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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 {\displaystyle X} in some category C {\displaystyle {\mathcal {C}}} . The dual notion is that of an undercategory (also called a coslice category).

• Definition

• Properties

• Examples

• Overcategories on a site

• Category of algebras as an undercategory

• Overcategories of spaces

• See also

• References

Let C {\displaystyle {\mathcal {C}}} be a category and X {\displaystyle X} a fixed object of C {\displaystyle {\mathcal {C}}} cite-ref-1[1]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 π ) {\displaystyle (A,\pi )} where π π : A → → X {\displaystyle \pi :A\to X} is a morphism in C {\displaystyle {\mathcal {C}}} . Then, a morphism between objects f : ( A , π π ) → → ( A ′ , π 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 ↓ ↓ ↓ ↓ 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 ψ ) {\displaystyle (B,\psi )} where ψ 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 ↓ ↓ ↓ ↓ 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 theorycite-ref-2[2] and higher category theorycite-ref-3[3]pg 43, with definitions either analogous or essentially the same.

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.

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} .

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}} .

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 (e.g. the scheme-theoretic intersection), given the objects are subobjects of the fixed object.

• Comma category

cite-note-11. ↑ citerefleinster2016Leinster, Tom (2016-12-29). "Basic Category Theory". arXiv:1612.09375 [math.CT].

cite-note-22. ↑ "Section 4.32 (02XG): Categories over categories—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-16.

cite-note-33. ↑ citereflurie2008Lurie, Jacob (2008-07-31). "Higher Topos Theory". arXiv:math/0608040.