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Stream: learning: questions

Topic: "Highschool maths" categorically

Thibaut Benjamin (Apr 09 2020 at 12:54):

I just realized today that the formula for integrals $\int_a^b f + \int _b^c f = \int_a^c f$ sounded very much like a composition in some category, and this got me thinking :

• On one hand, we can construct the groupoid $\mathbb{R}$, whose objects are real numbers, and such that there is exactly one morphism between $a$ and $b$ for all $a$ and $b$ (hence they are all invertible)
• On the other hand, we can construct the "delooping category" $\mathbf{B}\mathbb{R}_+$, which has just one object, whose hom-set is $\mathbb{R}$, with the composition given by the addition. This is the usual "a group is a groupoid with one object thing"

The relation I stated above really says that given a function $f$, which is sufficiently integrable, $\int f : \mathbb{R} \to \mathbf{B}\mathbb{R}_+$ is a functor!

As a bonus I can think of one other (French people will understand why I think of these two). Given an $\mathbb{R}$-affine space $E$ of dimension $n$, i can assemble it into a groupoid $\mathbf{E}$ by letting the objects of $\mathbf{E}$ be the points of $E$, and imposing one morphism between every two points. I also consider $\mathbf{B}\mathbb{R}^n$ to be the one-point category whose hom-set is $\mathbb{R}^n$, and whose composition is given by addition. Then there is a functor $\mathbf{E} \to \mathbf{B}\mathbb{R}^n$, which sends the unique morphism between $A$ and $B$ onto the vector $\overrightarrow{AB}$. This is just the relation
$\overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC}$.

The reason I said French will understand why, is that these two formulas, along with others are known under the same name of a "Chasles relation" in French https://fr.wikipedia.org/wiki/Relation_de_Chasles (unfortunately, I don't think there is an English equivalent, which is a shame, because learning these under the same name at 15yo really unveils the beautiful unity that one can see in mahts). So now I can give an (overly complicated)
formal definition of what is a Chasles relation : it is a functor from a groupoid with only one morphisms between each objects to the delooping of an abelian group (or maybe the delooping of any monoid would do, as you wish).

Of course this is all pretty basic stuff, but I had never seen these examples before despite working with categories for quite some time, so I was wondering if anyone had seen or written this sort of cute reinterpretation of highschool pieces of maths categorically? And what are other examples that you people can think of where a very elementary piece of maths could me made unnecessarily complicated (but also feel very natural) by giving a category theoretic formulation?

Thibaut Benjamin (Apr 09 2020 at 12:57):

I should add : don't try to find something deep in this question, it is only for entertainment purposes

Min Ro (Apr 09 2020 at 15:13):

I seem to recall a paper about thinking about Riemann integration categorically, though not with groups and groupoids (as far as I can recall). Though I can't find it and don't recall the details well enough.

Paolo Capriotti (Apr 09 2020 at 15:32):

The integral example is related to deRham cohomology. More generally, if you have a smooth manifold $M$, and a 1-form $\omega$ on $M$, you can define the integral of $\omega$ over a smooth curve in $M$, and that gives you a map from (the homotopy type of) $M$ to $\mathrm{B}\mathbb R$, i.e. a cohomology class in $H^1(M, \mathbb R)$.

Joachim Kock (Apr 09 2020 at 15:53):

This is nice! It is always nice to see categorical interpretations of geometry.

I think the observation can be summarised by saying that an affine space is a torsor over a vector space. Taking classifying spaces, you get the categories E -> B with the Chasles relation.

Joachim Kock (Apr 09 2020 at 15:54):

Here is a simplicial interpretation: for G a group (written additively), let X=BG be the classifying space, seen as a simplicial set. Then the corresponding torsor Y=EG is obtained by taking decalage of X. That means shifting down all simplicial degrees, so that the old degree 1 becomes the new degree 0. (And removing all top face and degeneracy maps.) So Y_0 are the group elements, and a 1-cell from g to h is a commutative triangle g=x+h. (So necessarily there is a unique such: x=g-h). Now Dec(X) has a simplicial map back to X, the dec map dec:Dec(X)->X given by the original top face maps. That's the projection map (the 'Chasles functor'), which sends a 1-cell (g,h) (given by an original triangle g=x+h) to x=g-h. So the Chasles functor is the dec map.

John Baez (Apr 09 2020 at 17:59):

Paolo Capriotti said:

The integral example is related to deRham cohomology. More generally, if you have a smooth manifold $M$, and a 1-form $\omega$ on $M$, you can define the integral of $\omega$ over a smooth curve in $M$, and that gives you a map from (the homotopy type of) $M$ to $\mathrm{B}\mathbb R$, i.e. a cohomology class in $H^1(M, \mathbb R)$.

John Baez (Apr 09 2020 at 18:00):

There's a 2-category $X(M)$ where objects are points of $M$, morphisms are smooth paths in $M$, and 2-morphisms from $\gamma : x \to y$ to $\gamma' : x \to y$ are 2-chains $c$ with $\partial c = \gamma' - \gamma$.

John Baez (Apr 09 2020 at 18:01):

There's also a 2-category $C$ with one object, one morphism from this object to itself for each number in $\mathbb{R}$, composition being addition, and only identity 2-morphisms.

John Baez (Apr 09 2020 at 18:02):

Then integrating a closed 1-form along a path gives a 2-functor $\int : X(M) \to C$.

John Baez (Apr 09 2020 at 18:03):

There are other, better variants of this idea.

Tim Hosgood (Apr 09 2020 at 18:26):

just while we're talking about de Rham cohomology and integration from a categorical point of view, this tiny thread is a nice little amuse-bouche

Jules Hedges (Apr 09 2020 at 18:48):

This whole idea reminds me of the great book "Mathematics made difficult" https://en.wikipedia.org/wiki/Mathematics_Made_Difficult

Thibaut Benjamin (Apr 09 2020 at 21:10):

Joachim Kock said:

This is nice! It is always nice to see categorical interpretations of geometry.

I think the observation can be summarised by saying that an affine space is a torsor over a vector space. Taking classifying spaces, you get the categories E -> B with the Chasles relation.

That sounds great, could you elaborate on how you take the classifying spaces? I am not completely sure about what that means

Thibaut Benjamin (Apr 09 2020 at 21:21):

John Baez said:

Then integrating a closed 1-form along a path gives a 2-functor $\int : X(M) \to C$.

I see, that's prettier than what I had in mind, considering paths in the manifold as morphism to model integration along a path!

This makes me wonder (I had to dust off old memories for this so I might be a bit rusty) : A vector field is the gradient of a scalar field if and only if its integral along any path only depends on the endpoints of the path. From the point of view of the category $X(M)$, the latter condition sounds like the $2$-functor it defines is locally constant (ie, constant on each homs). I wonder if there is a nice categorical way to frame this entire theorem. It's too late and about things I remember too vaguely for me to try doing it now, but I need to think about that

John Baez (Apr 09 2020 at 21:23):

Yes, there are nice ways to formulate this theorem using category theory. I bet you can have fun inventing one.

Gershom (Apr 09 2020 at 22:12):

Paolo Capriotti said:

The integral example is related to deRham cohomology. More generally, if you have a smooth manifold $M$, and a 1-form $\omega$ on $M$, you can define the integral of $\omega$ over a smooth curve in $M$, and that gives you a map from (the homotopy type of) $M$ to $\mathrm{B}\mathbb R$, i.e. a cohomology class in $H^1(M, \mathbb R)$.

The simple thing I'm thinking of here is differential forms on a manifold give a differential graded algebra, and that functor should suffice?

(perhaps i'm only stating the first half of what Paolo is talking about here...)

Paolo Capriotti (Apr 09 2020 at 22:50):

Gershom said:

The simple thing I'm thinking of here is differential forms on a manifold give a differential graded algebra, and that functor should suffice?

I was thinking of something else, avoiding chain complexes.

Fix a 1-form $\omega$, and consider the simplicial set $S(M)$ of smooth simplices in $M$ (which I'm assuming is weakly equivalent to the singular complex of $M$, hence to $M$ itself). Then integration gives you a simplicial map $S(M) \to B\mathbb{R}$, where $B\mathbb{R}$ is the classifying space of the additive group of real numbers. This is the same construction as @Thibaut Benjamin suggested, but for general smooth paths on $M$. So you map every point to the base point of $B\mathbb R$ and every path to the integral of $\omega$ on it, and verify that it can be extended to higher simplices. Such a map determines a cohomology class of degree 1 with coefficients in $\mathbb R$. I expect this to be the class corresponding to $\omega$ under the isomorphism of singular and de Rham cohomology.

You could also play this game with $n$-forms, in which case you would get maps into the $n$-fold delooping $B^n\mathbb R$, giving cohomology classes in $H^n(M, \mathbb R)$.

John Baez (Apr 09 2020 at 23:00):

If you let yourself take finite linear combinations of smooth simplices in $M$ you get a simplicial abelian group. The category of simplicial abelian groups is equivalent to the category of (nonnegatively graded) chain complexes of abelian groups by the Dold-Kan theorem. Then there's a way to turn a chain complex of abelian groups into a strict $\infty$-category. You get an abelian group object in the category of strict $\infty$-categories. There is probably something like an equivalence here too:

[chain complexes of abelian groups] $\sim$ [abelian group objects in $\infty$-Cat]

Tim Hosgood (Apr 10 2020 at 00:07):

John Baez said:

[...] by the Dold-Thom theorem. [...]

not Dold-Kan or Dold-Puppe?

John Baez (Apr 10 2020 at 00:24):

I meant Dold-Kan. I'll correct my comment.

Thibaut Benjamin (Apr 10 2020 at 09:45):

Thanks for your answers, I learnt a lot! This makes me want to open an elementary math book and try to categorize everything!

Joachim Kock (Apr 10 2020 at 11:09):

Thibaut Benjamin said:

could you elaborate on how you take the classifying spaces? I am not completely sure about what that means

I just meant for the underlying abelian groups, and then the simplicial viewpoint of my following comment.

Johannes Drever (Apr 10 2020 at 19:13):

Hinze and James wrote Reason Isomorphically! on high-scool algebra:

Now is a good opportunity to cast our minds back to simpler
times and recall our high-school algebra. For nostalgia’s sake, these
basic laws are listed in Table 1. We shall derive these laws from
first principles, showcasing two important tools: the Yoneda lemma
and adjunctions. We hope to demonstrate to the reader why these
concepts should be in their reasoning toolbox.

Joachim Kock (Apr 11 2020 at 14:14):

I wanted to follow up with another little geometry gem:

Define a category structure on the projective plane (or indeed any projective space): the objects are the points. The hom set Hom(P,Q) is the set of points of the line PQ. Composition is given as follows. Hom(P,Q) x Hom(Q,R) -> Hom(P,R) takes a point X on the line PQ and a point Y on the line QR, and returns the point of intersection of the line XY with the line PR. (There is an issue here with identity arrows. This can be fixed (I forgot how), but at the moment, in the spirit of projective geometry, let us just consider the generic situation.)

THEOREM. The associative law is equivalent to Desargues's theorem.

[Anders Kock: The category aspect of projective space, Aarhus Preprint Series (1974/75)]

John Baez (Apr 11 2020 at 16:37):

Very nice! As you probably know, an axiomatic projective plane comes from a division ring iff it's Desarguesian (i.e. it obeys Desargues' theorem). For a Desarguesian projective plane there's a way to cook up the division ring from operations on lines. I'm wondering if the theorem you mention is connected to this. It would be neat if the associativity of composition was connected to associativity of multiplication in the division ring, or something like that.