than the class of all morphisms from all $X$ into all $Y$. It's that class that's a kind of red flag: you don't want to use classes when you could easily use sets.

Julius Hamilton (Sep 26 2024 at 13:36):

Perhaps someone could give me a clue to help me understand the motivation to use a category of surjections.

For example, might we say, that since the image of a function has an equivalent definition as a surjection, we want to see the relationship between the surjection whose codomain is $I$, and other surjections?