Naturality of the lie bracket.

Question

I heard the term natural transformation for functors. Does the naturality of the lie bracket has something to do with that?
Naturality of the lie bracket : $F_*[V_1,V_2]=[F_*V_1,F_*V_2]$ where the star denotes the pushforward and $V_1$, $V_2$ are vector fields.

Answer 1

Yeah, these ideas can be expressed in the language of category theory. For instance, the fact that the star is a subscript (instead of a superscript) on $F_*$ indicates that there is a covariant functor (instead of a contravariant functor) behind the scenes. To be precise, the naturality you seek involves the category Diff with smooth manifolds as objects and diffeomorphisms as morphisms, as well as the category Vect$_\mathbf{R}$ with real vector spaces as objects and linear maps as morphisms.

In this language, $\mathfrak{X}$ is a functor from Diff to Vect$_\mathbf{R}$ sending diffeomorphisms $F\colon M\to N$ of smooth manifolds to linear maps $F_*\colon\mathfrak{X}(M)\to\mathfrak{X}(N)$, where $\mathfrak{X}(M)$ denotes the space of smooth vector fields on $M$. That is, $\mathfrak{X}\colon F\mapsto F_*$.

Similarly, we may define a functor $\mathfrak{X}\times\mathfrak{X}$ from Diff to Vect$_\mathbf{R}$ by sending $F$ to $F_*\times F_*$, where $(F_*\times F_*)\colon\mathfrak{X}(M)\times\mathfrak{X}(M)\to\mathfrak{X}(N)\times\mathfrak{X}(N)$ is defined as one may expect — namely by sending $(X,Y)$ to $(F_*X,F_*Y)$.

The Lie bracket $[\cdot,\cdot]\colon\mathfrak{X}\times\mathfrak{X}\to\mathfrak{X}$ is then a natural transformation from the functor $\mathfrak{X}\times\mathfrak{X}$ to the functor $\mathfrak{X}$, as the identity $F_*[X,Y]=[F_*X,F_*Y]$ is precisely the statement that the diagram

$$\require{AMScd}\begin{CD}\mathfrak{X}(M)\times \mathfrak{X}(M) @>F_*\times F_*>> \mathfrak{X}(N)\times \mathfrak{X}(N)\\@VV[\cdot,\cdot]V @VV[\cdot,\cdot]V\\\mathfrak{X}(M) @>F_*>> \mathfrak{X}(N)\end{CD}$$

commutes.

Answer 2

One possible view which would make the use of the term "natural" here refer to natural transformation is to think of the space of vector fields as a sheaf of Lie-algebras. Thinking of a sheaf as a functor and viewing the pushforward associated to a smooth map on the base manifolds as a natural transformation between these functors may are may not serve as a satisfying interpretation, depending on your tastes.