mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in the field of

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

, a discrete category is a category whose only

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s are the

identity morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Alt ...

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 to such a category.

Simple facts

Any

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

of objects defines a discrete category when augmented with identity maps. Any subcategory 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 Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

from a discrete category into another category is called a product, while the colimit is called a coproduct. Thus, for example, the discrete category with just two objects can be used as a

diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age o ...

or diagonal functor 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 i ...

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 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