Overcategory

In mathematics, specifically category theory, an overcategory (and undercategory) is a distinguished class of categories used in multiple contexts, such as with covering spaces (espace etale). They were introduced as a mechanism for keeping track of data surrounding a fixed object formula_1 in some category formula_2. There is a dual notion of undercategory, which is defined similarly.
Definition.
Let formula_2 be a category and formula_1 a fixed object of formula_2pg 59. The overcategory (also called a slice category) formula_6 is an associated category whose objects are pairs formula_7 where formula_8 is a morphism in formula_2. Then, a morphism between objects formula_10 is given by a morphism formula_11 in the category formula_2 such that the following diagram commutesformula_13There is a dual notion called the undercategory (also called a coslice category) formula_14 whose objects are pairs formula_15 where formula_16 is a morphism in formula_2. Then, morphisms in formula_14 are given by morphisms formula_19 in formula_2 such that the following diagram commutesformula_21These 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 formula_2 are inherited by the associated over and undercategories for an object formula_1. For example, if formula_2 has finite products and coproducts, it is immediate the categories formula_6 and formula_14 have these properties since the product and coproduct can be constructed in formula_2, and through universal properties, there exists a unique morphism either to formula_1 or from formula_1. In addition, this applies to limits and colimits as well.
Examples.
Overcategories on a site.
Recall that a site formula_2 is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category formula_31 whose objects are open subsets formula_32 of some topological space formula_1, and the morphisms are given by inclusion maps. Then, for a fixed open subset formula_32, the overcategory formula_35 is canonically equivalent to the category formula_36 for the induced topology on formula_37. This is because every object in formula_35 is an open subset formula_39 contained in formula_32.
Category of algebras as an undercategory.
The category of commutative formula_41-algebras is equivalent to the undercategory formula_42 for the category of commutative rings. This is because the structure of an formula_41-algebra on a commutative ring formula_44 is directly encoded by a ring morphism formula_45. If we consider the opposite category, it is an overcategory of affine schemes, formula_46, or just formula_47.
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 formula_48, formula_49. Fiber products in these categories can be considered intersections, given the objects are subobjects of the fixed object.