Overcategory

Short description: Category theory concept

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 [math]\displaystyle{ X }[/math] in some category [math]\displaystyle{ \mathcal{C} }[/math]. There is a dual notion of undercategory, which is defined similarly.

Definition

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

[math]\displaystyle{ \begin{matrix} A & \xrightarrow{f} & A' \\ \pi\downarrow \text{ } & \text{ } &\text{ } \downarrow \pi' \\ X & = & X \end{matrix} }[/math]

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

[math]\displaystyle{ \begin{matrix} X & = & X \\ \psi\downarrow \text{ } & \text{ } &\text{ } \downarrow \psi' \\ B & \xrightarrow{g} & B' \end{matrix} }[/math]

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

Properties

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

Examples

Overcategories on a site

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

Category of algebras as an undercategory

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

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 [math]\displaystyle{ S }[/math], [math]\displaystyle{ \text{Sch}/S }[/math]. Fiber products in these categories can be considered intersections, given the objects are subobjects of the fixed object.

See also

• Comma category