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Homotopy theory in algebraic topology
within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical
Homotopy_lifting_property
Concept category theory (mathematics)
the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of
Lifting_property
Continuous deformation between two continuous functions
homotopy lifting property is used to characterize fibrations. Another useful property involving homotopy is the homotopy extension property, which characterizes
Homotopy
smooth map satisfies an infinitesimal lifting property. Lifting property in categories Monsky–Washnitzer cohomology lifts p-adic varieties to characteristic
Lift_(mathematics)
Concept in algebraic topology
mapping p : E → B {\displaystyle p\colon E\to B} satisfies the homotopy lifting property for a space X {\displaystyle X} if: for every homotopy h : X × [ 0
Fibration
Mathematical category with weak equivalences, fibrations and cofibrations
third. Lifting: acyclic cofibrations have the left lifting property with respect to fibrations, and cofibrations have the left lifting property with respect
Model_category
Property in algebraic topology
homotopy extension property of cofibrations is dual to the homotopy lifting property that is used to define fibrations. Let X {\displaystyle X\,\!} be a
Homotopy_extension_property
Branch of mathematics
{\displaystyle s} is called the path lifting associated to p {\displaystyle p} . Conversely, if there is a path lifting s {\displaystyle s} , then p {\displaystyle
Homotopy_theory
Direct summand of a free module (mathematics)
Eilenberg. The usual category theoretical definition is in terms of the property of lifting that carries over from free to projective modules: a module P is
Projective_module
theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions Λ i n ⊂ Δ n , 0 ≤ i < n {\displaystyle
Fibration_of_simplicial_sets
Category theory generalization of fumction factorization
a category C. Then e has the left lifting property with respect to m (respectively m has the right lifting property with respect to e) when for every
Factorization_system
Topics referred to by the same term
plants Lift (mathematics), an kind of morphism in category theory Homotopy lifting property, a unique path over a map Covering graph or lift Shoe lifts, a
Lift
Algebraic geometry
algebraic geometry, a morphism is called formally étale if it has a lifting property that is analogous to being a local diffeomorphism. Let A be a topological
Formally_étale_morphism
Type of continuous map in topology
fundamental group: for one, since all coverings have the homotopy lifting property, covering spaces are an important tool in the calculation of homotopy
Covering_space
French: Formellement lisse) if it satisfies the following infinitesimal lifting property: Suppose B is given the structure of an A-algebra via the map f. Given
Formally_smooth_map
Associative algebra with lifting property
a quasi-free algebra is an associative algebra that satisfies the lifting property similar to that of a formally smooth algebra in commutative algebra
Quasi-free_algebra
Mathematical concept
approximate fibration is a sort of fibration such that the homotopy lifting property holds only approximately. The notion was introduced by Coram and Duvall
Approximate_fibration
Notion in measure theory
monograph of the Ionescu Tulceas. Lifting theory continued to develop since then, yielding new results and applications. A lifting on a measure space ( X , Σ
Lifting_theory
Type of pump using high pressure fluid to entrain a lower pressure fluid
Other key properties of an injector include the fluid inlet pressure requirements i.e. whether it is lifting or non-lifting. In a non-lifting injector
Injector
right lifting property with respect to all injective fibrations and the projective trivial fibrations also have to have the left lifting property with
Injective and projective model structure
Injective_and_projective_model_structure
Method of factorization, especially in category theory
the left lifting property with respect to F {\displaystyle F} . Similarly, we write r ( F ) {\displaystyle r(F)} for the right lifting property. Then Theorem—
Small_object_argument
commutative k-algebra A is said to be 0-smooth if it satisfies the following lifting property: given a k-algebra C, an ideal N of C whose square is zero and a k-algebra
Smooth_algebra
Concept in homotopy theory
to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton
Cofibration
Force perpendicular to flow of surrounding fluid
Foil (fluid mechanics) Küssner effect Lift-to-drag ratio Lifting-line theory Spoiler (automotive) "What is Lift?". Glenn Research Center | NASA. NASA
Lift_(force)
Heavy natural stone lifted by people in strength competitions
Lifting stones are heavy natural stones which people are challenged to lift, proving their strength. They are common throughout Northern Europe, particularly
Lifting_stone
Technique for wavelet analysis
The lifting scheme factorizes any discrete wavelet transform with finite filters into a series of elementary convolution operators, so-called lifting steps
Lifting_scheme
Concept in topological group theory
the group law on G can be constructed by lifting the group law H × H → H to G, using the lifting property of the covering map G × G → H × H. The non-connected
Covering_group
fibration, is a homomorphism of directed graphs that satisfies a unique lifting property analogous to that of a fibration in topology. Intuitively, the target
Fibrations_of_graphs
Weight training equipment
Safety Weight Lifting Belts: Purposes, Types, Comparison to Industrial Back Belts Forum discussion debating the value of a weight lifting belt during deadlifts
Weightlifting_belt
Concept in mathematical category theory
{\displaystyle {\overline {f}}=(f,\operatorname {id} _{a}).} The required lifting property then holds trivially. Next, if μ : F → G {\displaystyle \mu :F\to G}
Category_of_elements
Mathematical concept
characterized by having a right lifting property with respect to any trivial cofibration in the category. This property makes fibrant objects the "correct"
Fibrant_object
Map between simplicial sets with lifting property
very similar to that of fibrations in topology (see also homotopy lifting property), whence the name "fibration". Using the correspondence between n {\displaystyle
Kan_fibration
map is a projection. Since p is a fibration, by the homotopy lifting property, h lifts to a homotopy g : p − 1 ( b ) × I → E {\displaystyle g:p^{-1}(b)\times
Change_of_fiber
Elevator for use in a residential home
with 194 parameters of safety for a lift to be installed inside a private property. Home lifts are compact lifts for 2 to 4 persons which typically run
Home_lift
Mathematical behavior near singularities
homotopy lifting property to "follow" paths on the base space X {\displaystyle X} (we assume it path-connected for simplicity) as they are lifted up into
Monodromy
Continuous surjection satisfying a local triviality condition
homotopy-theoretic properties in common with fiber bundles. Specifically, under mild technical assumptions a fiber bundle always has the homotopy lifting property or
Fiber_bundle
Construction in homological algebra
extension 0 → B → X → A → 0 {\displaystyle 0\to B\to X\to A\to 0} , the lifting property of P {\displaystyle P} gives a map τ : P → X {\displaystyle \tau :P\to
Ext_functor
Mathematical model to quantify lift
The Lanchester–Prandtl lifting-line theory is a mathematical model in aerodynamics that predicts lift distribution over a three-dimensional wing from the
Lifting-line_theory
Lighter-than-air aircraft
improve its lifting power. Helium is the only lifting gas which is both non-flammable and non-toxic, and it has almost as much (about 92%) lifting power as
Aerostat
Engineered Lifting Systems & Equipment Inc (ELS) is a Canadian manufacturing company specializing in standard and custom overhead material lifting systems
Engineered Lifting Systems & Equipment
Engineered_Lifting_Systems_&_Equipment
Romanian-American mathematician (1935–2025)
at the age of 89. Some of her early work involved properties and consequences of lifting. Lifting theory, which had started with the pioneering papers
Alexandra_Bellow
Theorem on polynomial roots modulo prime powers
{I}}.} The lifting process is the inverse of reduction. That is, given objects depending on elements of R / I , {\displaystyle R/I,} the lifting process
Hensel's_lemma
{\displaystyle p:E\to B} . Then the mapping fiber Mp has the homotopy lifting property and it follows that Mp and the fiber F = p − 1 ( b 0 ) {\displaystyle
Puppe_sequence
Algebraic topology uses abstract algebra to study topological spaces
Winding number Simply connected Universal cover Monodromy Homotopy lifting property Mapping cylinder Mapping cone (topology) Wedge sum Smash product Adjunction
List of algebraic topology topics
List_of_algebraic_topology_topics
Mathematical theories
from the n-sphere to p−1(Δ). Because fibrations satisfy the homotopy lifting property, and Δ is contractible; p−1(Δ) is homotopy equivalent to F. So this
Obstruction_theory
Topological group that is in a certain sense assembled from a system of finite groups
abelian torsion groups. A profinite group is projective if it has the lifting property for every extension. This is equivalent to saying that G {\displaystyle
Profinite_group
connected Semi-locally simply connected Path (topology) Homotopy Homotopy lifting property Pointed space Wedge sum Smash product Cone (topology) Adjunction space
List of general topology topics
List_of_general_topology_topics
Concept from mathematics
Serre fibration is a quasifibration. This follows from the Homotopy lifting property. The projection of the letter L onto its base interval is a quasifibration
Quasi-fibration
Type of topological space
discrete space with two points to X {\displaystyle X} has the left lifting property with respect to the map from the finite topological space with two
Hausdorff_space
weak equivalences determine the fibrations) the maps having the right lifting property with respect to the cofibrations in M which are also C-local equivalences
Bousfield_localization
Topological space in which all singleton sets are closed
trivial. The map from the Sierpiński space to the single point has the lifting property with respect to the map from X {\displaystyle X} to the single point
T1_space
Theory in algebraic topology
B {\displaystyle p\colon E\to B} is defined by having the homotopy lifting property, represented by the following diagram and a cofibration i : A → X {\displaystyle
Eckmann–Hilton_duality
Special kind of model structure
they generate all cofibrations and trivial cofibrations using the lifting property: Cofib = ⊥ ( I ⊥ ) ; {\displaystyle \operatorname {Cofib} ={}^{\perp
Cisinski_model_structure
Type of machine
between lifted object and lifting equipment at the time of pick-up, and C {\displaystyle C} is the stiffness of the crane system at the lifting point.
Crane_(machine)
On the homotopy groups of the infinite symmetric product of a connected CW complex
multiple of a, hence different from the basepoint, so the Homotopy lifting property fails to be fulfilled. Verifying the fourth axiom can be done quite
Dold–Thom_theorem
Application of homotopy to algebraic varieties
cofibration if it is a monomorphism. f is a fibration if it has the right lifting property with respect to any cofibration which is a weak equivalence. The homotopy
A¹_homotopy_theory
Type of topological space
in A. The map ∅ → X {\displaystyle \emptyset \rightarrow X} has the lifting property with respect to a map from a certain finite topological space with
Normal_space
French philosopher and anarchist (1809–1865)
language. His best-known assertion is that "property is theft!", contained in his first major work, What Is Property? Or, an Inquiry into the Principle of Right
Pierre-Joseph_Proudhon
14th-century Scottish clan battle
Glenlui and Loch Arkaig. Each side had raided each other's lands, lifting property. In 1370, it is recorded in the Mackintosh MSS (manuscript), that around
Battle_of_Invernahavon
Rings admitting weak inverses
element y such that xyx = x and yxy = y). R is a V-ring. R has the right-lifting property against the ring homomorphism Z[t] → Z[t±] × Z determined by t ↦ (t
Von_Neumann_regular_ring
Puzzle game involving sliding pieces
Unlike tour puzzles, a sliding block puzzle prohibits lifting any pieces off the board. This property separates sliding puzzles from rearrangement puzzles
Sliding_puzzle
Bridge in Kyūshū, Japan
Railway Museum (Saitama). The bridge was designated an Important Cultural Property in 2003, and in 2007 it was included in the Mechanical Engineering Heritage
Chikugo_River_Lift_Bridge
1850s and 1860s engineering project in Chicago
to lift the city out of its low-lying swampy ground. Buildings and sidewalks were raised on jackscrews. The work was funded by both private property owners
Raising_of_Chicago
American mathematician
doi:10.1002/cpa.3160180402. Donoghue, William F. (1965). "On the lifting property". Proceedings of the American Mathematical Society. 16 (5): 913–914
William_F._Donoghue_Jr.
Skyscraper mansion in Mumbai, India
12 November 2017. Retrieved 22 September 2024. "Legality of orphanage property sold to Mukesh Ambani's Antilia in question". India Today. 29 November
Antilia_(building)
1914 film by Charlie Chaplin
onto the old man. Eventually Charlie calls for help in lifting the trunk, which Garlico easily lifts off. Back in the dressing room, Garlico makes his bride
The_Property_Man
Indian manufacturer of lifts and escalators
properties typically required escalators in addition to lifts, and many developers preferred to award contracts to a single supplier for both lifts and
Johnson_Lifts
Vessel designed to move very large loads
Hansa became the world's largest heavy lift shipping company. In terms of lifting capacity it reached its maximum in 1978 with refitting the Japanese-built
Heavy-lift_ship
Theft of goods from a retail establishment
valuable, enjoyable, and disposable". Shoplifting, originally called "lifting", is as old as shopping. The first documented shoplifting started to take
Shoplifting
Process of developing an opinion of value for real property
estate appraisal, home appraisal, property valuation or land valuation is the process of assessing the value of real property (usually market value). Appraisal
Real_estate_appraisal
Process to level concrete by levelling its underlying foundation
lifting the concrete back into place. Once in place, the holes are filled with a color-matching grout. Benefits of mudjacking: Low-pressure lifting of
Concrete_leveling
Type of firearm propellant
burst charges, and rescue-line launches. It is also used in fireworks for lifting shells, in rockets as fuel, and in certain special effects. Combustion
Gunpowder
Planned family of US military helicopters
carry four crew and 12 troops, and have a 13,000-pound (5,900 kg) external lifting capacity. It has a six-by-six-foot (1.8 m × 1.8 m) cabin, which is twice
Future_Vertical_Lift
Video-sharing platform
specific videos is sometimes prevented due to copyright and intellectual property protection laws (e.g. in Germany), violations of hate speech, and preventing
YouTube
Business that provides packing and moving services for relocation
is the largest household goods shipper in the world, with the Personal Property Program accounting for 20% of all moves. A 2020 OnePoll survey showed that
Moving_company
Historic resort in Palm Beach, Florida, US
of maintaining the property exceeded the funds provided by Post, and because it was difficult to secure the facility, the property was returned to the
Mar-a-Lago
Recovering a ship or cargo after a maritime casualty
and inside a sunk vessel. External lifting involves lifting units that can be synchronized to achieve the desired lift throughout the operation; it can
Marine_salvage
President of the United States (2017–2021; since 2025)
of Trump administration, some GOP lawmakers advance measure to oppose lifting Russian sanctions". The Washington Post. Retrieved October 5, 2021. Baker
Donald_Trump
Mathematics glossary
paths are equivalent if they are homotopic to each other). path lifting A path lifting function for a map p: E → B is a section of E I → P p {\displaystyle
Glossary of algebraic topology
Glossary_of_algebraic_topology
American multinational technology company
businesses had the right to know if the government searches or seizes their property. On October 23, 2017, Microsoft said it would drop the lawsuit as a result
Microsoft
Theme park in Hengqin, Zhuhai, China
Resort is primarily accessed by bus or car and guests travel around the property by walking or by taking a boat ride on the resort's central canal. Small
Chimelong International Ocean Tourist Resort
Chimelong_International_Ocean_Tourist_Resort
Leader of China since 2012
COVID-19 pandemic, he imposed zero-COVID policies for two years before lifting them following widespread protests. Xi has pursued an assertive foreign
Xi_Jinping
Lifting force due to light
optical lift". Nature Photonics. 5 (1): 48–51. Bibcode:2011NaPho...5...48S. doi:10.1038/nphoton.2010.266. Edwards, Lin (7 December 2010). "Optical lifting demonstrated
Optical_lift
Sporadic simple group
double cover can be defined abstractly, by starting with the graph and lifting Ru to 2Ru in the double cover 2A4060. This is because 1 of the conjugacy
Rudvalis_group
1909–1963 German/French car manufacturer
Monge who had already applied Bugatti Brescia engines in his "Type 7.5" lifting body. Ettore Bugatti also designed a successful motorised railcar, the
Bugatti
American investor (born 1980)
Goodman, David (July 14, 2016). "New York Officials Were Warned About Lifting Nursing Home's Deed Limits, Report Says". The New York Times. Klein, Melissa;
Joel_Landau
Prime Minister of Ethiopia since 2018
these confrontations turned violent and resulted in the loss of life and property. Abiy, as an elected member of parliament took a proactive role in working
Abiy_Ahmed
Urban planning concept
the conscious use of artists to be the vanguard of gentrification, to lift property values and to make areas safe before others move in, otherwise referred
Creative_city
1855 tornado in Illinois
completely destroyed; the tornado lifted a short distance to the south of the property. In addition, furniture on the property was carried upward by the tornado
1855_Des_Plaines_tornado
Aerial cable transport with looped track
A gondola lift (cable car) is a means of cable transport and type of aerial lift which is supported and propelled by cables from above. It consists of
Gondola_lift
Equipment manufacturer company
Telegraph, 18 August 2010. Craneception: Watch a Crane Lifting a Crane Lifting a Crane Lifting a Crane Liebherr Undertake Renovation work inside Passau
Liebherr
Criminal acts against private property
in order to do so. Lifting: If the bike is locked to an insecure structure such as a small sign or tree, the thief is able to lift the bike along with
Property_crime
Vehicle or machine that can fly by gaining support from the air
solely fixed-wing lift in horizontal flight is not a rotorcraft but a convertiplane. A lifting body is an aircraft that produces lift through the shape
Aircraft
Chemical element with atomic number 1 (H)
as a lifting gas in balloons and airships. The first hydrogen-filled balloon was invented by Jacques Charles in 1783. Hydrogen provided the lift for the
Hydrogen
American actor and bodybuilder (born 1997)
practices and his familial bonding with his father. Upon taking up weight lifting, his father gave him the book The Encyclopedia of Modern Bodybuilding,
Joseph_Baena
American mathematician (1937-2015)
(link) with R. J. Sacker: Sacker, Robert J.; Sell, George R. (1977). Lifting properties in skew-product flows with applications to differential equations
George_Roger_Sell
Performances by Austrian and American actor
20, 2014.[dead link] Hiltzik, Michael (December 3, 2014). "A true 'hot property': Elliott Gould's 'Long Goodbye' apartment is for rent!". Los Angeles Times
Arnold Schwarzenegger filmography
Arnold_Schwarzenegger_filmography
Strongest type of permanent magnet from an alloy of neodymium, iron and boron
Benchtop NMR spectrometers Electric motors Cordless tools Servomotors Lifting and compressor motors Synchronous motors Spindle and stepper motors Electrical
Neodymium_magnet
American serial killer (1942–1994)
in the crawl space of his home, and three were buried elsewhere on his property; four were discarded in the Des Plaines River. Gacy had previously been
John_Wayne_Gacy
Bridge in Cleveland, Ohio
vertical lift bridge over the Cuyahoga River in Cleveland, Ohio. It is one of several moveable bridges within the city, and is a contributing property to the
Columbus_Road_Bridge
travel, tourism, insurance
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travel, tourism, insurance