Overcategory

Template:Short description In mathematics, an overcategory (also called a slice category) is a construction from category theory used in multiple contexts, such as with covering spaces (espace étalé). They were introduced as a mechanism for keeping track of data surrounding a fixed object $X$ in some category $𝒞$. The dual notion is that of an undercategory (also called a coslice category).

Both can be expressed in terms of the more general construction of a comma category.

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.

By construction, $(X,\mathrm{id})$ is a terminal object of $𝒞/X$ and an initial object of $X/𝒞$.

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

Template:See also 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 (e.g. the scheme-theoretic intersection), given the objects are subobjects of the fixed object.