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.

I'm re-editing the question with the hope of getting some more feedback.

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?

Are there conditions under which it becomes degenerate?

Any pointers would be appreciated.

I'm re-editing the question with the hope of getting some more feedback.

The idea is that

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

I'm curious to know what kind of structure I'm looking at. I have the feeling originally felt that I've defined I was defining some (possibly degenerate) strict kind of $n$-category (or $\omega$-cat), where there is with one object at each level(I guess , namely the endomorphisms)endomorphism, but I'm not entirely sure how to cast this categorically.

Have I defined a know structure? What additional (degenerate) properties does it satisfy?

Any pointers would be appreciated.

EDIT: Note one reader commented that I say nothing about the structure of $P$ because generally I'm only interested in the hierarchy of monoids/modificationsmy structures were too floppy, meaning that there were not enough equations.

EDIT 2:

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 perhaps rather than automorphism, I'm being indirect by operating via 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?

Are there conditions under which it becomes degenerate?

Any pointers would be appreciated.

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): $m\bullet_{i+1}(n\circ_{i}n')=(m\bullet_{i+1}n)\circ_{i} (m\bullet_{i+1} n')$.

The idea is that $P$ is some space we are interested in, $M_0$ can be thought of as modifications of $P$, and each $M_{i+1}$ are modifications of the modifications in $M_i$, thus higher-order modifications.

The hierarchy isn't necessarily infinite.

I'm curious to know what kind of structure I'm looking at. I have the feeling that I've defined some (possibly degenerate) strict $n$-category (or $\omega$-cat), where there is one object at each level (I guess the endomorphisms), but I'm not entirely sure how to cast this categorically.

Have I defined a know structure? What additional (degenerate) properties does it satisfy?

Any pointers would be appreciated.

EDIT: Note that I say nothing about the structure of $P$ because generally I'm only interested in the hierarchy of monoids/modifications.

EDIT 2: 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 perhaps I'm being indirect by operating via monoid actions.

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): $m\bullet_{i+1}(n\circ_{i}n')=(m\bullet_{i+1}n)\circ_{i} (m\bullet_{i+1} n')$.

The idea is that $P$ is some space we are interested in, $M_0$ can be thought of as modifications of $P$, and each $M_{i+1}$ are modifications of the modifications in $M_i$, thus higher-order modifications.

The hierarchy isn't necessarily infinite.

I'm curious to know what kind of structure I'm looking at. I have the feeling that I've defined some (possibly degenerate) strict $n$-category (or $\omega$-cat), where there is one object at each level (I guess the endomorphisms), but I'm not entirely sure how to cast this categorically.

Have I defined a know structure? What additional (degenerate) properties does it satisfy?

Any pointers would be appreciated.

EDIT: Note that I say nothing about the structure of $P$ because generally I'm only interested in the hierarchy of monoids/modifications.

EDIT 2: 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 perhaps I'm being indirect by operating via monoid actions.

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): $m\bullet_{i+1}(n\circ_{i}n')=(m\bullet_{i+1}n)\circ_{i} (m\bullet_{i+1} n')$.

The idea is that $P$ is some space we are interested in, $M_0$ can be thought of as modifications of $P$, and each $M_{i+1}$ are modifications of the modifications in $M_i$, thus higher-order modifications.

The hierarchy isn't necessarily infinite.

I'm curious to know what kind of structure I'm looking at. I have the feeling that I've defined some (possibly degenerate) strict $n$-category (or $\omega$-cat), where there is one object at each level (I guess the endomorphisms), but I'm not entirely sure how to cast this categorically.

Have I defined a know structure? What additional (degenerate) properties does it satisfy?

Any pointers would be appreciated.

EDIT: Note that I say nothing about the structure of $P$ because generally I'm only interested in the hierarchy of monoids/modifications.

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): $m\bullet_{i+1}(n\circ_{i}n')=(m\bullet_{i+1}n)\circ_{i} (m\bullet_{i+1} n')$.

The idea is that $P$ is some space we are interested in, $M_0$ can be thought of as modifications of $P$, and each $M_{i+1}$ are modifications of the modifications in $M_i$, thus higher-order modifications.

The hierarchy isn't necessarily infinite.

I'm curious to know what kind of structure I'm looking at. I have the feeling that I've defined some (possibly degenerate) strict $n$-category (or $\omega$-cat), where there is one object at each level (I guess the endomorphisms), but I'm not entirely sure how to cast this categorically.

Have I defined a know structure? What additional (degenerate) properties does it satisfy?

Any pointers would be appreciated.