In category theory, a branch of mathematics, a

subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitively, ...

\mathcal

of a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

\mathcal

is said to be isomorphism closed or replete if every

\mathcal

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...

h:A\to B

with

A\in\mathcal

belongs to

\mathcal.

https://www.cs.cornell.edu/courses/cs6117/2018sp/Lectures/Subcategories.pdf 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 In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again con ...

one usually studies properties that are invariant under

homeomorphism In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function ...

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