Cofiltered

Cofiltered is a term used in category theory to describe a property of categories, dual to the notion of filtered categories. A small category I is said to be cofiltered if it satisfies three conditions: it is nonempty; for any finite collection of objects X1, ..., Xn in I there exists an object Y together with morphisms Y → Xi for each i (a common source or cone from Y to the Xi); and for any pair of parallel arrows f, g: X → Y in I, there exists an object Z and a morphism h: Z → X such that f ∘ h = g ∘ h. Intuitively, a cofiltered category admits common sources for finite families of objects and can equalize parallel arrows by precomposing with a suitable map.

Conceptually, cofiltered categories are the dual notion of filtered categories; a category is cofiltered if and only if its opposite category is filtered. This duality is often used when discussing limits (as opposed to colimits) because many inverse limit constructions are indexed by cofiltered categories.

Applications and examples, brief overview: cofiltered indexing categories naturally arise when forming inverse limits in various settings, such as topological spaces, groups, or modules, where one takes a limit over a system directed “from the top down.” A canonical example is the category whose objects are natural numbers and where there is a unique morphism i → j whenever i ≥ j; this category is nonempty, has a cone from a larger index to any finite family, and satisfies the parallel-arrow condition, illustrating a simple cofiltered shape. The opposite of a filtered category is cofiltered, and pro-objects are built from cofiltered diagrams.

In summary, cofiltered categories provide the indexing framework for inverse limits and are central to dual constructions in category theory.