Definition
Let be a category and a fixed object of pg 59. The overcategory (also called a slice category) is an associated category whose objects are pairs where is a morphism in . Then, a morphism between objects is given by a morphism in the category such that the following diagram commutesThere is a dual notion called the undercategory (also called a coslice category) whose objects are pairs where is a morphism in . Then, morphisms in are given by morphisms in such that the following diagram commutesThese two notions have generalizations in 2-category theory and higher category theorypg 43, with definitions either analogous or essentially the same.Properties
Many categorical properties of are inherited by the associated over and undercategories for an object . For example, if has finite products and coproducts, it is immediate the categories and have these properties since the product and coproduct can be constructed in , and through universal properties, there exists a unique morphism either to or from . In addition, this applies to limits andExamples
Overcategories on a site
Recall that a site is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category whose objects are open subsets of some topological space , and the morphisms are given by inclusion maps. Then, for a fixed open subset , the overcategory is canonically equivalent to the category for the induced topology on . This is because every object in is an open subset contained in .Category of algebras as an undercategory
The category of commutative - algebras is equivalent to the undercategory for the category of commutative rings. This is because the structure of an -algebra on a commutative ring is directly encoded by a ring morphism . If we consider the opposite category, it is an overcategory of affine schemes, , or just .Overcategories of spaces
Another common overcategory considered in the literature are overcategories of spaces, such as schemes, smooth manifolds, or topological spaces. These categories encode objects relative to a fixed object, such as the category of schemes over , . Fiber products in these categories can be considered intersections, given the objects are subobjects of the fixed object.See also
*References
{{Reflist Category theory