I'm trying to come up with a concise definition of a complicated category that arises from repeated application of functors to a "seed category" of objects.
The following example isn't the actual category, but I'm hoping that it will generalize.
The goal is to define some category $C$ with the following requirement:
Given:
- a posetal category $D$
- a finite subcategory $X\leq D$
- injective functors $F:D\Rightarrow D$ and $G:D\Rightarrow D$
- neither have fixed points
- they do not commute with each other for any values
define $C$ as exactly the set of finite compositions of $F$ and $G$ applied to all $x\in X$, e.g., $F(G(F(x)))$, $G(x)$, $x$.
In other words:
- $v\in C$ iff $\exists x\in X,n\in\mathbb{N},\{f_i\in\{F,G\}\}_{i\in\mathbb{Z}_n}$ such that $v=\left(\bigcirc_{i\in\mathbb{Z}_n}f_i\right)(x)$.
- Given $u,v\in C$, $u=v$ iff they have the same $\left(x,\{f_i\}_{i\in\mathbb{Z}_n}\right)$, i.e., all expansions are unique.
I'd like to define $C$ without explicitly referring to the sequence of functor applications or starting object.
I don't think $D$ needs to be fully defined, since the only structure we need comes from $X$, $F$, and $G$. (In fact, I think we don't really need to know anything about $F$ and $G$ outside of the requirements listed in point 3.)
Attempt 1: Closure
Maybe something like this:
define $C$ as the closure of $X$ under finite applications of $F$ and $G$
This is the closest to how the actual category I'm working with actually arises, but again, I'm not sure if "finite" and "closure" are rigorous enough here.
Attempt 2: Self-reference
We can loosely rephrase the definition of $C$ as:
$C = X\coprod F[C]\coprod G[C]$
However, I don't think this is well-defined because it could either include or exclude infinite applications of $F$ and/or $G$. We therefore need some additional constraint on $C$.
Attempt 3: Recurrence relation
Given $C_n:=F[C_{n-1}]\coprod G[C_{n-1}]$ and $C_0:=X$
define $C:= \coprod_{n\in\mathbb{N}} C_n$
Maybe this allows compositions to be unbounded, but without actually including the infinite cases?
This doesn't look to me like a question about categories per se; it seems you're only trying to define a set (or class) of objects, and then presumably you're content with taking the full subcategory on those objects (unless you have something more sophisticated in mind for the morphisms but then you should say so). So we are really asking a question about a set $D$ (or class, whatever), two functions $F, G : D \to D$, and a subset $X \subseteq D$, and trying to describe the closure of $X$ in $D$ under repeated application of $F$ and $G$.
The recursive definition you suggest will work fine: we can set $D_0 = D$ and then
$$D_{n+1} = F(D) \cup G(D)$$ $$D_{\infty} = \bigcup_{n=0}^{\infty} D_n$$
(I don't see any reason to require that the union be disjoint; in this construction I have not made any assumptions about $F$ and $G$). This allows arbitrarily long compositions but they are still finite.
Alternatively, we can consider the intersection of all subsets of $D$ which contain $X$ and are closed under $F$ and $G$. The intersection of subsets closed under $F$ and $G$ remains closed under $F$ and $G$ so this is the smallest subset of $D$ containing $X$ closed under $F$ and $G$, and it agrees with $D_{\infty}$ as constructed above.
I don't understand this bit: without $D$ how are you defining $F$ and $G$? Without $D$ how do you know what the morphisms in the category are?