We define a recursive map as maps, $\chi \to \xi^{'}, \, \chi^{'} \to \xi^{''}, \, \chi^{''} \to \xi^{'''}, \ldots, \chi^{n} \to \xi^{n+1} \wedge \xi \to \chi, \, \xi^{'} \to \chi{'}, \xi^{''} \to \chi^{''}, \ldots \xi^{n} \to \chi^{n} $. Explicitly the maps are $\psi_1 : \chi^{n} \to \xi^{n+1}$ and $\psi_2 : \xi^{n} \to \chi^{n}$ between objects, $\chi$ and $\xi$, such that the induced maps $\chi \to \chi^{k}$ and $\xi \to \xi^{k}$ have the property that, $\forall j \leq k$, $\chi \simeq \chi^{j}$ and $\xi \simeq \xi^{j}$.
$$ \chi \underset{\psi_1,\psi_2}{\overset{k}{\circlearrowright}} \xi := \chi \overset{\psi_1}{\to} \xi^{'} \overset{\psi_2}{\to} \chi^{'} \overset{\psi_1}{\to} \xi^{''} \overset{\psi_2}{\to} \cdots \overset{\psi_1}{\to} \xi^{k} \overset{\psi_2}{\to} \chi^{k} $$
$$ \chi \underset{\psi^{-1}_2,\psi^{-1}_1}{\overset{k}{\circlearrowleft}} \xi := \chi^{k} \overset{\psi^{-1}_2}{\to} \xi^{k} \overset{\psi^{-1}_1}{\to} \cdots \overset{\psi^{-1}_2}{\to} \xi^{''} \overset{\psi^{-1}_1}{\to} \chi^{'} \overset{\psi^{-1}_2}{\to} \xi^{'} \overset{\psi^{-1}_1}{\to} \chi $$ and
Thus, our notation for a recursive map requires two maps $\psi_1, \, \psi_2$, two objects on which these maps are defined, as well as the number of recursions, $k$
This was written in an old paper I came across, and I'd like to see how it can be formalized using category theory, as the relation seems relatively transparent.
If it helps, the laymans example given is between an interval, $\chi$ and a square, $\xi$. So given an interval $[0,x_0]$, we map into a square by $[0,x_0+a] \times [0,x_0+a]$, then back to an interval as $[0,x_0+a -b]$ and once again back into the square by $[0,x_0+2a-b] \times [0,x_0+2a-b]$ and back into the interval as $[0,x_0+2a-2b]$, $k$ times. Clearly $\chi \simeq \chi^j$ and $\xi \simeq \xi^j$ So here I would think loosely the categories are the squares and the intervals, with the defined morphisms...The author intends $\simeq$ to be homotopic, but I don't believe this is necessary, I believe a more general categorical equivalence could be suitable.
Any comments are appreciated.
I would imagine it would go a little something like this:
Let $\mathscr{C}$ category with a class of objects $\mathscr{A} := \{ A, A^{'}, A^{''}, \ldots , A^n \}$ and $\mathscr{B} := \{ B, B^{'}, B^{''}, \ldots, B^{n} \}$ with morphisms $a := \{ a_1, a_2, \ldots , a_n \}$ and $b := \{ b_1, b_2, \ldots, b_n \}$ where $a_i : A^{i} \to B^{i+1}$ and $b_i : A^{i} \to B^{i}$ with the condition that $A \simeq A^{j}$ and $B \simeq B^{j}$, for some equivalence relation $\simeq$. We have the condition that all compositions $a \circ b$ and $b \circ a$ represent this equivalence relation, for example, a homeomorphism.
Thus we define our recursive map as
$$ \mathscr{A} \underset{a,b}{\overset{k}{\circlearrowright}} \mathscr{B} := A \overset{a_1}{\to} B^{'} \overset{b_1}{\to} A^{'} \overset{a_2}{\to} B^{''} \overset{b_2}{\to} \cdots \overset{a_{k-1}}{\to} B^{k} \overset{b_k}{\to} A^{k}$$
with inverse
$$ \mathscr{A} \underset{a^{-1}, b^{-1}}{\overset{k}{\circlearrowleft}} \mathscr{B} := A^{k} \overset{b^{-1}_k}{\to} B^{k} \overset{a^{-1}_{k-1}}{\to} \cdots \overset{b^{-1}_2}{\to} B^{''} \overset{a^{-1}_2}{\to} A^{'} \overset{b^{-1}_1}{\to} B^{'} \overset{a^{-1}_1}{\to} A $$
This is where I'm confused in terms of defintion. How to define this categorically, properly?
There are a few ways I can think of that would use categories and I'm not sure which one is the best here, but I'll give it a try:
Let $\overline{A}$ be the diagram with objects $A,A',A'', \cdots$, with a unique isomorphism between each pair of objects (in the example you give, this would be the scaling map - if you want a more general homotopy equivalence I suppose you would have to move to some sort of homotopy category to make the inverses work out right), and let $\overline{B}$ be the diagram whose objects are $B,B',B'', \cdots$ with similarly defined arrows.
Then, a recursive map is just a natural transformation from $\overline{A}$ to $\overline{B}$ along with a natural transformation $\overline{B} \to \overline{A}$ such that the composition $\overline{A} \to \overline{B} \to \overline{A}$ is one of the arrows in the diagram.
(Assuming I'm interpretting recursive maps the right way, at least)