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Stream: deprecated: analysis

Topic: tangent bundle as a comonad

Joshua Meyers (Mar 12 2021 at 02:03):

So I think that tangent bundle is a comonad and vector fields are its coalgebras, though I don't know enough differential topology to check all the details.

Let $\textbf{Mfold}$ be some category of smooth manifolds and smooth maps.

There is a functor $T:\textbf{Mfold}\to\textbf{Mfold}$ sending $M$ to the tangent bundle $TM$ and sending a smooth map $f:M\to N$ to $df:TM\to TN$.

Counit is the projection map $TM\to M$.
Comultiplication is more tricky, but I think it is the map $TM\to TTM$ which sends $(x,v)$ to $((x,v),w)$ where $w$ is a vector which projects to $v$ and is orthogonal to $T_{(x,v)}T_xM$. If you have trouble seeing what I'm saying, write down the coherence laws and you'll see what $w$ has to be.

A coalgebra of this comonad is a vector field $\phi:M\to TM$, i.e. a section of the projection map.

Is this true? Is it known?

John Baez (Mar 12 2021 at 02:52):

This stuff is fun.

I don't think "orthogonal" makes sense because I don't think $T_{(x,v)} T_x M$ comes with an inner product. So, I'm unclear about your map $T M \to T T M$. Maybe it makes sense somehow.

On the other hand, there's definitely a map $T TM \to TM$, namely $d p : T TM \to TM$ where $p : TM \to M$ is the projection sending $(x,v) \in TM$ to $x \in M$.

John Baez (Mar 12 2021 at 02:55):

The best book on this stuff that I know is

• Iva Kolár, Peter W. Michor and Jan Slovák, Natural Operations in Differential Geometry.

John Baez (Mar 12 2021 at 03:05):

However, this doesn't seem to talk about monads or comonads.

Maybe $T$ is a monad: there's a "zero section" map $M \to TM$ and a map $TTM \to TM$ which I described, and it looks like they obey the monad laws.

John Baez (Mar 12 2021 at 03:06):

However, there is definitely also the projection $p: TM \to M$.

Joshua Meyers (Mar 12 2021 at 03:08):

Thanks for the book recommendation!

Joshua Meyers (Mar 12 2021 at 03:21):

Some stuff I know so far about the comultiplication $\delta:TM\to TTM,(m,v)\mapsto ((m,v),w)$, if it exists:

From one of the unit laws, $dp((m,v),w)=(m,v)$. You're right that there is no inner product on $T_{(m,v)}TM$, but we can still decompose it into $T_{(m,v)}T_mM\oplus W$ where $W$ is some subspace independent of $T_{(m,v)}T_mM$. If we consider the foliation of $TM$ into leaves $T_mM$ where $m$ runs over $M$, then $T_{(m,v)}T_mM$ is the subspace of $T_{(m,v)}TM$ which is along a leaf, and $W$ is the subspace along some transversal. So $w=\overline{v}+u$ where $\overline{v}\in W$ and $u\in T_{(m,v)}T_mM$, and $\overline{v}$ projects down to $v$.

The coassociativity square is too complicated for me to parse with my current state of knowledge, but it probably says something useful?

For a vector field $\phi:M\to TM$ which is a coalgebra for this putative comonad, we have that if $\phi(m)=(m,v)$, $\delta(m,v)=d\phi(m,v)$. Not sure what this tells us exactly, any ideas? $d\phi(m,v)$ feels kind of like an "acceleration".

John Baez (Mar 12 2021 at 03:24):

Sure. I'm not sure how much it will help you with your current question. But it's cool. It talks a lot about "functorial bundles", i.e. ways to functorially turn a manifold $M$ into a vector bundle over $M$, and "natural operations" between functorial bundles, i.e. natural transformations between such functors.

If you've heard about the "exterior derivative" of a differential form, or the "Lie bracket" of two vector fields, those are classic examples of natural operations. But the special feature of this book is that it aims to really classify such natural operations... instead of just bump into them, the way most geometers do.

Joshua Meyers (Mar 12 2021 at 03:30):

John Baez said:

However, this doesn't seem to talk about monads or comonads.

Maybe $T$ is a monad: there's a "zero section" map $M \to TM$ and a map $TTM \to TM$ which I described, and it looks like they obey the monad laws.

I think this fails one of the unit laws: if we do $TM\xrightarrow{\eta_{TM}}TTM\xrightarrow{dp} TM$ we get $(m,v)\mapsto ((m,v),0)\mapsto (m,0)$ which is not the identity.

Joshua Meyers (Mar 12 2021 at 03:39):

So apparently there is something called the "canonical vector field" $V:TM\to TTM$ but I don't understand the definition -- is this what I am attempting to describe?

John Baez (Mar 12 2021 at 03:47):

There's something like this. The kernel of $dp: TTM \to TM$ is a vector sub-bundle of $TTM$ called the vertical bundle, say $VTM$. It's a vector bundle over $TM$. To orient ourselves: if the dimension of $M$ is $m$, the dimension of $TM$ is $2m$, the dimension of $TTM$ is $4m$, and the dimension of $VTM$ is $2m$.

More later...

John Baez (Mar 12 2021 at 05:18):

So, we have an inclusion of vector bundles over $TM$, a mono

$i : VTM \to TTM$

and as always that means we can choose a splitting

$TTM \cong VTM \oplus HTM$

for example by choosing a smoothly varying inner product on the fibers of $TTM$ and choosing the fiber of $HTM$ at $p \in TM$ to be orthogonal complement of $VTM$. This reminds me of what you were trying to do!

But there's no natural splitting! (If $M$ were a Riemannian manifold, we'd get a splitting.)

Joshua Meyers (Mar 12 2021 at 05:31):

Apparently $T$ is a monad with the unit being the zero section, as you suggested, but where the multiplication is $dp_P+p_{TM}$

Joshua Meyers (Mar 12 2021 at 05:34):

The story about this seems to be a lot more complicated

Tobias Fritz (Mar 12 2021 at 06:48):

The tangent bundle functor has been axiomatized in terms of tangent categories, so the literature on that may be relevant for cross-checking your thoughts and providing inspiration.

John Baez (Mar 12 2021 at 06:55):

Joshua Meyers said:

Apparently $T$ is a monad with the unit being the zero section, as you suggested, but where the multiplication is $dp_P+p_TM$

Nice! If you look at the other formula for the multiplication that they give, you'll see it's a lot like "addition of tangent vectors" (though not quite literally that). It makes some sense that if the unit is "zero" then the multiplication is "addition".

John Baez (Mar 12 2021 at 06:56):

They also show their way is the unique way to to make $T$ into a monad!

John Baez (Mar 12 2021 at 07:36):

It's really quite a fun-filled paper. They also discuss "affine manifolds", where all of the transition functions between charts are affine maps.

JS PL (he/him) (Mar 12 2021 at 07:52):

Joshua Meyers said:

Apparently $T$ is a monad with the unit being the zero section, as you suggested, but where the multiplication is $dp_P+p_TM$

Proposition 2.2.4 of Jubin's thesis (the paper you linked) says that there is no comonad structure on the tangent bundle functor. Essentially, there can be no natural transformation $\delta: T \Rightarrow TT$ which satisfies the comonad identities with the projection $p: T \Rightarrow 1$

JS PL (he/him) (Mar 12 2021 at 07:53):

Vector fields are instead special maps in the Kleisli category of the tangent bundle monad.

JS PL (he/him) (Mar 12 2021 at 07:55):

Tobias Fritz said:

The tangent bundle functor has been axiomatized in terms of tangent categories, so the literature on that may be relevant for cross-checking your thoughts and providing inspiration.

Yes! See "Differential structure, tangent structure, and SDG" by Cockett and Cruttwell (https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/sman3.pdf) where they talk about the monad structure of these abstract tangent bundle functors.

JS PL (he/him) (Mar 12 2021 at 07:59):

This paper (https://arxiv.org/abs/1807.09554) by Blute, Cruttwell, and Lucyshyn-Wright, studies a nice subcategory where the tangent category does have a comonad structure.

Matteo Capucci (he/him) (Mar 12 2021 at 08:36):

JS Pacaud Lemay said:

Joshua Meyers said:

Apparently $T$ is a monad with the unit being the zero section, as you suggested, but where the multiplication is $dp_P+p_TM$

Proposition 2.2.4 of Jubin's thesis (the paper you linked) says that there is no comonad structure on the tangent bundle functor. Essentially, there can be no natural transformation $\delta: T \Rightarrow TT$ which satisfies the comonad identities with the projection $p: T \Rightarrow 1$

Isn't there, in the same thesis, a proof that $T^*$ is instead one? I recall reading a thesis about that

JS PL (he/him) (Mar 12 2021 at 12:33):

Matteo Capucci (he/him) said:

Isn't there, in the same thesis, a proof that $T^*$ is instead one? I recall reading a thesis about that

Quoting from the thesis: "the cotangentfunctor seems to give no satisfactory notion of (co)monad on it...." and "Finally, the cotangent functor is contravariant, and there is no notion of (co)monad ona contravariant “endofunctor”."

JS PL (he/him) (Mar 12 2021 at 12:33):

So no it does not seem like it...

JS PL (he/him) (Mar 12 2021 at 12:35):

But the tangent bundle functor on the subcategory of affine manifolds does have a comonad structure on it (in fact it is a Hopf monad)

Ben MacAdam (Mar 12 2021 at 19:36):

Joshua Meyers said:

So I think that tangent bundle is a comonad and vector fields are its coalgebras, though I don't know enough differential topology to check all the details.

Let $\textbf{Mfold}$ be some category of smooth manifolds and smooth maps.

There is a functor $T:\textbf{Mfold}\to\textbf{Mfold}$ sending $M$ to the tangent bundle $TM$ and sending a smooth map $f:M\to N$ to $df:TM\to TN$.

Counit is the projection map $TM\to M$.
Comultiplication is more tricky, but I think it is the map $TM\to TTM$ which sends $(x,v)$ to $((x,v),w)$ where $w$ is a vector which projects to $v$ and is orthogonal to $T_{(x,v)}T_xM$. If you have trouble seeing what I'm saying, write down the coherence laws and you'll see what $w$ has to be.

A coalgebra of this comonad is a vector field $\phi:M\to TM$, i.e. a section of the projection map.

Is this true? Is it known?

The comultiplication is coassociative, so you get the notion of a semi-comonad (co-semi-monad?). The category of smooth vector bundles is a full subcategory of the associative coalgebras of this comonad.

Ben MacAdam (Mar 12 2021 at 19:41):

I wonder if there's a special property of vector bundles in the category affine manifolds that also satisfy the unit law. @JS Pacaud Lemay What is the counit in that case?

dusko (Mar 13 2021 at 07:16):

Joshua Meyers said:

So I think that tangent bundle is a comonad and vector fields are its coalgebras, though I don't know enough differential topology to check all the details.

Let $\textbf{Mfold}$ be some category of smooth manifolds and smooth maps.

There is a functor $T:\textbf{Mfold}\to\textbf{Mfold}$ sending $M$ to the tangent bundle $TM$ and sending a smooth map $f:M\to N$ to $df:TM\to TN$.

Counit is the projection map $TM\to M$.
Comultiplication is more tricky, but I think it is the map $TM\to TTM$ which sends $(x,v)$ to $((x,v),w)$ where $w$ is a vector which projects to $v$ and is orthogonal to $T_{(x,v)}T_xM$. If you have trouble seeing what I'm saying, write down the coherence laws and you'll see what $w$ has to be.

A coalgebra of this comonad is a vector field $\phi:M\to TM$, i.e. a section of the projection map.

Is this true? Is it known?

bertfried fauser and i worked out the same idea here:
https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/smooth-coalgebra-testing-vector-analysis/EC6F285BC1B6EF74BCFA6D4D359969B2
--- though, of course, there is no reason why a different workout would not be better for some purposes. our purpose was to get close to the examples in the end.

dusko (Mar 13 2021 at 07:18):

(also https://arxiv.org/abs/1402.4414)

dusko (Mar 13 2021 at 08:36):

... i just remembered that cotangent bundle is a comonad, and that tangent bundle is i think a monad, with vector fields as algebras. what you are describ ing as the counit is the algebra structure on the manifolds, which you are canonically capturing by the monad. i think. it was a long time ago. but we worked it out in detail in the paper.

John Baez (Mar 13 2021 at 16:39):

Cool! I'm going to teach classical mechanics as a math grad course starting at the end of March, and there $T^\ast T^\ast M$ and $T^\ast TM$ play big roles. For example the tautological 1-form on $T^\ast M$ is a section of $T^\ast T^\ast M$ that makes the world go round as far as Hamiltonian mechanics is concerned. For some reason I'd never thought of trying to understand this stuff in terms of monads and comonads. It's fancier than the usual situation (just a monad, or just a comonad) because there are lots of interesting interactions between $T$ and $T^\ast$.

dusko (Mar 13 2021 at 22:01):

what i like about this story is how the entire landscape opens up by embedding manifolds into the gros topos of all spaces. this is i think in sec 4. ok, the heart of it is in sec 4.2. that's where everything interacts with everything else in the primordial gros topos of all spaces. and we can just put a little frame on the structure there. the little pictures arise from the big picture.

dusko (Mar 13 2021 at 22:17):

it displays how grothendieck's view of the world came about. we are, of course, on the territory of his thesis, the TVS. he extracted all his tensor products, the various continuity structures, from the various embeddings into Top. then he quit the TVS and proceeded to apply the approach to everything else. alg geometry: embed a topological space into Top to get a generalized space, le petit topos. alg topology: go back along such an embedding to determine the invariants... he was all about how we simplify hard problems by generalizing: abstract away the implementation details. that is the first thing they tell yo to try in any math circle, and you soon find out that it doesn't always work, but the TVS was i think where grothendieck did it first. the general terms of how to evaluate algebras and coalgebras on one another were not available yet, but the story is his.

Matteo Capucci (he/him) (Mar 14 2021 at 17:42):

dusko said:

... i just remembered that cotangent bundle is a comonad, and that tangent bundle is i think a monad, with vector fields as algebras. what you are describ ing as the counit is the algebra structure on the manifolds, which you are canonically capturing by the monad. i think. it was a long time ago. but we worked it out in detail in the paper.

A few messages above @JS Pacaud Lemay said $T^*$ isn't... maybe we are talking about different functors on morphisms?

dusko (Mar 15 2021 at 01:08):

Matteo Capucci (he/him) said:

dusko said:

... i just remembered that cotangent bundle is a comonad, and that tangent bundle is i think a monad, with vector fields as algebras. what you are describ ing as the counit is the algebra structure on the manifolds, which you are canonically capturing by the monad. i think. it was a long time ago. but we worked it out in detail in the paper.

A few messages above JS Pacaud Lemay said $T^*$ isn't... maybe we are talking about different functors on morphisms?

like i said, all structure is spelled out in the paper. $T^*$ has both a covariant and a contravariant version. the arrow parts are in sections 4.4.1 and 4.4.2. the comonad structure is in sec 4.4.5.

in general, proving that something has a structure boils down to displaying a construction, whereas proving that it does not requires deriving a contradiction from all possible constructions.

dusko (Mar 15 2021 at 01:35):

... maybe it is useful to remember that functors are not monads or comonads because they happen to carry this or that structure for this or that family of morphisms. they are monads because they inductively adjoin some algebraic structure to objects, and they are comonads because the add state of some sort. and in this case, the monad and the comonad structure express the fact that dynamic systems are described by applying some algebras of tests to some coalgebras of stateful behaviors. so the tangent and the cotangent bundle functors don't just happen to carry the monad and the comonad structure because someone invents some morphisms that happen to support that. these structures capture how we observe and describe dynamical systems.

JS PL (he/him) (Mar 15 2021 at 08:41):

Matteo Capucci (he/him) said:

A few messages above JS Pacaud Lemay said $T^*$ isn't... maybe we are talking about different functors on morphisms?

Yes that seems like what's happening! In @dusko 's paper, the contravariant cotangent bundle functor is denoted $T^\ast$, which does not have a comonad structure since it's a contravariant functor (which is what Jubin states in his thesis). But also in @dusko's paper, the covariant cotangent bundle functor is denoted $T_\#$, and this is a comonad! Which is super cool!

JS PL (he/him) (Mar 15 2021 at 08:44):

The kicker is that on objects, $T^\ast(M) = T_\#(M)$, but they differ on maps $f: M \to N$, since $T^\ast(f): T^\ast(N) \to T^\ast(M)$ and $T^\#(f): T^\#(M) \to T^\#(N)$.

JS PL (he/him) (Mar 15 2021 at 08:46):

(of course, I'm just repeating some of the above comments)

JS PL (he/him) (Mar 15 2021 at 08:46):

Ben MacAdam said:

I wonder if there's a special property of vector bundles in the category affine manifolds that also satisfy the unit law. JS Pacaud Lemay What is the counit in that case?

The counit is still just the projection $p: T(M) \to M$, but the comultiplication $\delta: T(M) \to TT(M)$ is not the vertical lift from the tangent category structure.