See also compact object.
An object in an AB5-category? is of finite type if one of the following equivalent conditions hold:
(i) any complete directed set of subobjects of is stationary
(ii) for any complete directed set of subobjects of an object the natural morphism is an isomorphism.
An object is finitely presented if it is of finite type and if for any epimorphism where is of finite type, it follows that is also of finite type. An object in an AB5 category is coherent if it is of finite type and for any morphism of finite type is of finite type.
For an exact sequence in an AB5 category the following hold:
(a) if and are finitely presented, then is finitely presented;
(b) if is finitely presented and of finite type, then is finitely presented;
(c) if is coherent and of finite type then is also coherent.
For a module over a ring this is equivalent to being finitely generated -module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form where and are finite.
The terminology is used specifically in rational homotopy theory.