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 $\mathcal$. There is a dual notion of undercategory, which is defined similarly.

Definition

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

$\begin A & \xrightarrow & A' \\ \pi\downarrow \text & \text &\text \downarrow \pi' \\ X & = & X \end$

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

$\begin X & = & X \\ \psi\downarrow \text & \text &\text \downarrow \psi' \\ B & \xrightarrow & B' \end$

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 $\mathcal$ are inherited by the associated over and undercategories for an object $X$. For example, if $\mathcal$ has finite products and coproducts, it is immediate the categories $\mathcal/X$ and $X/\mathcal$ have these properties since the product and coproduct can be constructed in $\mathcal$, 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 $\mathcal$ is a categorical generalization of a topological space first introduced by Grothendieck. One of the canonical examples comes directly from topology, where the category $\text(X)$ 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 $\text(X)/U$ is canonically equivalent to the category $\text(U)$ for the induced topology on $U \subseteq X$. This is because every object in $\text(X)/U$ 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 $A/\text$ 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, $\text/\text(A)$, or just $\text_A$.

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

See also

* Comma category

References

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Category theory