Fractal Category Theory

This deposit contains the manuscript Fractal Category Theory: Scale as a Frobenius (Co)Monad and Ind–Pro Bicompletion with Stable Equivariant Kan Extensions. The work models “scale” via a single endofunctor T:C→CT:\mathcal{C}\to\mathcal{C}T:C→C carrying compatible monad and comonad structures (Frobenius compatibility). Starting from a small seed C0\mathcal{C}_0C0, we construct a TTT-generated Ind–Pro bicompletion FracT(C0)⊂Pro(Ind(C0))\mathrm{Frac}_T(\mathcal{C}_0)\subset \mathrm{Pro}(\mathrm{Ind}(\mathcal{C}_0))FracT(C0)⊂Pro(Ind(C0)): TTT lifts to T^=LanJ(J ⁣∘T)\widehat T=\mathrm{Lan}_J(J\!\circ T)T=LanJ(J∘T) on Ind(C0)\mathrm{Ind}(\mathcal{C}_0)Ind(C0) and levelwise to T~\widetilde TT on Pro-objects. A mixed indexing category records η,ε,μ,δ\eta,\varepsilon,\mu,\deltaη,ε,μ,δ; by a finality argument, actual (bi)limit computations reduce to an ηε\eta\varepsilonηε-subpresentation, yielding presentation-independent formulas and an identification FracT≃(IndT∩ProT)repl\mathrm{Frac}_T\simeq(\mathrm{Ind}_T\cap \mathrm{Pro}_T)^{\mathrm{repl}}FracT≃(IndT∩ProT)repl under explicit accessibility and preservation hypotheses.

In a Lawvere-metric enriched setting, uniform contractivity implies algebraic compactness: the initial TTT-algebra equals the terminal TTT-coalgebra on a canonical “ambifixpoint” obeying Frobenius diagrams. On presheaves, TTT lifts via precomposition and (lax-to-strong) Day convolution; we define TTT-equivariant left Kan extensions computed on the ηε\eta\varepsilonηε-subpresentation and prove difference-based stability: for 0<q<10<q<10<q<1, ∥F−G∥≤11−q∥F0−G0∥\|F-G\|\le \frac{1}{1-q}\|F_0-G_0\|∥F−G∥≤1−q1∥F0−G0∥ with truncation error ≤11−qqk+1\le \frac{1}{1-q}q^{k+1}≤1−q1qk+1 (and 1-Lipschitz stability when q=1q=1q=1). Bicategorical and (∞,1)(\infty,1)(∞,1)-categorical variants are given with strictification/truncation hypotheses. Applications include substitution systems, contractive IFS via hyperspaces (separating functorial scale from Hutchinson dynamics), and coarse-graining channels (classical/CPTP).

Keywords
Category Theory; Frobenius monad; comonad; Ind-completion; Pro-completion; Ind–Pro bicompletion; Kan extension; Day convolution; Lawvere metric; algebraic compactness; ambifixpoint; equivariant functor; higher category; bicategory; (∞,1)(\infty,1)(∞,1)-category; restricted Yoneda; final/initial semantics; iterated function systems (IFS); Hutchinson operator; coarse-graining; stochastic maps; CPTP channels.