# Category: Objects (category theory)

Initial and terminal objects
In category theory, a branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X. The dual notion is that
Zero object (algebra)
In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure,
Monoid (category theory)
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is an object M together with two morphisms * μ: M ⊗
Subobject
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion is a generalization of concepts such as subset
List object
In category theory, an abstract branch of mathematics, and in its applications to logic and theoretical computer science, a list object is an abstract definition of a list, that is, a finite ordered s
Natural numbers object
In category theory, a natural numbers object (NNO) is an object endowed with a recursive structure similar to natural numbers. More precisely, in a category E with a terminal object 1, an NNO N is giv
Projective object
In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a proj
Strict initial object
In the mathematical discipline of category theory, a strict initial object is an initial object 0 of a category C with the property that every morphism in C with codomain 0 is an isomorphism. In a Car
Subobject classifier
In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typica
Exponential object
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and expone
Fibrant object
In mathematics, specifically in homotopy theory in the context of a model category M, a fibrant object A of M is an object that has a fibration to the terminal object of the category.
Global element
In category theory, a global element of an object A from a category is a morphism where 1 is a terminal object of the category. Roughly speaking, global elements are a generalization of the notion of
Group object
In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topol