# Object in a category

A term used to denote elements of an arbitrary category, playing the role of sets, groups, topological spaces, etc. An object in a category is an undefined concept. Every category consists of elements of two classes, the class of objects and the class of morphisms. The class of objects of a category $ \mathfrak K $
is usually denoted by $ \mathop{\rm Ob} \mathfrak K $.
With any object $ A $
of $ \mathfrak K $
there is associated a unique identity morphism $ 1 _ {A} $,
so that different identity morphisms correspond to different objects. Hence the concept of a category can be formally defined by means of morphisms alone. However, the term "object in a category" is a linguistic convenience which is practically always used. The division of the elements of a category into objects and morphisms is only meaningful within a fixed category, since the objects of one category can be the morphisms of another. Thanks to the presence of morphisms, interrelations can be defined between the objects of a category, allowing one to single out special classes of objects (cf. Integral object of a category; Null object of a category; Small object; Projective object of a category; Injective object; etc.).

#### Comments

Cf. also Generator of a category.

**How to Cite This Entry:**

Object in a category.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Object_in_a_category&oldid=48035