# Chain/Hierarchy of Monoids

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.

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What you have is a chain of monoids, each acting by endomorphisms of the previous monoid, but you haven't specified any relationship between monoids of distance greater than one. For example, there is no constraint relating a monoid's endomorphisms (arising from the next monoid) and its action on the previous monoid. This seems too floppy to be anything like an n-category. Also, if you had a higher categorical structure with one k-morphism for each k, then that structure would be the point. –  S. Carnahan May 15 '10 at 15:16
@Scott: Do you have a specific example or reference to the kind of relationships I'm lacking? –  supercooldave May 15 '10 at 15:33