** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

** for all morphisms and

** for all morphisms and

** Category of abelian groups Ab has abelian groups as objects and group homomorphisms as morphisms

** morphisms: ;

** Eunectes murinus, the green anaconda, the largest species, is found east of the Andes in Colombia, Venezuela, the Guianas, Ecuador, Peru, Bolivia, Brazil and on the island of Trinidad.

** Eunectes notaeus, the yellow anaconda, a smaller species, is found in eastern Bolivia, southern Brazil, Paraguay and northeastern Argentina.

** Eunectes deschauenseei, the dark-spotted anaconda, is a rare species found in northeastern Brazil and coastal French Guiana.

** Eunectes beniensis, the Bolivian anaconda, the most recently defined species, is found in the Departments of Beni and Pando in Bolivia.

** Well-ordering theorem: Every set can be well-ordered.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** Trichotomy: If two sets are given, then either they have the same cardinality, or one has a smaller cardinality than the other.

** The Cartesian product of any family of nonempty sets is nonempty.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Every surjective function has a right inverse.

** Zorn's lemma: Every non-empty partially ordered set in which every chain ( i. e. totally ordered subset ) has an upper bound contains at least one maximal element.

** Hausdorff maximal principle: In any partially ordered set, every totally ordered subset is contained in a maximal totally ordered subset.

** Tukey's lemma: Every non-empty collection of finite character has a maximal element with respect to inclusion.

** Antichain principle: Every partially ordered set has a maximal antichain.

** Every vector space has a basis.

** Every unital ring other than the trivial ring contains a maximal ideal.

** For every non-empty set S there is a binary operation defined on S that makes it a group.

** The closed unit ball of the dual of a normed vector space over the reals has an extreme point.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

** In the product topology, the closure of a product of subsets is equal to the product of the closures.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

** Any union of countably many countable sets is itself countable.

For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms ( usually called a small category ), or even locally small categories, whose hom-objects are sets, then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets.

Category theory deals with abstract objects and morphisms between those objects.

This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets.

In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

* Associativity: composition of morphisms is always associative.

In category theory, n-ary functions generalise to n-ary morphisms in a multicategory.

The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category.

The construction of the derived morphisms of one variable will work in a closed monoidal category.

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows ( also called morphisms, although this term also has a specific, non category-theoretical meaning ), where these collections satisfy some basic conditions.

One of the simplest examples of a category is that of groupoid, defined as a category whose arrows or morphisms are all invertible.

Instead of focusing merely on the individual objects ( e. g., groups ) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects ; by studying these morphisms, we are able to learn more about the structure of the objects.

In the case of groups, the morphisms are the group homomorphisms.

A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps ( morphisms ) between topological spaces in topology ( the associated category is called Top ), and the study of smooth functions ( morphisms ) in manifold theory.

In fact, what we have done is define a category of categories and functors – the objects are categories, and the morphisms ( between categories ) are functors.

By studying categories and functors, we are not just studying a class of mathematical structures and the morphisms between them ; we are studying the relationships between various classes of mathematical structures.

* A class hom ( C ), whose elements are called morphisms or maps or arrows.

Each morphism f has a unique source object a and target object b. The expression, would be verbally stated as " f is a morphism from a to b ". The expression — alternatively expressed as,, or — denotes the hom-class of all morphisms from a to b.

* A binary operation ∘, called composition of morphisms, such that for any three objects a, b, and c, we have.

Relations among morphisms ( such as ) are often depicted using commutative diagrams, with " points " ( corners ) representing objects and " arrows " representing morphisms.