In mathematics, in the field of

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a discrete category is a category whose only

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...

s are the

identity morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...

s: :hom_''C''(''X'', ''X'') = {id_''X''} for all objects ''X'' :hom_''C''(''X'', ''Y'') = ∅ for all objects ''X'' ≠ ''Y'' Since by axioms, there is always the identity morphism between the same object, we can express the above as condition on the cardinality of the hom-set :, hom_''C''(''X'', ''Y'') , is 1 when ''X'' = ''Y'' and 0 when ''X'' is not equal to ''Y''. Some authors prefer a weaker notion, where a discrete category merely needs to be

equivalent Equivalence or Equivalent may refer to: Arts and entertainment *Album-equivalent unit, a measurement unit in the music industry *Equivalence class (music) *''Equivalent VIII'', or ''The Bricks'', a minimalist sculpture by Carl Andre *'' Equival ...

to such a category.

Simple facts

Any class of objects defines a discrete category when augmented with identity maps. Any

subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...

of a discrete category is discrete. Also, a category is discrete if and only if all of its subcategories are full. The limit of any

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ...

from a discrete category into another category is called a

product Product may refer to: Business * Product (business), an item that serves as a solution to a specific consumer problem. * Product (project management), a deliverable or set of deliverables that contribute to a business solution Mathematics * Produ ...

, while the colimit is called a

coproduct In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...

. Thus, for example, the discrete category with just two objects can be used as a

diagram A diagram is a symbolic representation of information using visualization techniques. Diagrams have been used since prehistoric times on walls of caves, but became more prevalent during the Enlightenment. Sometimes, the technique uses a three-d ...

diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct a ...

to define a product or coproduct of two objects. Alternately, for a general category C and the discrete category 2, one can consider the

functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in ...

C². The diagrams of 2 in this category are pairs of objects, and the limit of the diagram is the product. The

from Set to Cat that sends a set to the corresponding discrete category is

left adjoint In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kn ...

to the functor sending a small category to its set of objects. (For the right adjoint, see indiscrete category.)

References

* Robert Goldblatt (1984). ''Topoi, the Categorial Analysis of Logic'' (Studies in logic and the foundations of mathematics, 98). North-Holland. Reprinted 2006 by Dover Publications, and availabl
online
a
Robert Goldblatt's homepage
Categories in category theory