Lift (mathematics)

Lift (mathematics)

"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 "gh" = "f".

A basic example in topology is lifting a 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:egin{align} fcolon& [0,1] o mathbb{RP}^2 , &qquad& ext{(projective plane path)} \ gcolon& S^2 o mathbb{RP}^2 , &qquad& ext{(covering map)} \ hcolon& [0,1] o S^2 . &qquad& ext{(sphere path)} end{align}

Lifts are ubiquitous; for example, the definition of fibrations (see homotopy lifting property) and the valuative criteria of separated and proper maps of schemes are formulated in terms of existence and (in the last case) unicity of certain lifts.


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать курсовую

Look at other dictionaries:

  • Lift — may mean:*Lift (force), a mechanical force generated by a solid object moving through a fluid *Lift (soaring), rising air used by soaring birds and glider, hang glider and paraglider pilots for soaring flight *Lift (soft drink), a brand of… …   Wikipedia

  • Mathematics and Physical Sciences — ▪ 2003 Introduction Mathematics       Mathematics in 2002 was marked by two discoveries in number theory. The first may have practical implications; the second satisfied a 150 year old curiosity.       Computer scientist Manindra Agrawal of the… …   Universalium

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • Lifting — may refer to:*Weightlifting *Shoplifting *Facelift *An undesirable type of movement in the sport of racewalking. *Taking an inference rule in propositional logic and adapting it for predicate logic *Lift (mathematics) *Lifting Scheme (wavelets) …   Wikipedia

  • List of common misconceptions — This incomplete list is not intended to be exhaustive. This is a list of current, widely held, false ideas and beliefs about notable topics which have been reported by reliable sources from around the world. Each has been discussed in published… …   Wikipedia

  • Holonomy — Parallel transport on a sphere depends on the path. Transporting from A → N → B → A yields a vector different from the initial vector. This failure to return to the initial vector is measured by the holonomy of the connection. In differential… …   Wikipedia

  • Signed graph — In the area of graph theory in mathematics, a signed graph is a graph in which each edge has a positive or negative sign.Formally, a signed graph Sigma; is a pair ( G , sigma;) that consists of a graph G = ( V , E ) and a sign mapping or… …   Wikipedia

  • Crystalline cohomology — In mathematics, crystalline cohomology is a Weil cohomology theory for schemes introduced by Alexander Grothendieck (1966, 1968) and developed by Pierre Berthelot (1974). Its values are modules over rings of Witt vectors over the base… …   Wikipedia

  • Spin structure — In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical …   Wikipedia

  • Gravity assist — In orbital mechanics and aerospace engineering, a gravitational slingshot, gravity assist or swing by is the use of the relative movement and gravity of a planet or other celestial body to alter the path and speed of a spacecraft, typically in… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”