In mathematics, specifically category theory, an overcategory (also called a slice category), as well as an undercategory (also called a coslice category), 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
There is a dual notion of undercategory, which is defined similarly.
The overcategory (also called a slice category)
is an associated category whose objects are pairs
such that the following diagram commutes
There is a dual notion called the undercategory (also called a coslice category)
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
has finite products and coproducts, it is immediate the categories
have these properties since the product and coproduct can be constructed in
, and through universal properties, there exists a unique morphism either to
In addition, this applies to limits and colimits as well.
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
is canonically equivalent to the category
for the induced topology on
The category of commutative
-algebras is equivalent to the undercategory
for the category of commutative rings.
is directly encoded by a ring morphism
If we consider the opposite category, it is an overcategory of affine schemes,
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.