From Darwin to teleonomy: A categorical final-cause calculus for evolution.

We propose a teleonomical calculus for evolution that generalizes the classical Darwin-Fisher picture by making final causes - what systems keep true about themselves - into mathematical objects with universal properties. In our framework, the state space is a category C acted upon (laxly) by time T, and viability constraints live in a fibration p:E→C. An endogenous functor Gt:C→E extracts invariants from the system (e.g. topological features via persistent homology, sheaf gluing compatibilities, symmetry/conservation laws, or behavioral attractors). The present compatible with realizing these constraints at horizon t is the right Kan extension [Formula: see text] equivalently a (possibly enriched) limit or a largest invariant subcoalgebra. Passing to concrete dynamics xt yields an endogenous bias that selects among feasible futures without introducing exogenous rewards: [Formula: see text] where the coherence deficitLt is built from Gt (e.g. PH witness distances and sheaf mismatch penalties). Classical selection appears as the scalar collapse L=-fitness (replicator-mutator). Richer choices of Gt produce a ladder of mechanisms: multi-objective Pareto teleonomy, morphogenetic teleonomy (global sections; dwell-time ∼eαΔW/D), behavioral attractors (final coalgebras), niche-construction holonomy (order-of-operations gaps), multi-level coherence (homotopy limits; synergy/variance bounds), and meta-teleonomy (doctrine updates for new codes). We derive measurable predictions, provide a unifying categorical spine, and outline algorithms to infer Gt from data. This reframes evolution as selection by endogenous invariants, with Darwinian fitness as a special case.