What "given a foliation $f:L \to M$ " means?

Question

I'm reading this paper. And on page $4$ the author writes the following paragraph:

I am not sure about what the phrase "given a foliation $f:L\to M$" means. A foliation is an atlas, so it is awkward to say that a function is a foliation. At first sight, I thought that the phrase "$f:L\to M$" was simply an inaccurate way to say "the foliation induced by the submersion $f$". However as we can see over the underlined line there is the phrase "so that $f$ must be a $\mathcal{C}^r$-immersion", then my assumption is wrong.

Can anyone help me to understand what "given $f:L\to M$" actually means?.

Just for the records the definition of foliation that I use is the same that we can find on the wikipedia.

Answer 1

I'm not sure, but is it possible that to say that $f : A \rightarrow B$ is a foliation is to say that the leaves of the fioliation under question are the fibers of $f$, that is, the elements of $$\{f^{-1}(b):b \in B\}$$

? This seems possible.

According to wikipedia:

If $M^n → N^q (q \leq n)$ is a submersion of manifolds, it follows from the inverse function theorem that the connected components of the fibers of the submersion define a codimension $q$ foliation of $M$. Fiber bundles are an example of this type.

So it seems that foliations can be viewed special cases of surjective mappings between manifolds, and/or as generalizations of submersions.