Autopoiesis as viability-localized self-production in a topos.

Autopoiesis characterizes living organization by closure: a system maintains itself by producing and repairing the components-and crucially the boundary/interface conditions-that realize the network of production that produces them. We propose a topos-theoretic definition that (i) treats viability as an internal modality (a Lawvere-Tierney topology and its induced left-exact localization), (ii) models production/repair as a monad equipped with realization data, and (iii) makes boundary maintenance explicit via a boundary extractor, a boundary-production map, and an admissibility predicate. An autopoietic system is defined by four axioms (AP1-AP4): viability-local closure (realization becomes invertible after viability localization), produced boundaries in the viability semantics, constraint-respecting production via admissibility, and existence of at least one viable global regime. We also clarify that (AP3) is j-density of admissibility under (AP1). Finally, we illustrate consequences of the axioms on cellular sheaves over finite tissues (graphs/cell complexes), where closure reduces to equalizer/kernel tests and loopy tissues exhibit structured local-to-global obstruction signals.