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, ...
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) ...
is said to be isomorphism closed or replete if every
-
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 ...
with
belongs to
This implies that both
and
belong to
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
-object that is isomorphic to an
-object is also an
-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 cont ...
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 isom ...
s—so-called
topological properties. Every topological property corresponds to a strictly full subcategory of
References
{{PlanetMath attribution, id=8112, title=Isomorphism-closed subcategory
Category theory