(This entry describes two distinct notions, one in the theory of inner product spaces, and the second in a more purely categorical context.)
Two elements in an inner product space are orthogonal to each other, if
there exists a unique diagonal filler making both triangles commute:
Given a class of maps , the class is denoted or . Likewise, given , the class is denoted or . These operations form a Galois connection on the poset of classes of morphisms in the ambient category. In particular, we have and .
A pair such that and is sometimes called a prefactorization system. If in addition every morphism factors as an -morphism followed by an -morphism, it is an (orthogonal) factorization system.
A strong epimorphism in any category is, by definition, an epimorphism in , where is the class of monomorphisms. (If the category has equalizers, then every map in is epic.) Dually, a strong monomorphism is a monomorphism in .
|type of subspace of inner product space||condition on orthogonal space|
|Lagrangian subspace||(for symplectic form)|
|symplectic space||(for symplectic form)|