Objects
An object A in a category is said to be:
- isomorphic to an object B provided that there is an isomorphism between A and B.
- initial provided that there is exactly one morphism from A to each object B; e.g., empty set in Set.
- terminal provided that there is exactly one morphism from each object B to A; e.g., singletons in Set.
- a zero object if it is both initial and terminal, such as a trivial group in Grp.
An object A in an abelian category is:
- simple if it is not isomorphic to the zero object and any subobject of A is isomorphic to zero or to A.
- finite length if it has a composition series. The maximum number of proper subobjects in any such composition series is called the length of A.
Read more about this topic: Glossary Of Category Theory
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