[note dump] Iterations and Actions

[MphLee]

16/04/2026 - road to a global theory
I apology for the raw notes. This is just a reminder.

Let \(I\) be a category of indexing structures/objects \(i\), i.e. structured spaces of degrees/indexes of some kind; Let \(C\) be a category of "function spaces" \(X\), they are the supporting objects in the theory.
The starting point is a general profunctor (or distributor) \(H:I^{op}\times C\to {\rm Set}\) that takes an indexing object \(i\in I\) and a support object \(X\in C\) and output the set of "\(H\)-kind" of heteromaps \({\phi}:i \rightsquigarrow X\). This is meant to be abstract and \(\phi\) can be interpreted into many concrete cases and is to be thought as a process computing for each degree-index in the structure \(i\) an element in the support space \(X\).

I conjecture that the canonical way of representing those \(H\)-heteromorphisms as actual homomorphisms of a specific category, i.e. as homomorphisms in the category \(\hat I\) ( of preasheaves over \(I\)) via the yoneda map \(y:I\to \hat I\)

\(H(i,X)\cong \hat I(y(i),H_X)\)

, naturally upgrades to a way of seeing the categories of \(H\)-heteromaps over a fixed \(X\in C\), i.e. the

category of elements \({\bf H}_X:=\displaystyle \int_I H(-,X)\) as a full subcategory of the slice category \(\hat I_{/H_{X}}\).




Also, each profunctor \(H\) induces by currying a functor \( H_{-}\in [C, \hat I]\) and this covariant functor is sent to  \({\bf H}_{-}\in [C, \rm{Cat}_{/I}]\) via post-composition by the "category of elements functor".

\(\displaystyle (\int _I)_!:[C, \hat I]\to [C, \rm{Cat}_{/I}]\)

applying this time something similar to the grothendieck construction \((\int _C):[C, \rm{Cat}_{/I}]\to {\rm Cat}_{/C}\) we should get the real "global category" of \({\bf H}\) over \(I^{op}\times C\)

\(\displaystyle {\bf H}:=\int _C{\bf H}_{-}= \int _C\int_I H_{-}\)

Question: do we have \({\bf H}:=\displaystyle\int _C\int_I H_{-}\cong \int _{I^{op}\times C} H(-,-)\)

examples: 
i) Let \(I=C={\rm Mon}\), and \(H={\rm Mon}\). Processes \({\phi}:M \rightsquigarrow L\) are just monoid morphisms. \({\bf H}_L={\rm Mon}_{/L}\) is the category of \(L\)-graded monoids, 
for \(L={\rm End}(X)\) processes \({\phi}:M \rightsquigarrow {\rm End}(X) \) are just \(L\)-actions opver the set (or space) \(X\). \({\bf H}_{ {\rm End}(X)}={\rm ACT}(X)\) is the category of actions previously discussed in this thread.
The global category \({\bf H}=\rm Mon ^{\downarrow}\) turns out to be the category of graded monoids, also called arrow category of monoids, or the category of monoid co-presheaves over the walking arrow.

ii) Let \(I=\mathbb N{\rm Set}\) and \(C={\rm Mon}\), and \(H={\rm Goo}\) the goodstein profunctor described elsewhere in this forum. Processes \({\phi}:J \rightsquigarrow M\) are just \(J\)-ranked families of \(M\) elements satisfying Goodstein functional equation, i.e. goodstein maps with ranks in the dynamical system \(J\).
\({\bf Goo}_M\) is the category of all the possible Goodstein hyperoperations families over the monoid \(M\), with all possible kind of ranks. 
for \(M={\rm End}(X)\) processes \({\alpha}:J \rightsquigarrow {\rm End}(X) \) are just \(J\)-ranked Ackermann functions over the set \(X\). In other words functions \( \alpha:J\times X\to X\) satisfying, for each \(j\in J\) the equation \[\alpha (j^+,f(x))=\alpha(j,\alpha(j^+,x))\] and \({\bf Goo}_{ {\rm End}(X)}={\rm Ack}(X)\) is the category of ackermann functions over \(X\).
The global category \({\bf Goo}\) seems much mysterious, maybe it is the key to unlocking real equivariant ranks, aka. non-discrete ranks.