Rational homotopy theory
See also compact object.
In algebraic geoemtry
See at morphism of finite type for the notion in algebraic geometry.
In Abelian categories
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.
In homotopical algebra
A graded object is often said to be of finite type if it is degreewise of finite dimension/rank, in some sense.
The terminology is used specifically in rational homotopy theory.
Notably a rational space is said to be of finite type if all its rational homotopy groups are finite dimensional vector spaces over the rational numbers.
Accordingly, chain complex of vector spaces, possibly that generating a semifree dga is said to be of finite type if it is degreewise finite dimensional.
Beware however that the terminology clashes with the use in homotopy theory, there the concept of finite homotopy type is crucially different from homotopy type with finite homotopy groups.
In stable homotopy theory
A spectrum (in the sense of stable homotopy theory) is one whose cohomology is finitely generated in each degree, but could exist in infinitely many degrees.