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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 \mathcalpg 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 theorypg 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
coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The cop ...
s, 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 Site most often refers to: * Archaeological site * Campsite, a place used for overnight stay in an outdoor area * Construction site * Location, a point or an area on the Earth's surface or elsewhere * Website, a set of related web pages, typi ...
\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 In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and additio ...
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 In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become ob ...


References

{{Reflist Category theory