Isomorphism-closed subcategory
Introduction

In category theory, a branch of mathematics, a [subcategory](/wiki/Subcategory "Subcategory") \\mathcal{A} of a [category](/wiki/Category_%28mathematics%29 "Category (mathematics)") \\mathcal{B} is said to be **isomorphism closed** or **replete** if every \\mathcal{B}\-[isomorphism](/wiki/Isomorphism "Isomorphism") h:A\\to B with A\\in\\mathcal{A} belongs to \\mathcal{A}. <https://www.cs.cornell.edu/courses/cs6117/2018sp/Lectures/Subcategories.pdf> This implies that both B and h^{\-1}:B\\to A belong to \\mathcal{A} as well. 

A subcategory that is isomorphism closed and [full](/wiki/Full_subcategory "Full subcategory") is called **strictly full**. In the case of full subcategories it is sufficient to check that every \\mathcal{B}\-object that is isomorphic to an \\mathcal{A}\-object is also an \\mathcal{A}\-object.

This condition is very natural. For example, in the [category of topological spaces](/wiki/Category_of_topological_spaces "Category of topological spaces") one usually studies properties that are invariant under [homeomorphisms](/wiki/Homeomorphism "Homeomorphism")—so\-called [topological properties](/wiki/Topological_property "Topological property"). Every topological property corresponds to a strictly full subcategory of \\mathbf{Top}.


What is category theory?