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Overcategory

Category theory concept

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Source updated September 15, 2026

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).

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

01Definition

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.

02Properties

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 that the category ${\mathcal {C}}/X$ has finite coproducts and the category $X/{\mathcal {C}}$ has finite products, 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.

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

03Examples

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.

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Sources and credits

This article is adapted from the Wikipedia article “Overcategory”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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