In category theory, a branch of mathematics, a subcategory

\mathcal

of a

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

\mathcal

is said to be isomorphism closed or replete if every

\mathcal

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

h:A\to B

with

A\in\mathcal

belongs to

\mathcal.

This implies that both

B

and

h^:B\to A

belong to

\mathcal

as well. A subcategory that is isomorphism closed and full is called strictly full. In the case of full subcategories it is sufficient to check that every

\mathcal

-object that is isomorphic to an

\mathcal

-object is also an

\mathcal

-object. This condition is very natural. For example, in the category of topological spaces one usually studies properties that are invariant under

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomor ...

s—so-called

topological properties In topology and related areas of mathematics, a topological property or topological invariant is a property of a topological space that is invariant under homeomorphisms. Alternatively, a topological property is a proper class of topological space ...

. Every topological property corresponds to a strictly full subcategory of

\mathbf.

References

{{PlanetMath attribution, id=8112, title=Isomorphism-closed subcategory Category theory