Post-modern portfolio theory

2014-01-15T18:35:39Z

71.202.19.247:

[[Image:Lifting diagram.png|right|thumb|100px|Lift of ''f'' ([[commutative diagram]])]] In the branch of [[mathematics]] called [[category theory]], given a [[morphism]] ''f'' from an object ''X'' to an object ''Y'', and a morphism ''g'' from an object ''Z'' to ''Y'', a '''lift''' (or '''lifting''') of ''f'' to ''Z'' is a morphism ''h'' from ''X'' to ''Z'' such that ''g <math>\circ</math> h'' = ''f''.

A basic example in [[topology]] is lifting a [[path (topology)|path]] in one space to a path in a [[covering space]]. Consider, for instance, mapping opposite points on a [[sphere]] to the same point, a continuous map from the sphere covering the [[projective plane]]. A path in the projective plane is a continuous map from the unit interval, [0,1]. We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the category of topological spaces with continuous maps as morphisms, we have
:<math>\begin{align}
f\colon& [0,1] \to \mathbb{RP}^2 , &\qquad&\text{(projective plane path)} \\
g\colon& S^2 \to \mathbb{RP}^2 , &\qquad&\text{(covering map)} \\
h\colon& [0,1] \to S^2 . &\qquad&\text{(sphere path)}
\end{align}</math>

Lifts are ubiquitous; for example, the definition of [[fibration]]s (see [[homotopy lifting property]]) and the valuative criteria of [[separated morphism|separated]] and [[proper map]]s of [[scheme (mathematics)|schemes]] are formulated in terms of existence and (in the last case) unicity of certain lifts.

[[Category:Category theory]]

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Post-modern portfolio theory