Cocontinuumit

Cocontinuumit is a hypothetical mathematical construct in category theory, used in speculative discussions about cocontinuity and colimits. The term denotes a structured object that combines a category with a designated, diagram-dependent method for forming colimits, together with coherence conditions to ensure compatibility with morphisms.

Formally, a cocontinuumit consists of a category C with a designated colimit operation L. For every small diagram D: J -> C, L assigns a colimit Colim_J D in C, and these assignments come with natural isomorphisms that render L functorial in the shape J and in the diagram D. A morphism f: C -> C' preserves the designated colimits, meaning that for every D: J -> C, f(Colim_J D) ≅ Colim_J (f ∘ D), functorially in D and J.

Typical properties attributed to a cocontinuumit include closure of C under the designated colimits, stability of the colimit operation under composition of diagrams, and compatibility with standard constructions such as pushouts when specialized to the ambient setting. In this sense, a cocontinuumit generalizes a cocomplete category, but with a focus on a specified colimit structure.

Relation to established concepts: the notion formalizes how a category could carry an intrinsic, functorial colimit procedure that is preserved by structure-preserving mappings. In ordinary theory, a cocomplete category with its standard colimits or a cocontinuous functor between such categories serves as the corresponding object. A cocontinuumit may be viewed as a narrower or differently constrained analogue.

Uses and status: cocontinuumit appears mainly in thought experiments, pedagogical illustrations, or speculative writings. It is not part of standard practice and has no widely accepted formal status in mainstream literature. See also colimit, cocomplete category, and cocontinuous functor.