The isomorphism theorems provide canonical isomorphisms that are not unique. For a detailed discussion of relational homomorphisms and isomorphisms see. There are similar isomorphisms on symplectic manifolds. In category theory, a localization of a category amounts to making some morphisms isomorphisms. The isomorphisms in this case are bijections with the chosen degree of differentiability. However, the bijections are not always the isomorphisms for more complex categories. Homeomorphisms are the isomorphisms in the category of topological spaces. So the long exact cohomology sequences themselves ultimately separate into a chain of isomorphisms. There are also the isomorphisms of the Lie algebras and . Logarithmic functions are the only continuous isomorphisms between these groups. The transitions denoted by the arrows obey isomorphisms. The indiscrete topology is generated by the pretopology that has only isomorphisms for covering families. Order isomorphisms are functions that define such a renaming. This is also functorial for isomorphisms of vector spaces. There are three well-characterized isomorphisms that transform from the Aitchison simplex to real space.