mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 a ...

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 *''Equiva ...

to such a category.

Simple facts

Any

class Class 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 differentl ...

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 Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...

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 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 In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...

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- ...

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 ...

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 t ...

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 kno ...

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

indiscrete category An indiscrete category is a category (mathematics), category ''C'' in which every hom-set ''C''(''X'', ''Y'') is a singleton (mathematics), singleton. Every class (set theory), class ''X'' gives rise to an indiscrete category whose objects are the ...

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