Editing Direct limit (section)

==Examples==
*A collection of subsets <math>M_i</math> of a set <math>M</math> can be [[Partial order|partially ordered]] by inclusion. If the collection is directed, its direct limit is the union <math>\bigcup M_i</math>. The same is true for a directed collection of [[Subgroup|subgroups]] of a given group, or a directed collection of [[Subring|subrings]] of a given ring, etc.
*The [[weak topology]] of a [[CW complex]] is defined as a direct limit.
*Let <math>X</math> be any directed set with a [[greatest element]] <math>m</math>. The direct limit of any corresponding direct system is isomorphic to <math>X_m</math> and the canonical morphism <math>\phi_m: X_m \rightarrow X</math> is an isomorphism.
*Let ''K'' be a field. For a positive integer ''n'', consider the [[general linear group]] GL(''n;K'') consisting of invertible ''n'' x ''n'' - matrices with entries from ''K''. We have a group homomorphism GL(''n;K'') → GL(''n''+1;''K'') that enlarges matrices by putting a 1 in the lower right corner and zeros elsewhere in the last row and column. The direct limit of this system is the general linear group of ''K'', written as GL(''K''). An element of GL(''K'') can be thought of as an infinite invertible matrix that differs from the infinite identity matrix in only finitely many entries. The group GL(''K'') is of vital importance in [[algebraic K-theory]].
*Let ''p'' be a [[prime number]]. Consider the direct system composed of the [[Quotient group|factor groups]] <math>\mathbb{Z}/p^n\mathbb{Z}</math> and the homomorphisms <math>\mathbb{Z}/p^n\mathbb{Z} \rightarrow \mathbb{Z}/p^{n+1}\mathbb{Z}</math> induced by multiplication by <math>p</math>. The direct limit of this system consists of all the [[roots of unity]] of order some power of <math>p</math>, and is called the [[Prüfer group]] <math>\mathbb{Z}(p^\infty)</math>.
*There is a (non-obvious) injective ring homomorphism from the ring of [[Symmetric polynomial|symmetric polynomials]] in <math>n</math> variables to the ring of symmetric polynomials in <math>n + 1</math> variables. Forming the direct limit of this direct system yields the [[ring of symmetric functions]].
*Let ''F'' be a ''C''-valued [[sheaf (mathematics)|sheaf]] on a [[topological space]] ''X''. Fix a point ''x'' in ''X''. The open neighborhoods of ''x'' form a directed set ordered by inclusion (''U'' ≤ ''V'' if and only if ''U'' contains ''V''). The corresponding direct system is (''F''(''U''), ''r''<sub>''U'',''V''</sub>) where ''r'' is the restriction map. The direct limit of this system is called the ''[[stalk (mathematics)|stalk]]'' of ''F'' at ''x'', denoted ''F''<sub>''x''</sub>. For each neighborhood ''U'' of ''x'', the canonical morphism ''F''(''U'') → ''F''<sub>''x''</sub> associates to a section ''s'' of ''F'' over ''U'' an element ''s''<sub>''x''</sub> of the stalk ''F''<sub>''x''</sub> called the ''[[germ (mathematics)|germ]]'' of ''s'' at ''x''.
*Direct limits in the [[category of topological spaces]] are given by placing the [[final topology]] on the underlying set-theoretic direct limit.
*An [[ind-scheme]] is an inductive limit of schemes.