Self-Similarity along Compact Seams
Metric completion and final coalgebras in a condensed infinity-topos
The attached 42-page manuscript extends Ratnayake and Manokaran, arXiv:2611.07584v1, from one sort and finitely many point identifications to graph-directed gluing along compact marked boundaries. The source archive includes the manuscript in LaTeX and the kernel-checked Lean specialization.
Specification
For each sort $s$ take a compact nonempty metric boundary $B_s$. Each sort has a finite nonempty family of edges $e$ from $s$ to a target sort $t(e)$. A continuous surjection $q_s:\coprod_e B_{t(e)}\to K_s$ into a compact Hausdorff seam records the overlap relation. An embedding $b_s:B_s\to K_s$ supplies output marks. The gluing functor $F$ on monically marked families is the pushout $$ (FX)_s=K_s\sqcup_{\coprod_e B_{t(e)}}\coprod_e X_{t(e)}. $$ An unmarked branch representative remains a singleton class. Two different representatives agree precisely when both are marked and their $q_s$-images agree. Each branch has weight $r_e\in(0,1]$. The maximum $\epsilon_n$ of products of weights along length-$n$ paths tends to zero. On a finite graph this is equivalent to requiring strictly contractive products around directed cycles.
Metric and completion theorem
On the branch coproduct, steps in branch $e$ cost $r_e$ times their input distance, steps across branches cost one, and jumps between equal representatives are free. The quotient metric is the infimum of paid chain scores. At the seed $X=B$, let $\delta_s$ be the resulting seam pseudometric. Admissibility requires that $\delta_s$ separate the points of $K_s$ and that $b_s$ be an isometry.
For isometrically marked inputs, any chain between seam points can be shortened to a boundary chain by combining successive paid steps inside one branch. The seam subspace metric therefore stays equal to $\delta_s$. A distance between arbitrary representatives is the minimum of the cap one, the direct within-copy cost, and the infimum of an entry cost, a distance in the seam, and an exit cost. Compactness gives the minimum in the entry and exit variables. This formula proves separation, agreement with the closed topological quotient, preservation of isometries, and preservation of completeness. For completeness, a Cauchy sequence approaching the seam has a limit there; a sequence staying a positive distance from the seam eventually stays in one copy and converges there. Dense isometric inclusions remain dense and isometric, so completion commutes with gluing.
The initial chain starts at $B$ and has isometric inclusions by admissibility. Its union $G$ is the marked-set initial algebra with isometric structure map $FG\to G$. At depth $n$, the images of the branches have diameter at most $\epsilon_n$, and a marked tail provides a center in the initial chain for each of these finitely many images. Thus $G$ is totally bounded. Its Cauchy completion $P$ is compact and the algebra extends to an isometric isomorphism $a:FP\to P$.
Finality
Given a marked-set coalgebra $c:X\to FX$, the operator $u\mapsto aF(u)c$ on marked maps into $P$ has an $n$-step Lipschitz bound of $\epsilon_n$ in the supremum metric. The mapping space is nonempty and complete in this metric. A strict iterate has a unique fixed point, which is also the unique fixed point of the original operator. Hence the inverse of $a$ is final. For a continuous coalgebra, at depth $n$ the inverse images of branch pieces form a finite closed cover. Values of the final map on each cover element lie in the same piece of diameter at most $\epsilon_n$. Removing the cover elements that miss a given point proves continuity. A quantitative comparison with the marked boundary proves shortness for short coalgebras with isometric marks and gives the source estimate $1/\Delta(X)$ for a short coalgebra with arbitrary distinct finite marks.
Attractors and exact overlaps
A compatible graph-directed system has complete bounded ambient spaces, an embedded geometric image of each $B_s$, and branch maps $\phi_e$ whose products of Lipschitz bounds shrink uniformly along paths. The marked branch equations factor through $q_s$ and agree with the output marks. The compact attractor $A$ is the Hausdorff-metric limit of the increasing marked orbit stages. The orbit algebra is injective exactly when its stage maps are injective, equivalently when the initial-algebra comparison $G\to O$ is bijective.
There are continuous surjective evaluations from the same compact space of infinite directed addresses to $P$ and to $A$. All addresses of an output mark give its geometric marked value by the branch equations. If two distinct addresses agree in $P$, discard their common transitions until two different branch representatives of one point are reached. Both are boundary representatives of the same seam point. Their geometric values then agree. This gives a continuous quotient $P\to A$ extending the initial-algebra comparison. It is a homeomorphism exactly when the geometric gluing algebra $FA\to A$ is injective, equivalently when the canonical and geometric address relations coincide. The bi-Lipschitz orbit comparison of the source paper implies this equivalence and gives the same constants on the completions.
The source's point-glued carpet has additional orbit coincidences. With the whole outer perimeter as boundary, the new seam identifies complete shared edges. The manuscript constructs the resulting completion as the usual carpet with its intrinsic path metric, bi-Lipschitz to the ambient metric. Other examples recover the interval, the Cantor set, the gasket, both marked presentations of the $p$-adic integers, and the continuous conjugacy for nested radicals. A two-sort interval example allows a weight-one edge and unequal weights on the other branches.
Higher finality and descent
Let $y$ denote the representable sheaf on compact Hausdorff probes with finite jointly surjective covers. The branch family of the homotopy pushout is an effective epimorphism. For compact marked inputs the pushout agrees with the representable topological quotient. Surjectivity follows by pulling back the finite closed cover by branches along a plot. The kernel equality follows by a closed refinement separating the equality locus from the seam locus. Both refinements are allowed by the compact-cover topology.
Evaluation at the singleton preserves limits and colimits. Its right adjoint sends a space $V$ to the sheaf of all maps from the underlying discrete set of a probe into $V$. The pointwise coalgebra map is therefore determined by the set theorem, using $0$-truncation for monically marked spaces. For a compact plot of a higher coalgebra over any base object $T$, lift each depth-$n$ observation along a finite compact cover to one branch on each covering probe. The images of the cover maps are closed and the pointwise values on each of them lie in a piece of diameter at most $\epsilon_n$. This proves continuity on the original plot. The plot maps assemble into the coalgebra map to $T\times yP$ over $T$. Its mapping space is contractible since the relevant maps into the relatively $0$-truncated targets are determined by singleton evaluation. Descent transports the result along locally constant specifications, including twisted families with monodromy. The compact kernel pairs of the address evaluations give effective groupoid objects presenting $yP$ and $yA$, and their comparison realizes the attractor quotient.
Equivalence with the source and proof evidence
In one sort take the discrete boundary $I$, common weight $\rho$, seam $K=(M\times I)/\overline R$ and output $b(j)=[D_j]$. The comparison sends each branch representative to the original class $m\otimes x$. The two presentations have the same fibers and the same paid step costs, hence the same chain scores and quotient metrics for every non-degenerate finite marked input. The natural isomorphism identifies the initial chain, the canonical metric, the completion, the structure maps, and the universal mediating maps. In the fixed-point IFS specialization the orbit and completed branch maps agree on every representative; density then identifies the completed orbit algebra. Section 8 provides the comparison with the source theorems and supporting propositions.
The Lean 4.19.0 companion checks both inverse laws, the marking correspondence, naturality, the pushout universal property, equality of the predicates of admissible chain costs, and transport of algebra and coalgebra equations and uniqueness. The analytic, geometric, and higher-topos arguments are given as mathematical proofs in the manuscript.
The responsible submitter uses the anonymous byline. AI assistance was used in developing and drafting the paper and in producing and checking the Lean companion. Key background references include Hutchinson, Leinster, Clausen and Scholze's condensed mathematics lectures, and Lurie's Higher Topos Theory. The PDF gives the complete bibliography.