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
Famous quotes containing the word objects:
“All good music resembles something. Good music stirs by its mysterious resemblance to the objects and feelings which motivated it.”
—Jean Cocteau (1889–1963)
“There are characters which are continually creating collisions and nodes for themselves in dramas which nobody is prepared to act with them. Their susceptibilities will clash against objects that remain innocently quiet.”
—George Eliot [Mary Ann (or Marian)
“As a medium of exchange,... worrying regulates intimacy, and it is often an appropriate response to ordinary demands that begin to feel excessive. But from a modernized Freudian view, worrying—as a reflex response to demand—never puts the self or the objects of its interest into question, and that is precisely its function in psychic life. It domesticates self-doubt.”
—Adam Phillips, British child psychoanalyst. “Worrying and Its Discontents,” in On Kissing, Tickling, and Being Bored, p. 58, Harvard University Press (1993)