In mathematics, a direct limit (also called inductive limit) is a colimit of a "directed family of objects". We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category.
Other articles related to "direct limit, limit, limits, direct limits, direct":
... The categorical dual of the direct limit is called the inverse limit (or projective limit) ... More general concepts are the limits and colimits of category theory ... The terminology is somewhat confusing direct limits are colimits while inverse limits are limits ...
... that K0(A) is finitely generated and, since K0 respects direct limits, K0(B) = ∪n βn* K0 (Bn) ... between dimension groups, one constructs a diagram of commuting triangles between the direct systems of A and B by applying the second lemma ... Using the property of the direct limit and moving A2 further down if necessary, we obtain diagram 4, a commutative triangle on the level of K0 ...
... In mathematics, a direct limit of groups is the direct limit of a direct system of groups ... "unstable" groups, the groups occurring in the limit ... are generally infinite-dimensional, constructed as limits of groups with finite-dimensional representations ...
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... two properties of Riesz groups are preserved by direct limits, assuming the order structure on the direct limit comes from those in the inductive system ... the fact that every Riesz group is the direct limit of Zk 's, each with the canonical order structure ... Corollary Every Riesz group (G, G+) can be expressed as a direct limit where all the connecting homomorphisms in the directed system on the right hand side are positive ...
Famous quotes containing the words limit and/or direct:
“Berowne they call him, but a merrier man,
Within the limit of becoming mirth,
I never spent an hours talk withal.”
—William Shakespeare (15641616)
“One merit in Carlyle, let the subject be what it may, is the freedom of prospect he allows, the entire absence of cant and dogma. He removes many cartloads of rubbish, and leaves open a broad highway. His writings are all unfenced on the side of the future and the possible. Though he does but inadvertently direct our eyes to the open heavens, nevertheless he lets us wander broadly underneath, and shows them to us reflected in innumerable pools and lakes.”
—Henry David Thoreau (18171862)