Let's assume that we have the following collection of structures:

- Some
*space*$P$. - Monoids $(M_{i+1},\circ_{i+1})$, and
- Actions $\bullet_{i+1}:M_{i+1}\times M_i\to M_i$, for $i\ge 0$
- And $\bullet_{0}:M_0\times P\to P$.

satisfying

- ($\bullet$ is a monoid action): $(m\circ_{i+1}m')\bullet_{i+1} n = m\bullet_{i+1}(m'\bullet_{i+1} n)$ and
- ($m\bullet-$ is a homomorphism): $m\bullet_{i+1}(n\circ_{i}n')=(m\bullet_{i+1}n)\circ_{i} (m\bullet_{i+1} n')$.

In my application, $P$ corresponds to computer programs. $M_0$ are modifications to elements of $P$. If you wish, you can think of $M_0$ as some kind of structured patch. Then each $M_{i+1}$ are *higher-order modifications* of the modifications in $M_i$.

The hierarchy isn't necessarily infinite.

I'm curious to know what kind of structure I'm looking at. I originally felt that I was defining some kind of $n$-category with one object at each level, namely the endomorphism, but one reader commented that my structures were too floppy, meaning that there were not enough equations.

It seems that the structure I'm interested in is related to the automorphism tower for groups, except that I'm interest in monoids, and rather than automorphism, I'm only concerned with endomorphism, and I am working indirectly through monoid actions, rather than having the endomorphism apply to the morphisms at the level below.

Have I defined a known structure?

What natural equations would one expect to link the various levels with each other?

What additional properties does it satisfy? What reasonable properties should it satisfy?

Are there conditions under which it becomes degenerate?

Any pointers would be appreciated.