nLab object of finite type

This entry is about “objects of finite type” in algebra, homological algebra and rational homotopy theory. For finite homotopy types and π-finite homotopy types in homotopy theory see there. For related notions in category theory see at compact object. For finite types in type theory and in homotopy type theory see at inductive family. For more disambigation see at finite type.


Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Schanuel's lemma

Homology theories


Rational homotopy theory



The terminology of “objects of finite type” or of objects that “have finite type” is common in various contexts and usually means something like finitely generated object; but beware that conventional usage across contexts is not fully systematic.


In algebraic geometry

See at morphism of finite type for the notion in algebraic geometry.

In Abelian categories

An object XX in an AB5-category CC is of finite type if one of the following equivalent conditions hold:

(i) any complete directed set {X i} iI\{X_i\}_{i\in I} of subobjects of XX is stationary

(ii) for any complete directed set {Y i} iI\{Y_i\}_{i\in I} of subobjects of an object YY the natural morphism colim iC(X,Y i)C(X,Y)colim_i C(X,Y_i) \to C(X,Y) is an isomorphism.

An object XX is finitely presented if it is of finite type and if for any epimorphism p:YXp:Y\to X where YY is of finite type, it follows that kerpker\,p is also of finite type. An object XX in an AB5 category is coherent if it is of finite type and for any morphism f:YXf: Y\to X of finite type kerfker\,f is of finite type.

For an exact sequence 0XXX00\to X'\to X\to X''\to 0 in an AB5 category the following hold:

(a) if XX' and XX'' are finitely presented, then XX is finitely presented;

(b) if XX is finitely presented and XX' of finite type, then XX'' is finitely presented;

(c) if XX is coherent and XX' of finite type then XX'' is also coherent.

For a module MM over a ring RR this is equivalent to MM being finitely generated RR-module. It is finitely presented if it is finitely presented in the usual sense of existence of short exact sequence of the form R IR JM0R^I\to R^J\to M\to 0 where II and JJ are finite.

In homotopical algebra and rational homotopy theory

A graded object is often said to be of finite type if it is degreewise of finite dimension/rank, in some sense. This terminology is used specifically in rational homotopy theory.

Notably a rational topological space is said to be of finite type if all its rational homotopy groups are finite dimensional vector spaces over the rational numbers.

Accordingly, a chain complex of vector spaces, possibly those generating a semifree dga is said to be of finite type if it is degreewise finite dimensional.

Beware however that the terminology clashes somewhat 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 of finite type (in the sense of stable homotopy theory) is one whose cohomology is finitely generated in each degree, but could exist in infinitely many degrees.


See also/instead at finite spectrum.


Last revised on July 9, 2021 at 07:22:10. See the history of this page for a list of all contributions to it.