What is that thing that keeps showing in papers on different fields?

Solution 1:

First, this is not the only spot where bizarre squiggly lines are used to represent deep concepts. There are things such as Feynmann diagrams,

circuit diagrams,

Petri nets,

and others. So this isn't the only spot where this type of thing is used. These things are called string diagrams and were first used by Roger Penrose in Combinatorial Mathematics and its Applications and Spinors and Spacetime for calculating with tensors.

The concept uses Poincaré duality to represent structures of dimension $d$ by structures of dimension 2-$d$. Objects are represented by portions of a plane, a 1-cell is represented by a vertical segment (or string) separating the plane into two, and a 2-cell is represented by an intersection of strings. Monoidal categories can be pictured this way, and the graphical language may be extended to represent expressions in categories with other structures, such as symmetric monoidal categories, and dagger categories.

Hope this helps!

References:

These are a few sites/papers that have good information (and good citations) about this topic; I'll be adding more as I find them:

• Wikipedia article
• The Geometry of Tensor Calculus, I by André Joyal

Solution 2:

The whole field of operad theory (and properads, and props) studies this kind of things. I think you could take a look at the theory of algebraic operads. If you are interested in learning more, here are some references:

1. A very gentle introduction is the paper Algebra + Homotopy = Operad by B. Vallette.
2. More technical, but a lot more complete, almost up to date with current research is the book by Loday-Vallette, which you can find here in pdf version.

Let me know if you want some details on specific topics, and I'll see if I'm able to answer.

PS: The last diagrams you wrote down remind me a lot of Hopf algebras. (Of course, there is a prop - if I'm not wrong, could be a properad; I'm not really sure of the difference between the two, I only know operads decently enough - coding them.)