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Stream: theory: philosophy

Topic: Tensions between local data and global data

Keith Elliott Peterson (Jul 01 2023 at 05:42):

I'm not sure if I'm asking this properly, but why is it that there is sometimes a tension, in math, sciences, and philosophy, between data that is constructed and "glued" locally and data that is "built down" from a global point-of-view? One would expect them to come to some congruence and align, but it doesn't always seem to be the case.

For instance, as I understand, the microeconomics tendencies and leanings of the Austrian school of economics, and the macroeconomics tendencies and leanings of the Neoclassical school, seem to diverge and give drastically different recommendations for policy, even though they are both studying the same object: market economies. Though, I'm sure there are far better examples in the sciences, mathematics, and philosophy.

John Baez (Jul 01 2023 at 13:55):

I don't know why there is this tension, but a bunch of mathematicians are a bit unhappy with objects that are defined using pieces (e.g. 'charts') that are glued together (using 'transition functions'), since this seems artificially constructed, and one often needs to prove that various things don't really depend too much on the choice of those pieces. For example I think Alain Connes has expressed his dissatisfaction with the concept of 'smooth manifold' as one of the reasons he likes spectral triples instead.

John Baez (Jul 01 2023 at 14:21):

One might say that things defined as quotients (like a bunch of charts stuck together) look artificial if you're used to things defined as subobjects (like a submanifold of $\mathbb{R}^n$) and vice versa. Certainly the approach to defining manifolds using charts was developed to develop what they 'intrinsic geometry' - geometry that doesn't depend on a choice of embedding of your manifold in $\mathbb{R}^n$.

John Baez (Jul 01 2023 at 14:28):

Regardless of what's 'artificial' or not - that could be a subjective judgement - there's a real distinction built into category theory between working with subobjects and working with quotient objects. There's a kind of 'paradise' when you can show every quotient object of an object $X$ is also a subobject: for example in set theory we reach this paradise only by assuming the axiom of choice.

John Baez (Jul 01 2023 at 14:29):

In algebra, we reach this paradise when 'every short exact sequence splits'. That basically just means every quotient object is a subobject, and vice versa. That's one reason people love [[semisimple objects]]: for these objects, every quotient object is a subobject, and vice versa.

John Baez (Jul 01 2023 at 14:33):

For example in algebra we love [[semisimple rings]], which are defined to be rings whose modules are all semisimple, meaning that every quotient module of any module is also a submodule, and vice versa. But this situation is so nice that it's now considered boring; one could say, with only a little exaggeration, that the whole machinery of homological algebra was developed to understand what's going when not every quotient object is a subobject.

Naso (Jul 02 2023 at 01:30):

John Baez said:

for example in set theory we reach this paradise only by assuming the axiom of choice.

John, is this about saying saying quotients of $X$ are equivalent to equialence relations (i.e. certain subsets of $X \times X$)? Where does AOC come in? Is it about choosing a representative from each class?

John Baez said:

That basically just means every quotient object is a subobject, and vice versa. That's one reason people love [[semisimple objects]]: for these objects, every quotient object is a subobject, and vice versa.

The nLab definition says semisimple (for an abelian category?) means a coproduct of simple objects, simple meaning it has exactly two quotient objects, $0$ and itself.
Can you please say how this is equivalent to "every quotient object is a subobject and vice versa"?

David Michael Roberts (Jul 02 2023 at 09:29):

AC comes because you need it to find a section of a surjection.

David Michael Roberts (Jul 02 2023 at 09:31):

The correspondence between quotients and equivalence relations doesn't need AC, it holds in any pretopos, for instance.

John Baez (Jul 02 2023 at 13:03):

Naso said:

John Baez said:

That basically just means every quotient object is a subobject, and vice versa. That's one reason people love [[semisimple objects]]: for these objects, every quotient object is a subobject, and vice versa.

The nLab definition says semisimple (for an abelian category?) means a coproduct of simple objects, simple meaning it has exactly two quotient objects, $0$ and itself.
Can you please say how this is equivalent to "every quotient object is a subobject and vice versa"?

I didn't say they were equivalent: it's a one-way implication.

John Baez (Jul 02 2023 at 13:05):

In fact I added a remark to the nLab pointing to a Mathoverflow discussion of examples where the converse fails.

John Baez (Jul 02 2023 at 13:08):

But you can show that in a semisimple (abelian) category every short exact sequence splits, and that's a precise way of saying "every subobject is a quotient object and vice versa".

John Baez (Jul 02 2023 at 13:10):

I'm too lazy to write up the proof here on my cell phone.

John Baez (Jul 02 2023 at 13:12):

If you look at the Wikipedia article on "semisimple module" it will give you the basic idea in an example.

Naso (Jul 03 2023 at 01:22):

David Michael Roberts said:

AC comes because you need it to find a section of a surjection.

Which surjection is that? If the correspondence between quotients and equivalence relations doesn't need AC, I am not sure what John meant by saying the paradise is reached only in Sets + AC. I think it cannot mean the subsets of $X$ correspond to quotients of $X$, since these are counted by different formulas ($2^n$ vs the bell number $B_n$).

Kevin Arlin (Jul 03 2023 at 21:19):

It just means that every quotient of $X$ is isomorphic to some subset of $X,$ although the more transparent connection to AC is when you ask that every quotient can be included as a subobject via map which splits the quotient map. I'm not sure what the axiomatic status of "every quotient is isomorphic to some subset" is, although it doesn't seem like a particularly natural condition in itself.

David Michael Roberts (Jul 04 2023 at 12:24):

In set theory there are two cardinality orderings, via injections and via surjections (the empty set being a special case for that one). Without AC they are not the same.

John Baez (Jul 04 2023 at 14:42):

Naso said:

David Michael Roberts said:

AC comes because you need it to find a section of a surjection.

Which surjection is that? If the correspondence between quotients and equivalence relations doesn't need AC, I am not sure what John meant by saying the paradise is reached only in Sets + AC.

It does need AC. @David Michael Roberts was explaining why. Let me say what he said, but much more lengthily:

Suppose you have a surjection of sets $p: A \to B$. Then $B$ is a quotient of $A$. But now suppose you want to see $B$ as a subobject of $A$. For this you want to find an injection of sets $i: B \to A$ such that $p \circ i = 1_B$. In other words, you want to choose for each $b \in B$ an element $a \in A$ such that $p(a) = b$, so you can define $i(b) = a$ and get $p(i(b)) = b$. You can do this "choosing" if the axiom of choice holds. In fact the axiom of choice is equivalent to saying that for every surjection $p: A \to B$ there exists an injection $i: B \to A$ such that $p \circ i = 1_B$.

You can say all this much more quickly as follows: "AC holds iff every surjection has a section".

David Michael Roberts (Jul 04 2023 at 23:46):

However, one might wonder about the existence of any injection, not just a section. There are examples where this also needs AC, for instance take a nontrivial double cover of the natural numbers in a setting where countable choice fails. This is like a nontrivial Z/2-bundle even though the base is discrete. The total space is bigger than the base in the $\leq^*$ ordering (which uses surjections), but the base is not smaller than the total space in the $\leq$ ordering, the one that uses injections. That is, there is no injection from the natural numbers to the total space (which is sometimes called a Russell set, after Bertrand Russell's thought experiment about an infinite number of pairs of socks).

Fun puzzle: why is it so?

Keith Elliott Peterson (Jul 05 2023 at 04:14):

@John Baez I was thinking more in line with Penrose's 'Cohomology of Impossible Figures'. Though, I won't stop the discussion with AC.

Keith Elliott Peterson (Jul 05 2023 at 04:15):

Unless of course I'm missing something in the discussion.

Martti Karvonen (Jul 05 2023 at 08:13):

This tension between local and global data shows up in the sheaf-theoretic approach to contextuality. This is the founding paper and this might be an easier entry-point. They even mention this Penrose paper and the impossible figures in some talks.

Naso (Jul 06 2023 at 01:09):

John Baez said:

It does need AC. David Michael Roberts was explaining why. Let me say what he said, but much more lengthily:

Suppose you have a surjection of sets $p: A \to B$. Then $B$ is a quotient of $A$. But now suppose you want to see $B$ as a subobject of $A$. For this you want to find an injection of sets $i: B \to A$ such that $p \circ i = 1_B$. In other words, you want to choose for each $b \in B$ an element $a \in A$ such that $p(a) = b$, so you can define $i(b) = a$ and get $p(i(b)) = b$. You can do this "choosing" if the axiom of choice holds. In fact the axiom of choice is equivalent to saying that for every surjection $p: A \to B$ there exists an injection $i: B \to A$ such that $p \circ i = 1_B$.

You can say all this much more quickly as follows: "AC holds iff every surjection has a section".

Thank you! I think this is kind of what I had in mind when I said "choosing a representative from each class".

John Baez (Jul 06 2023 at 10:41):

Right: for any surjection $p: A \to B$ there is a unique equivalence relation on $A$ such that the elements of $B$ are the equivalence classes, and $p$ sends any element to its equivalence class. Then $i: B\to A$ with $p \circ i = 1_B$ is a way of choosing a representative of each equivalence class.

dan pittman (Jul 06 2023 at 21:36):

This is a really interesting question! I have a "build up" example from philosophy: assemblages. I'm most familiar with Deleuze and Guattari's version of this—it is a model for libidinal forces from a mereological perspective. They draw from dynamical systems, but also when I came across Behavioral Mereology the first time, it felt like a way to formalize the lyrical and metaphorical thinking of D&G.

dan pittman (Jul 06 2023 at 21:39):

John Baez said:

Right: for any surjection $p: A \to B$ there is a unique equivalence relation on $A$ such that the elements of $B$ are the equivalence classes, and $p$ sends any element to its equivalence class. Then $i: A \to B$ with $p \circ i = 1_B$ is a way of choosing a representative of each equivalence class.

I think $i$ probably ought to be $i : B \to A$ here!

John Baez (Jul 07 2023 at 04:54):

Right, I'll fix it.

Keith Elliott Peterson (Jul 09 2023 at 00:28):

Martti Karvonen said:

This tension between local and global data shows up in the sheaf-theoretic approach to contextuality. This is the founding paper and this might be an easier entry-point. They even mention this Penrose paper and the impossible figures in some talks.

Thank you for the recommendations.