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 $X$ in some category $\mathcal{C}$ . The dual notion is that of an undercategory (also called a coslice category).

Definition

Let $\mathcal{C}$ be a category and $X$ a fixed object of $\mathcal{C}$ {{cite arXiv|last=Leinster|first=Tom|date=2016-12-29|title=Basic Category Theory|class=math.CT|eprint=1612.09375}}^{pg 59}. The overcategory (also called a slice category) $\mathcal{C}/X$ is an associated category whose objects are pairs $(A, \pi)$ where $\pi:A \to X$ is a morphism in $\mathcal{C}$ . Then, a morphism between objects $f:(A, \pi) \to (A', \pi')$ is given by a morphism $f:A \to A'$ in the category $\mathcal{C}$ such that the following diagram commutes

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

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

X/\mathcal{C}

whose objects are pairs

(B, \psi)

where

\psi:X\to B

is a morphism in

\mathcal{C}

. Then, morphisms in

X/\mathcal{C}

are given by morphisms

g: B \to B'

\mathcal{C}

such that the following diagram commutes

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

These two notions have generalizations in 2-category theory{{Cite web|title=Section 4.32 (02XG): Categories over categories—The Stacks project|url=https://stacks.math.columbia.edu/tag/02XG|access-date=2020-10-16|website=stacks.math.columbia.edu}} and higher category theory{{cite arXiv|last=Lurie|first=Jacob|date=2008-07-31|title=Higher Topos Theory|eprint=math/0608040}}^{pg 43}, with definitions either analogous or essentially the same.

Properties

Many categorical properties of $\mathcal{C}$ are inherited by the associated over and undercategories for an object $X$ . For example, if $\mathcal{C}$ has finite products and coproducts, it is immediate the categories $\mathcal{C}/X$ and $X/\mathcal{C}$ have these properties since the product and coproduct can be constructed in $\mathcal{C}$ , 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 $\mathcal{C}$ is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category $\text{Open}(X)$ 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 (e.g. the scheme-theoretic intersection), given the objects are subobjects of the fixed object.

References

Category:Category theory