• Source: Isomorphism-closed subcategory

In category theory, a branch of mathematics, a subcategory





A




{\displaystyle {\mathcal {A}}}

of a category





B




{\displaystyle {\mathcal {B}}}

is said to be isomorphism closed or replete if every





B




{\displaystyle {\mathcal {B}}}

-isomorphism



h
:
A
→
B


{\displaystyle h:A\to B}

with



A
∈


A




{\displaystyle A\in {\mathcal {A}}}

belongs to





A


.


{\displaystyle {\mathcal {A}}.}

This implies that both



B


{\displaystyle B}

and




h

−
1


:
B
→
A


{\displaystyle h^{-1}:B\to A}

belong to





A




{\displaystyle {\mathcal {A}}}

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





B




{\displaystyle {\mathcal {B}}}

-object that is isomorphic to an





A




{\displaystyle {\mathcal {A}}}

-object is also an





A




{\displaystyle {\mathcal {A}}}

-object.
This condition is very natural. For example, in the category of topological spaces one usually studies properties that are invariant under homeomorphisms—so-called topological properties. Every topological property corresponds to a strictly full subcategory of




T
o
p

.


{\displaystyle \mathbf {Top} .}





References



This article incorporates material from Isomorphism-closed subcategory on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

Kata Kunci Pencarian:

• Isomorphism-closed subcategory
• Subcategory
• Reflective subcategory
• Triangulated category
• Category of topological spaces
• Cartesian closed category
• Abelian category
• Category of rings
• Partially ordered set
• Exact category