Overcategory

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 ${\displaystyle X}$ in some category ${\displaystyle {\mathcal {C}}}$. 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.

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

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

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

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

These two notions have generalizations in 2-category theory² and higher category theory^{3pg 43}, with definitions either analogous or essentially the same.

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

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

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

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

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 ${\displaystyle S}$, ${\displaystyle {\text{Sch}}/S}$. Fiber products in these categories can be considered intersections (e.g. the scheme-theoretic intersection), given the objects are subobjects of the fixed object.