inverse system

inverse system (plural inverse systems)

1. (algebra) An indexed family of groups whose index set is a directed poset, together with a set of homomorphisms which are indexed by a subset of the Cartesian product of the index set with itself — which subset corresponds to the ordering relation of the directed poset — such that the domain and codomain of each homomorphism is obtained by reversing its ordered pair of indexes ${\displaystyle (e.g.,f_{ij}:G_{j}\rightarrow G_{i})}$ , and such that the homomorphisms compose in a way which reflects the reflexivity and transitivity of the order relation.
2. (category theory) A generalization of the above definition, where groups are replaced by objects, and homomorphisms are replaced by morphisms.