Overcategory

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

Definition
Let $$\mathcal{C}$$ be a category and $$X$$ a fixed object of $$\mathcal{C}$$ 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 $$g: B \to B'$$ in $$\mathcal{C}$$ 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 and higher category theory 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.

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, given the objects are subobjects of the fixed object.