In category theory, a branch of mathematics, a
subcategory 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 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
References
{{PlanetMath attribution, id=8112, title=Isomorphism-closed subcategory
Category theory