A mapping between two categories that preserves the structure of the categories.

A function between two sets that preserves the structure of the sets.

A relation between two sets that preserves the structure of the sets.

A mapping between two categories that preserves the structure of the categories and their morphisms.

A functor that is bijective on objects and morphisms.

A functor that is injective on objects and morphisms.

A functor that is surjective on objects and morphisms.

A functor that is bijective on objects but not necessarily on morphisms.

The category whose objects are sets and whose morphisms are functions between sets.

The category whose objects are sets and whose morphisms are relations between sets.

The category whose objects are sets and whose morphisms are equivalence relations between sets.

The category whose objects are sets and whose morphisms are partial functions between sets.

The category whose objects are groups and whose morphisms are homomorphisms between groups.

The category whose objects are groups and whose morphisms are isomorphisms between groups.

The category whose objects are groups and whose morphisms are endomorphisms between groups.

The category whose objects are groups and whose morphisms are automorphisms between groups.

The category whose objects are rings and whose morphisms are homomorphisms between rings.

The category whose objects are rings and whose morphisms are isomorphisms between rings.

The category whose objects are rings and whose morphisms are endomorphisms between rings.

The category whose objects are rings and whose morphisms are automorphisms between rings.

The category whose objects are modules and whose morphisms are homomorphisms between modules.

The category whose objects are modules and whose morphisms are isomorphisms between modules.

The category whose objects are modules and whose morphisms are endomorphisms between modules.

The category whose objects are modules and whose morphisms are automorphisms between modules.

The category whose objects are vector spaces and whose morphisms are linear transformations between vector spaces.

The category whose objects are vector spaces and whose morphisms are isomorphisms between vector spaces.

The category whose objects are vector spaces and whose morphisms are endomorphisms between vector spaces.

The category whose objects are vector spaces and whose morphisms are automorphisms between vector spaces.

The category whose objects are topological spaces and whose morphisms are continuous maps between topological spaces.

The category whose objects are topological spaces and whose morphisms are homeomorphisms between topological spaces.

The category whose objects are topological spaces and whose morphisms are endomorphisms between topological spaces.

The category whose objects are topological spaces and whose morphisms are automorphisms between topological spaces.

The category whose objects are smooth manifolds and whose morphisms are smooth maps between smooth manifolds.

The category whose objects are smooth manifolds and whose morphisms are diffeomorphisms between smooth manifolds.

The category whose objects are smooth manifolds and whose morphisms are endomorphisms between smooth manifolds.

The category whose objects are smooth manifolds and whose morphisms are automorphisms between smooth manifolds.

The category whose objects are schemes and whose morphisms are morphisms of schemes.

The category whose objects are schemes and whose morphisms are isomorphisms of schemes.

The category whose objects are schemes and whose morphisms are endomorphisms of schemes.

The category whose objects are schemes and whose morphisms are automorphisms of schemes.

A lemma that states that every functor from a category to the category of sets is representable by an object of the category.

A lemma that states that every functor from a category to the category of sets is faithful.

A lemma that states that every functor from a category to the category of sets is full.

A lemma that states that every functor from a category to the category of sets is essentially surjective.

A theorem that states that every functor has a left adjoint and a right adjoint.

A theorem that states that every functor has a left adjoint but not necessarily a right adjoint.

A theorem that states that every functor has a right adjoint but not necessarily a left adjoint.

A theorem that states that every functor has neither a left adjoint nor a right adjoint.

The category whose objects are categories and whose morphisms are functors between categories.

The category whose objects are categories and whose morphisms are natural transformations between functors.

The category whose objects are categories and whose morphisms are isomorphisms between categories.

The category whose objects are categories and whose morphisms are automorphisms between categories.

A set of axioms that characterize the category of homology groups.

A set of axioms that characterize the category of cohomology groups.

A set of axioms that characterize the category of K-theory groups.

A set of axioms that characterize the category of L-theory groups.

A theorem that relates the cohomology of a scheme to the geometry of the scheme.

A theorem that relates the homology of a scheme to the geometry of the scheme.

A theorem that relates the K-theory of a scheme to the geometry of the scheme.

A theorem that relates the L-theory of a scheme to the geometry of the scheme.