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 in some category . There is a dual notion of undercategory, which is defined similarly.

Definition

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Let   be a category and   a fixed object of  [1]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 commutes

 

There 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[2] and higher category theory[3]pg 43, with definitions either analogous or essentially the same.

Properties

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Many categorical properties of   are inherited by the associated over and undercategories for an object  . For example, if   has finite products and coproducts, 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

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Overcategories on a site

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Recall that a site   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

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The category of commutative  -algebras 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

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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

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References

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  1. ^ Leinster, Tom (2016-12-29). "Basic Category Theory". arXiv:1612.09375 [math.CT].
  2. ^ "Section 4.32 (02XG): Categories over categories—The Stacks project". stacks.math.columbia.edu. Retrieved 2020-10-16.
  3. ^ Lurie, Jacob (2008-07-31). "Higher Topos Theory". arXiv:math/0608040.