Overcategory Explained

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

X in some category l{C} . There is a dual notion of undercategory, which is defined similarly.

Definition

Let

l{C} be a category and X a fixed object of l{C} ^{[1] pg 59}. The overcategory (also called a slice category) l{C}/X is an associated category whose objects are pairs (A, \pi) where \pi:A \to X is a morphism in l{C} . Then, a morphism between objects f:(A, \pi) \to (A', \pi') is given by a morphism f:A \to A' in the category l{C} such that the following diagram commutes

\begin{matrix} A & \xrightarrow{f} & A' \\ \pi\downarrow & & \downarrow \pi' \\ X & = & X \end{matrix}

There is a dual notion called the undercategory (also called a coslice category) X/ l{C} whose objects are pairs (B, \psi) where \psi:X\to B is a morphism in l{C} . Then, morphisms in X/ l{C} are given by morphisms g: B \to B' in l{C} such that the following diagram commutes

\begin{matrix} X & = & X \\ \psi\downarrow & & \downarrow \psi' \\ B & \xrightarrow{g} & B' \end{matrix}

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.

Properties

Many categorical properties of

l{C} are inherited by the associated over and undercategories for an object X . For example, if l{C} has finite products and coproducts, it is immediate the categories l{C}/X and X/ l{C} have these properties since the product and coproduct can be constructed in l{C} , and through universal properties, there exists a unique morphism either to X or from X . In addition, this applies to limits and colimits as well.

Examples

Overcategories on a site

l{C} is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category Open (X) whose objects are open subsets U of some topological space X , and the morphisms are given by inclusion maps. Then, for a fixed open subset U , the overcategory Open (X)/U is canonically equivalent to the category Open (U) for the induced topology on U \subseteq X . This is because every object in Open (X)/U is an open subset V contained in U .

Category of algebras as an undercategory

The category of commutative

A -algebras is equivalent to the undercategory A/ CRing for the category of commutative rings. This is because the structure of an A -algebra on a commutative ring B is directly encoded by a ring morphism A \to B . If we consider the opposite category, it is an overcategory of affine schemes, Aff / Spec (A) , or just Aff _{A} .

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

S , Sch /S . Fiber products in these categories can be considered intersections, given the objects are subobjects of the fixed object.

See also

• Comma category

Notes and References

1. Leinster. Tom. 2016-12-29. Basic Category Theory. math.CT. 1612.09375.
2. Web site: Section 4.32 (02XG): Categories over categories—The Stacks project. 2020-10-16. stacks.math.columbia.edu.
3. Lurie. Jacob. 2008-07-31. Higher Topos Theory. math/0608040.