Discrete category

Pictorial representation of a discrete category consisting of five objects.

In the mathematical branch of category theory , a discrete category is a particularly trivial category . A category is called discrete if and only if it consists only of objects (and, if one differentiates between them, their respective identical morphisms ). Occasionally, categories that are equivalent to such a category are also permitted. In some constructions, discrete categories are an important special case. A category is discrete if and only if it is a groupoid and a partial order at the same time .

Functors

Every mapping between two discrete categories is a functor . Thus, the category of sets can be embedded in the category of ( small ) categories by means of a fully faithful functor , which assigns the discrete category, consisting of the elements of the set as objects, to each set.

Product category

For a discrete (small) category and any category , the category of functors from to with natural transformations as morphisms is nothing other than the product category . ${\ displaystyle I}$ ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle {\ mathcal {C}} ^ {I}}$ ${\ displaystyle I}$ ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle \ textstyle \ prod _ {i \ in I} {\ mathcal {C}}}$

Products and coproducts

The product of a family of objects (if it exists) in a category is the special case of the general concept of Limes : It is precisely the Limes of the functor , which is understood as a discrete category. Dual is the coproduct of that family of objects (if it exists) of the Kolimes of this functor. ${\ displaystyle (X_ {i}) _ {i \ in I}}$ ${\ displaystyle {\ mathcal {C}}}$ ${\ displaystyle I \ to {\ mathcal {C}}, i \ mapsto X_ {i}}$ ${\ displaystyle I}$

Web links

discrete category , entry in the nLab . (English)

Individual evidence

^ Saunders Mac Lane and Ieke Moerdijk: Sheaves in Geometry and Logic . A First Introduction to Topos Theory. Springer , New York 1992, ISBN 0-387-97710-4 , pp. 27-28 .
^ Saunders Mac Lane: Categories for the Working Mathematician . 2nd Edition. Springer , New York 1998, ISBN 0-387-98403-8 , pp. 64, 69 .

[1] Saunders Mac Lane and Ieke Moerdijk: Sheaves in Geometry and Logic . A First Introduction to Topos Theory. Springer , New York 1992, ISBN 0-387-97710-4 , pp. 27-28 .

[2] Saunders Mac Lane: Categories for the Working Mathematician . 2nd Edition. Springer , New York 1998, ISBN 0-387-98403-8 , pp. 64, 69 .