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
in some category
. There is a dual notion of undercategory, which is defined similarly.
Definition
Let
be a category and
a fixed object of
pg 59. The overcategory (also called a slice category)
is an associated category whose objects are pairs
where
is a
morphism in
. Then, a morphism between objects
is given by a morphism
in the category
such that the following diagram
commutesThere is a dual notion called the undercategory (also called a coslice category)
whose objects are pairs
where
is a morphism in
. Then, morphisms in
are given by morphisms
in
such that the following diagram commutes
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
are inherited by the associated over and undercategories for an object
. For example, if
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
and
have these properties since the product and coproduct can be constructed in
, and through universal properties, there exists a unique morphism either to
or from
. 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 ...
is a categorical generalization of a topological space first introduced by
Grothendieck. One of the canonical examples comes directly from topology, where the category
whose objects are open subsets
of some topological space
, and the morphisms are given by inclusion maps. Then, for a fixed open subset
, the overcategory
is canonically equivalent to the category
for the induced topology on
. This is because every object in
is an open subset
contained in
.
Category of algebras as an undercategory
The category of commutative
-
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
for the category of commutative rings. This is because the structure of an
-algebra on a commutative ring
is directly encoded by a ring morphism
. If we consider the opposite category, it is an overcategory of affine schemes,
, or just
.
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
,
.
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