Category theory; homological algebra

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• ConceptCategory theory

  Category theory is the part of mathematics that studies categories, functors between them and natural transformations between functors. It describes mathematical structures through the maps between them rather than through their elements. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the 1940s, first to make precise the word "natural" in algebraic topology. Today it is the common language of algebraic topology, algebraic geometry, homological algebra and parts of logic and computer science.

• ConceptCategory

  A category is made of objects, of morphisms (arrows) between pairs of objects, and of a composition of morphisms that is associative and has identities. It is the basic notion of category theory. Sets with functions, groups with group homomorphisms, and topological spaces with continuous maps are categories. A category can also be a single structure: a monoid is a category with one object. A preorder is a category with at most one morphism between any two objects.

• ConceptMorphism

  A morphism is an arrow $f \colon A \to B$ of a category, from its domain $A$ to its codomain $B$. Morphisms abstract structure-preserving maps such as group homomorphisms, linear maps and continuous maps. Isomorphisms, endomorphisms and automorphisms are special morphisms defined only through domains, codomains and composition. Much of category theory consists of describing objects by the morphisms into them and out of them.

• ConceptIsomorphism

  An isomorphism between two structures of the same signature is a bijection that preserves and reflects all operations, constants and relations. Two structures are isomorphic when an isomorphism between them exists; then they have the same properties expressible in their signature. Group isomorphisms, ring isomorphisms, linear isomorphisms and order isomorphisms are special cases. An isomorphism of a structure with itself is an automorphism.

• ConceptMonomorphism

  A monomorphism is a morphism $f$ that can be cancelled on the left: $f \circ g_1 = f \circ g_2$ implies $g_1 = g_2$. It is the categorical version of an injective function. In the categories of sets, groups, rings and vector spaces, the monomorphisms are exactly the injective homomorphisms. Its dual notion is the epimorphism.

• ConceptEpimorphism

  An epimorphism is a morphism $f$ that can be cancelled on the right: $g_1 \circ f = g_2 \circ f$ implies $g_1 = g_2$. It is the categorical version of a surjective function, and the dual notion of a monomorphism. In the categories of sets and of groups, the epimorphisms are exactly the surjective homomorphisms. In other categories an epimorphism need not be surjective: the inclusion of the integers into the rationals is an epimorphism of rings.