- Source: Overcategory
In mathematics, specifically category theory, an overcategory (also called a slice category), as well as an undercategory (also called a coslice category), 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
{\displaystyle X}
in some category
C
{\displaystyle {\mathcal {C}}}
. There is a dual notion of undercategory, which is defined similarly.
Definition
Let
C
{\displaystyle {\mathcal {C}}}
be a category and
X
{\displaystyle X}
a fixed object of
C
{\displaystyle {\mathcal {C}}}
pg 59. The overcategory (also called a slice category)
C
/
X
{\displaystyle {\mathcal {C}}/X}
is an associated category whose objects are pairs
(
A
,
π
)
{\displaystyle (A,\pi )}
where
π
:
A
→
X
{\displaystyle \pi :A\to X}
is a morphism in
C
{\displaystyle {\mathcal {C}}}
. Then, a morphism between objects
f
:
(
A
,
π
)
→
(
A
′
,
π
′
)
{\displaystyle f:(A,\pi )\to (A',\pi ')}
is given by a morphism
f
:
A
→
A
′
{\displaystyle f:A\to A'}
in the category
C
{\displaystyle {\mathcal {C}}}
such that the following diagram commutes
A
→
f
A
′
π
↓
↓
π
′
X
=
X
{\displaystyle {\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
/
C
{\displaystyle X/{\mathcal {C}}}
whose objects are pairs
(
B
,
ψ
)
{\displaystyle (B,\psi )}
where
ψ
:
X
→
B
{\displaystyle \psi :X\to B}
is a morphism in
C
{\displaystyle {\mathcal {C}}}
. Then, morphisms in
X
/
C
{\displaystyle X/{\mathcal {C}}}
are given by morphisms
g
:
B
→
B
′
{\displaystyle g:B\to B'}
in
C
{\displaystyle {\mathcal {C}}}
such that the following diagram commutes
X
=
X
ψ
↓
↓
ψ
′
B
→
g
B
′
{\displaystyle {\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 theorypg 43, with definitions either analogous or essentially the same.
Properties
Many categorical properties of
C
{\displaystyle {\mathcal {C}}}
are inherited by the associated over and undercategories for an object
X
{\displaystyle X}
. For example, if
C
{\displaystyle {\mathcal {C}}}
has finite products and coproducts, it is immediate the categories
C
/
X
{\displaystyle {\mathcal {C}}/X}
and
X
/
C
{\displaystyle X/{\mathcal {C}}}
have these properties since the product and coproduct can be constructed in
C
{\displaystyle {\mathcal {C}}}
, and through universal properties, there exists a unique morphism either to
X
{\displaystyle X}
or from
X
{\displaystyle X}
. In addition, this applies to limits and colimits as well.
Examples
= Overcategories on a site
=Recall that a site
C
{\displaystyle {\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
Open
(
X
)
{\displaystyle {\text{Open}}(X)}
whose objects are open subsets
U
{\displaystyle U}
of some topological space
X
{\displaystyle X}
, and the morphisms are given by inclusion maps. Then, for a fixed open subset
U
{\displaystyle U}
, the overcategory
Open
(
X
)
/
U
{\displaystyle {\text{Open}}(X)/U}
is canonically equivalent to the category
Open
(
U
)
{\displaystyle {\text{Open}}(U)}
for the induced topology on
U
⊆
X
{\displaystyle U\subseteq X}
. This is because every object in
Open
(
X
)
/
U
{\displaystyle {\text{Open}}(X)/U}
is an open subset
V
{\displaystyle V}
contained in
U
{\displaystyle U}
.
= Category of algebras as an undercategory
=The category of commutative
A
{\displaystyle A}
-algebras is equivalent to the undercategory
A
/
CRing
{\displaystyle A/{\text{CRing}}}
for the category of commutative rings. This is because the structure of an
A
{\displaystyle A}
-algebra on a commutative ring
B
{\displaystyle B}
is directly encoded by a ring morphism
A
→
B
{\displaystyle A\to B}
. If we consider the opposite category, it is an overcategory of affine schemes,
Aff
/
Spec
(
A
)
{\displaystyle {\text{Aff}}/{\text{Spec}}(A)}
, or just
Aff
A
{\displaystyle {\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
{\displaystyle S}
,
Sch
/
S
{\displaystyle {\text{Sch}}/S}
. Fiber products in these categories can be considered intersections, given the objects are subobjects of the fixed object.
See also
Comma category