The category of ind objects in the category is the full subcategory of the category of presheaves whose objects are small filtered colimits of representable presheaves. The category of pro objects is the opposite of the full subcategory of the category of functors whose objects small filtered limits of corepresentables on , i.e. of representables on . In other words, . In other words, the morphisms between pro objects are the opposites of the natural transformations.
Recall that the Yoneda embedding is right exact. It canonical splits as and the first part is right exact and right small. From that one obtains that is
the category of colimits of filtered diagrams of representables such that the hom is given by the formula
Regarding the well known descriptions of filtered limits and colimits in this formula can be written in terms of threads and germs as it is usual in the theory of direct and inverse systems.
If the category admits finite colimits, then a functor is left exact (flat) iff the category of elements (which consists of pairs , , ) is filtered. For a general , is in iff is filtered and cofinally small. Hence, in the particular case of the category with finite colimits, an element is in iff is left exact and the comma category is cofinally small.
Last revised on February 2, 2011 at 22:27:02. See the history of this page for a list of all contributions to it.