Overcategory

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

Definition

Let

${𝒞}$

be a category and

$X$

a fixed object of

${𝒞}$

^{pg 59}. The overcategory (also called a slice category)

${𝒞}/X$

is an associated category whose objects are pairs

$(A,\pi )$

where

$\pi :A\to X$

is a morphism in

${𝒞}$

. Then, a morphism between objects

$f:(A,\pi )\to (A^{\prime },{\pi }^{\prime })$

is given by a morphism

$f:A\to A^{\prime }$

in the category

${𝒞}$

such that the following diagram commutes

$\begin{array}{ccc}A& \stackrel{f}{\to }& A^{\prime }\\ \pi \downarrow & & \downarrow {\pi }^{\prime }\\ X& =& X\end{array}$

There is a dual notion called the undercategory (also called a coslice category)

$X/{𝒞}$

whose objects are pairs

$(B,\psi )$

where

$\psi :X\to B$

is a morphism in

${𝒞}$

. Then, morphisms in

$X/{𝒞}$

are given by morphisms

$g:B\to B^{\prime }$

in

${𝒞}$

such that the following diagram commutes

$\begin{array}{ccc}X& =& X\\ \psi \downarrow & & \downarrow {\psi }^{\prime }\\ B& \stackrel{g}{\to }& B^{\prime }\end{array}$

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 ${𝒞}$ are inherited by the associated over and undercategories for an object $X$. For example, if ${𝒞}$ has finite products and coproducts, it is immediate the categories ${𝒞}/X$ and $X/{𝒞}$ have these properties since the product and coproduct can be constructed in ${𝒞}$, 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

Recall that a site ${𝒞}$ is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category $\text{Open}(X)$ 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 $\text{Open}(X)/U$ is canonically equivalent to the category $\text{Open}(U)$ for the induced topology on $U\subseteq X$. This is because every object in $\text{Open}(X)/U$ 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 $A/\text{CRing}$ 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, $\text{Aff}/\text{Spec}(A)$, or just ${\text{Aff}}_A$.

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

See also

• Comma category