# Overcategory

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{{Short description|Category theory concept}}
In mathematics, an '''overcategory''' (also called a '''slice category''') is a construction from [category theory](/source/category_theory) used in multiple contexts, such as with [covering spaces (espace étalé)](/source/Sheaf_(mathematics)). They were introduced as a mechanism for keeping track of data surrounding a fixed object <math>X</math> in some [category](/source/Category_(mathematics)) <math>\mathcal{C}</math>. The [dual](/source/Duality_(mathematics)) 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](/source/comma_category).

== Definition ==
Let <math>\mathcal{C}</math> be a category and <math>X</math> a fixed object of <math>\mathcal{C}</math><ref>{{cite arXiv|last=Leinster|first=Tom|author-link=Tom Leinster|date=2016-12-29|title=Basic Category Theory|class=math.CT|eprint=1612.09375}}</ref><sup>pg 59</sup>. The '''overcategory''' (also called a '''slice category''') <math>\mathcal{C}/X</math> is an associated category whose objects are pairs <math>(A, \pi)</math> where <math>\pi:A \to X</math> is a [morphism](/source/morphism) in <math>\mathcal{C}</math>. Then, a morphism between objects <math>f:(A, \pi) \to (A', \pi')</math> is given by a morphism <math>f:A \to A'</math> in the category <math>\mathcal{C}</math> such that the following diagram [commutes](/source/Commutative_diagram)<blockquote><math>\begin{matrix}
A & \xrightarrow{f} & A' \\
\pi\downarrow \text{ } & \text{ } &\text{ } \downarrow \pi' \\
X & = & X
\end{matrix}</math></blockquote>There is a dual notion called the '''undercategory''' (also called a '''coslice category''') <math>X/\mathcal{C}</math> whose objects are pairs <math>(B, \psi)</math> where <math>\psi:X\to B</math> is a morphism in <math>\mathcal{C}</math>. Then, morphisms in <math>X/\mathcal{C}</math> are given by morphisms <math>g: B \to B'</math> in <math>\mathcal{C}</math> such that the following diagram commutes<blockquote><math>\begin{matrix}
X & = & X \\
\psi\downarrow \text{ } & \text{ } &\text{ } \downarrow \psi' \\
B & \xrightarrow{g} & B'
\end{matrix}</math></blockquote>These two notions have generalizations in [2-category theory](/source/2_category)<ref>{{Cite web|title=Section 4.32 (02XG): Categories over categories—The Stacks project|url=https://stacks.math.columbia.edu/tag/02XG|access-date=2020-10-16|website=stacks.math.columbia.edu}}</ref> and [higher category theory](/source/higher_category_theory)<ref>{{cite arXiv|last=Lurie|first=Jacob|author-link=Jacob Lurie|date=2008-07-31|title=[Higher Topos Theory](/source/Higher_Topos_Theory)|eprint=math/0608040}}</ref><sup>pg 43</sup>, with definitions either analogous or essentially the same.

== Properties ==
Many categorical properties of <math>\mathcal{C}</math> are inherited by the associated over and undercategories for an object <math>X</math>. For example, if <math>\mathcal{C}</math> has finite [products](/source/Product_(category_theory)) and [coproduct](/source/coproduct)s, it is immediate the categories <math>\mathcal{C}/X</math> and <math>X/\mathcal{C}</math> have these properties since the product and coproduct can be constructed in <math>\mathcal{C}</math>, and through [universal properties](/source/Universal_property), there exists a unique morphism either to <math>X</math> or from <math>X</math>. In addition, this applies to [limits](/source/Limit_(category_theory)) and [colimits](/source/colimits) as well.

By construction, <math>(X,\operatorname{id})</math> is a [terminal object](/source/terminal_object) of <math>\mathcal{C}/X</math> and an initial object of <math>X/\mathcal{C}</math>.

== Examples ==

=== Overcategories on a site ===
Recall that a [site](/source/Grothendieck_topology) <math>\mathcal{C}</math> is a categorical generalization of a [topological space](/source/topological_space) first introduced by [Grothendieck](/source/Alexander_Grothendieck). One of the canonical examples comes directly from topology, where the category <math>\text{Open}(X)</math> whose objects are open subsets <math>U</math> of some topological space <math>X</math>, and the morphisms are given by inclusion maps. Then, for a fixed open subset <math>U</math>, the overcategory <math>\text{Open}(X)/U</math> is canonically equivalent to the category <math>\text{Open}(U)</math> for the induced topology on <math>U \subseteq X</math>. This is because every object in <math>\text{Open}(X)/U</math> is an open subset <math>V</math> contained in <math>U</math>.

=== Category of algebras as an undercategory ===
The category of commutative <math>A</math>-[algebras](/source/Algebra_over_a_field) is equivalent to the undercategory <math>A/\text{CRing}</math> for the category of commutative rings. This is because the structure of an <math>A</math>-algebra on a commutative ring <math>B</math> is directly encoded by a [ring morphism](/source/ring_morphism) <math>A \to B</math>. If we consider the [opposite category](/source/opposite_category), it is an overcategory of [affine scheme](/source/affine_scheme)s, <math>\text{Aff}/\text{Spec}(A)</math>, or just <math>\text{Aff}_A</math>.

=== Overcategories of spaces ===
{{see also|Grothendieck's relative point of view}}
Another common overcategory considered in the literature are overcategories of spaces, such as [schemes](/source/scheme_(mathematics)), [smooth manifold](/source/smooth_manifold)s, or topological spaces. These categories encode objects relative to a fixed object, such as the category of schemes over <math>S</math>, <math>\text{Sch}/S</math>. [Fiber products](/source/Fiber_product_of_schemes) in these categories can be considered intersections (e.g. the [scheme-theoretic intersection](/source/scheme-theoretic_intersection)), given the objects are [subobject](/source/subobject)s of the fixed object.

== References ==
{{Reflist}}

{{Category theory}}

Category:Category theory

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Adapted from the Wikipedia article [Overcategory](https://en.wikipedia.org/wiki/Overcategory) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Overcategory?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.