Homology, Cohomology, Homotopy, Coverings, and Topos Theory of Sequence-Operator Spaces: Coarse Invariants for Branching Evolution

We develop a functorial topological theory of sequence-operator spaces from six preceding foundations: axiomatic sequence-operator systems, multivariate coupling trees, parity-progression dynamics, multiscale cylinder measure, application and transfer principles, and the exact arithmetic-progression cylinder calculus. Rather than placing an arbitrary topology on an unspecified operator collection, we pass from an operator system to its weighted-state transition relation, admissible-history category, realized-state path category, boundary cylinders, and compatible finite quotients. Nerves, realizations, graph and directed-path complexes, covering groupoids, and cylinder sites then provide canonical domains for homology, cohomology, homotopy, and sheaf theory. We prove functoriality under strict operator-system morphisms, contractibility of unmerged history realizations, cycle formulas for finite-state graphs, directed path-homology and cohomology comparison theorems, prism homotopy invariance, unique lifting and transfer for operator coverings, and the Grothendieck axioms for exact cylinder coverage. The resulting category of sheaves is a Grothendieck topos whose path points evaluate local operator data. We further construct coarse state metrics, ends, growth invariants, persistent modules, relative categories of weak equivalences, formal localizations, finite logical profiles, and inverse-limit completions. We apply the theory to unary and multivariate systems, dyadic arithmetic-progression cylinders, exact valuation trees, finite-state quotients, and computationally universal operator algebras. History trees are homologically trivial before state merger, whereas quotient recurrence, noncommuting activation, and directed obstruction cycles can survive in coarser reductions. Finite observations reconstruct states and one-step evolution under explicit residual-separation hypotheses, but they need not reconstruct unbounded-time reachability or select a concrete natural-number trajectory from a symbolic completion. In particular, no computable family of the coarse invariants developed here decides complete reachability on every operator system containing the universal cylinder calculus. The resulting invariants therefore measure branching evolution in a rigorous weak sense without identifying coarse equivalence with faithful orbit equivalence or arithmetic convergence. **Keywords** Sequence-operator system; history category; directed path homology; cohomology; homotopy; covering space; Grothendieck topology; topos; coarse geometry; persistent homology; inverse-limit completion.