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 in some category . There is a dual notion of undercategory, which is defined similarly.


Let be a category and a fixed object of [1]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 commutes

There 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 commutes

These two notions have generalizations in 2-category theory[2] and higher category theory[3]pg 43, with definitions either analogous or essentially the same.


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 and colimits as well.


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


  1. ^ Leinster, Tom (2016-12-29). "Basic Category Theory". arXiv:1612.09375 [math.CT].
  2. ^ "Section 4.32 (02XG): Categories over categories—The Stacks project". Retrieved 2020-10-16.
  3. ^ Lurie, Jacob (2008-07-31). "Higher Topos Theory". arXiv:math/0608040.