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Process in mathematics
In mathematics, a pullback is either of two related processes: precomposition and fiber-product; precomposition is a special case of the general fiber-product
Pullback
Mathematical operation
space of 1-forms on M {\displaystyle M} . This linear map is known as the pullback (by ϕ {\displaystyle \phi } ), and is frequently denoted by ϕ ∗ {\displaystyle
Pullback (differential geometry)
Pullback_(differential_geometry)
Most general completion of a commutative square given two morphisms with same codomain
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit
Pullback_(category_theory)
given a continuous map f: X → Y of topological spaces and a ring R, the pullback along f on cohomology theory is a grade-preserving R-algebra homomorphism:
Pullback_(cohomology)
Simple clockwork motor used in toy cars
A pullback motor (also pull back, pull back and go or pull-back) is a simple clockwork motor used in toy cars. A patent for them was granted to Bertrand
Pullback_motor
Fiber bundle induced by a map of its base space
In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π : E → B {\displaystyle
Pullback_bundle
Expression that may be integrated over a region
forms along with the exterior derivative defined on it is preserved by the pullback under smooth functions between two manifolds. This feature allows geometrically
Differential_form
non-autonomous. This requires one to consider the notion of a pullback attractor or attractor in the pullback sense. Consider a random dynamical system φ {\displaystyle
Pullback_attractor
Construction in algebraic topology
{\displaystyle f\colon X\to Y} . We will call the result the inverse image or pullback sheaf f − 1 G {\displaystyle f^{-1}{\mathcal {G}}} . If we try to imitate
Inverse_image_functor
Concept in algebraic geometry
of modules#Operations). The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle
Ample_line_bundle
Most general completion of a commutative square given two morphisms with same domain
{\displaystyle P=X+_{Z}Y} . The pushout is the categorical dual of the pullback. Explicitly, the pushout of the morphisms f and g consists of an object
Pushout_(category_theory)
Concepts in algebraic topology
A homotopy pullback (or homotopy fiber-product) is the dual concept of a homotopy pushout.It satisfies the universal property of a pullback up to homotopy
Homotopy_colimit_and_limit
American stock market index composed of 30 industry leaders
possibly coming to an end. For the decade, the Dow saw a rather substantial pullback for a negative return from 11,497.12 to 10,428.05, a loss of a 9.3%. During
Dow_Jones_Industrial_Average
and for all arrows f:c″→c′, S(f) is the restriction of Hom(f, c), the pullback by f (in the sense of precomposition, not of fiber products), to S(c′);
Sieve_(category_theory)
2005 video game
Bust-a-Move DS, known as Hippatte!! Puzzle Bobble (Japanese: ひっぱって!!パズルボブル, Hepburn: Hippatte!! Pazuru Boburu; "Pull Back!! Puzzle Bobble") in Japan, is
Bust-a-Move_DS
Mathematical technique for simplification
Radon–Nikodym derivatives of both the pullback and pushforward measures of m {\displaystyle m} under T {\displaystyle T} . The pullback measure in terms of a transformation
Change_of_variables
Japanese toy line released in the US as Penny Racers
Choro-Q is a line of Japanese 3–4 cm pullback car toys produced by Takara Tomy (formerly Takara). Known in North America as Penny Racers, they were introduced
Choro-Q
pullback square, and the morphism which goes from the domain of f × g {\displaystyle f\times g} to the domain of f is opposite to g in the pullback square
Fundamental theorem of topos theory
Fundamental_theorem_of_topos_theory
American venture capital firm
December 23, 2013. Retrieved June 13, 2023. Jennings, Katie. "Defying Pullback In Investments, General Catalyst Launches $670 Million Healthcare Fund"
General_Catalyst
Shogi opening
In shogi, Iijima Bishop Pullback or Iijima's Back Bishop (飯島流引き角 Iijima-ryuu hiki kaku) is a less common defensive Static Rook opening used against a Ranging
Iijima_Bishop_Pullback
Intentionally maximizing the use of AI service tokens
agentic AI eats up to 1,000x more tokens than standard AI, sparks corporate pullback at Microsoft, Meta, and Amazon". Tom's Hardware. Retrieved 2026-05-28.
Token_maxxing
Mathematical category
morphism A ′ → A {\displaystyle A'\to A} , the pullback is an I {\displaystyle I} -indexed coproduct of the pullbacks: ( ∐ i ∈ I B i ) × A A ′ ≅ ∐ i ∈ I ( B i
Topos
be exactly the same as pullback along f {\displaystyle f} followed by pullback along g {\displaystyle g} . However, pullback involves an arbitrary choice
Semantics_of_type_theory
Topics referred to by the same term
(category theory), a generalization of mathematical products Fibre product or pullback Coproduct or pushout Wick product of random variables Graph product Product
Product
1999 psychological thriller film by David Fincher
detailed using renderings by medical illustrator Katherine Jones. The pullback sequence from within the brain to the outside of the skull included neurons
Fight_Club
Category in which all small limits exist
from pullbacks and binary products (consider the pullback of (f, g) along the diagonal Δ), a category is complete if and only if it has pullbacks and products
Complete_category
Concept in algebraic topology
} {\displaystyle f^{*}(E)=\{(a,e)\in A\times E\ |\ f(a)=p(e)\}} is the pullback and the projections of f ∗ ( E ) {\displaystyle f^{*}(E)} onto A {\displaystyle
Fibration
Tap dance step
step, shuffle, leap, toe. The Maxie Ford is famous for its use of the pullback (or graboff) after the shuffle and best known as the Maxie Ford Break:
Maxie_Ford
Type of category in category theory
c_{X,Y,Z}(g,f)=g\circ f.} For a morphism p:X → Y, suppose the following pullback square exists, which defines the subobject of XY corresponding to maps
Cartesian_closed_category
Dual space to the tangent space in differential geometry
f_{*}^{}\colon T_{x}M\to T_{f(x)}N} every such map induces a linear map (called the pullback) between the cotangent spaces, only this time in the reverse direction:
Cotangent_space
Mathematical concept
the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions
Limit_(category_theory)
Injective homomorphism
categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks. The categorical dual of a monomorphism is an epimorphism, that is, a
Monomorphism
In mathematics, an algebraic cycle on an algebraic variety V is a formal linear combination of subvarieties of V. These are the part of the algebraic topology
Algebraic_cycle
Line of toy vehicles made by Takara Tomy
there were six models at that time. In the year 2002, Tomy again produced pullback models. From time to time, Tomy produces shorter runs of cars with sound
Tomica_(toy_line)
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Fundamental_groupoid
Ongoing armed conflict in Southeast Asia
The Irrawaddy. 2 October 2025. "Resource, manpower shortages drove TNLA pullback from Mogok and Momeik, senior leader says". Nora Pyae (21 April 2025).
Myanmar civil war (2021–present)
Myanmar_civil_war_(2021–present)
Generalizations of codimension-1 subvarieties of algebraic varieties
{\mathcal {M}}_{X}} , then this pullback can be used to define pullback of Cartier divisors. In terms of local sections, the pullback of { ( U i , f i ) } {\displaystyle
Divisor_(algebraic_geometry)
American dance drama multi-media franchise
'Wayne', Pilot 'Dark Cargo' Seek New Homes As YouTube Steps Up Scripted Pullback". Deadline Hollywood. Archived from the original on August 17, 2019. Retrieved
Step_Up_(franchise)
Construction in algebraic geometry
variety over one field determines a variety over a bigger field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change
Fiber_product_of_schemes
Theorem in vector calculus
v)={P_{u}}(u,v)\mathbf {e} _{u}+{P_{v}}(u,v)\mathbf {e} _{v}.} This is the pullback of F along ψ, and, by the above, it satisfies ∮ ∂ Σ F ( x ) ⋅ d l = ∮ γ
Stokes'_theorem
Concept in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Monoidal_functor
Topics referred to by the same term
back or pullback may refer to: Pullback, a name given to two different mathematical processes Pullback (cohomology), a term in topology Pullback (differential
Pull_back_(disambiguation)
Topics referred to by the same term
dictionary. The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different but closely related things. Pushforward
Pushforward
Linear approximation of smooth maps on tangent spaces
{\displaystyle d\varphi _{x}} is invertible, and the inverse gives the pullback of T φ ( x ) N . {\displaystyle T_{\varphi (x)}N.} The differential is
Pushforward_(differential)
Probability distribution
{\displaystyle {\tilde {p}}} is defined w.r.t. scaled Hausdorff measure. The pullback and Hausdorff measures agree, so that: p ( θ ∣ μ , Σ ) = p ~ ( v ∣ μ ,
Projected_normal_distribution
Overview of tap dance steps
landing on the foot/feet. Notable variations are single pullback, double pullback, alternating pullback, and other variations including other steps. riff:
Tap_dance_technique
'Wayne', Pilot 'Dark Cargo' Seek New Homes As YouTube Steps Up Scripted Pullback". Deadline Hollywood. August 16, 2019. Archived from the original on August
List of YouTube Premium original programming
List_of_YouTube_Premium_original_programming
exit by way of the southern end. Rutherford Road passes over the two hump pullback tracks at the north end of the yard, which curve to the west to end at
MacMillan_Yard
reportedly did not feel a sense of urgency in making the film amidst a wider "pullback" of the superhero genre. Feige confirmed that three or four iterations
Development of Blade (Marvel Studios film)
Development_of_Blade_(Marvel_Studios_film)
all pullbacks, it has pushouts along monomorphisms, and pushout squares of monomorphisms are also pullback squares and are stable under pullback. C has
Adhesive_category
Applications of category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Applied_category_theory
Generalisation of a sheaf; a fibered category that admits effective descent
topological basis. In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility
Stack_(mathematics)
Black's King's Head Vanguard Pawn (玉頭位取り gyokutou kuraidori). Iijima Bishop Pullback ☖ pieces in hand: – ☗ pieces in hand: – The Bishop-Exchange Fourth File
Fourth_File_Rook
most important one is the product of two smooth valuations. Together with pullback and pushforward, this operation extends to valuations on manifolds. Let
Valuation_(geometry)
BG, such that every bundle with the given structure group G over M is a pullback by means of a continuous map M → BG. When the definition of the classifying
Universal_bundle
Set of arguments where two or more functions have the same value
denotes the category-theoretic kernel. Any category with fibre products (pullbacks) and products has equalisers. In Top where objects are topological spaces
Equaliser_(mathematics)
D(X×Y). Most natural functors, including basic ones like pushforwards and pullbacks, are of this type. These kinds of functors were introduced by Mukai (1981)
Fourier–Mukai_transform
Concept in mathematics
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Tensor–hom_adjunction
Mathematical group occurring in algebraic geometry and the theory of complex manifolds
pullback. We say an L in Pic X / S ( T ) {\displaystyle \operatorname {Pic} _{X/S}(T)} has degree r if for any geometric point s → T the pullback s
Picard_group
System of moving vectors in differential geometry
that X {\displaystyle X} is parallel with respect to the pullback connection on the pullback bundle γ ∗ T M {\displaystyle \gamma ^{*}TM} . However
Parallel_transport
Symmetric monoidal category with a special involution
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Dagger symmetric monoidal category
Dagger_symmetric_monoidal_category
Type of map between algebraic groups
f : A → B be isogeny between two algebraic groups. This mapping induces a pullback mapping f* : K(B) → K(A) between their rational function fields. Since
Isogeny
Concept in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Point-surjective_morphism
spectral sequence addresses the calculation of the homology groups of a pullback over a fibration. The spectral sequence formulates the calculation from
Eilenberg–Moore spectral sequence
Eilenberg–Moore_spectral_sequence
Retrieved October 7, 2019. Nissenbaum, Dion (October 7, 2019). "U.S. Begins Pullback From Northern Syria, Clearing Way for Turkish Offensive". Wall Street Journal
Foreign policy of the first Trump administration
Foreign_policy_of_the_first_Trump_administration
French mathematician (born 1926)
principal bundles on algebraic varieties—those that become trivial after pullback by a finite étale map—are important. This acted as one important source
Jean-Pierre_Serre
2019 American action comedy television series
'Wayne', Pilot 'Dark Cargo' Seek New Homes As YouTube Steps Up Scripted Pullback". Deadline Hollywood. August 16, 2019. Archived from the original on August
Wayne_(TV_series)
Harley-Davidson cruiser motorcycle
shield; split five-spoke cast aluminum wheels with diamond cut highlights; pullback handlebar with polished finish; inverted front forks in silver and polished
Harley-Davidson_VRSC
Algebraic structure used in topology
many applications. At a basic level, this has to do with functions and pullbacks in geometric situations: given spaces X {\displaystyle X} and Y {\displaystyle
Cohomology
Category admitting tensor products
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Monoidal_category
President of Lebanon since 2025
Aoun unveils US proposal, demands disarmament of armed groups and Israeli pullback". LBCIV7. Retrieved 31 July 2025. Mroue, Bassem (13 August 2025). "Lebanon
Joseph_Aoun
Inclusion of one mathematical structure in another, preserving properties of interest
(pseudo-)metric in the sense that g {\displaystyle g} is equal to the pullback of h {\displaystyle h} by f {\displaystyle f} , i.e. g = f ∗ h {\displaystyle
Embedding
Concept in mathematical category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Symmetric_monoidal_category
estimate has “slim odds” of passing. Furthermore, this might lead to a pullback in credit card lending. Trump sought a great amount of control over US
Economic policy of the second Trump administration
Economic_policy_of_the_second_Trump_administration
Array of numbers describing a metric connection
called a pullback because it "pulls back" the gradient on R n {\displaystyle \mathbb {R} ^{n}} to a gradient on M {\displaystyle M} . The pullback is independent
Christoffel_symbols
principal G-bundle over a space S is (up to natural isomorphisms) the pullback of the universal bundle E G → B G {\displaystyle EG\to BG} along some map
Functor represented by a scheme
Functor_represented_by_a_scheme
Module over a sheaf of differential operators
finite rank). D-modules on different algebraic varieties are connected by pullback and pushforward functors comparable to the ones for coherent sheaves. For
D-module
Shogi opening in which the rook does not move
Bishop Pullback (hiki kaku), Gangi Castle Static Rook Anaguma (with Bishop Pullback, Static Bishop, or Bishop Exchange) Left Mino (with Bishop Pullback, Static
Static_Rook
History of maths
Internal category theory: Internalization of categories in a category V with pullbacks is replacing the category Set (same for classes instead of sets) by V
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Construction in homological algebra
{\displaystyle 0\to B\xrightarrow {f'} E'\xrightarrow {g'} A\to 0,} first form the pullback over A {\displaystyle A} , Γ = { ( e , e ′ ) ∈ E ⊕ E ′ | g ( e ) = g ′
Ext_functor
Type of loan in which the borrower uses a home as collateral
HELOC Mortgage?". Forbes. Retrieved 2022-09-22. "Two years after HELOC pullback, JPMorgan again eyes the business". American Banker. 2022-05-23. Retrieved
Home_equity_line_of_credit
Concept in differential geometry
X {\displaystyle X} to a diffeological space Y {\displaystyle Y} , the pullback diffeology on X {\displaystyle X} consists of those maps p : U → X {\displaystyle
Diffeology
American general (born 1952)
Julian E.; Spiegel, Peter (15 August 2007). "Top general may propose pullbacks". Los Angeles Times. Retrieved 22 May 2012. "U.S. Department of Defense"
David_Petraeus
Mathematical concept
→ W is called a symplectic map if the pullback preserves the symplectic form, i.e. f∗ρ = ω, where the pullback form is defined by (f∗ρ)(u, v) = ρ(f(u)
Symplectic_vector_space
Mathematical description of spacetime used in relativity
R 1 ( n ) = ι ∗ η , {\displaystyle h_{R}^{1(n)}=\iota ^{*}\eta ,} the pullback of the Minkowski metric η {\displaystyle \eta } under inclusion, is a Riemannian
Minkowski_spacetime
Subject of study in ergodic theory
whereas μ ( T ( A ) ) {\displaystyle \mu (T(A))} is generically called a pullback. Almost all properties and behaviors of dynamical systems are defined in
Measure-preserving dynamical system
Measure-preserving_dynamical_system
Set of topological invariants
{\displaystyle f\colon X'\to X} , where f ∗ E {\displaystyle f^{*}E} denotes the pullback vector bundle. The uniqueness of these classes is proved for example, in
Stiefel–Whitney_class
Coalition against the Islamic State
Retrieved 19 January 2020. "EXCLUSIVE: US military offers Iraq a partial pullback". Middle East Eye. 14 February 2020. "U.S. service members killed in Iraq"
US-led intervention in Iraq (2014–2021)
US-led_intervention_in_Iraq_(2014–2021)
US stockmarket milestones
the close. This was repeated the next trading day before another major pullback for the year. It took until Monday, June 4, 2018 (nearly three months)
Closing milestones of the Nasdaq Composite
Closing_milestones_of_the_Nasdaq_Composite
1994 follow-up treaty to the Oslo I Accord
Gaza Strip," "... Mr. Arafat had asked the Israelis to delay their total pullback for the 21 days allowed in the accord because the Palestinians were not
Gaza–Jericho_Agreement
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Smooth_functor
1998 video game
by Takara for the Sega Saturn. It is based on Takara's Choro Q line of pullback racer toys. Reception Two reviewers from Sega Saturn Magazine found the
Choro_Q_Park
isomorphism. One can define the pullback of vector-valued forms by smooth maps just as for ordinary forms. The pullback of an E-valued form on N by a smooth
Vector-valued differential form
Vector-valued_differential_form
2009 film by Gavin Hood
Digital creating five different mattes for the final scene of the film—a pullback depicting the destroyed Three Mile Island—and Gavin Hood handing company
X-Men_Origins:_Wolverine
Military field formation of the Indian Army
Satkhira axis. The corps moved to its present location in January 1985. The pullback of over four lakh troops along with heavy armored and artillery formations
II_Corps_(India)
Mathematical structure
by f {\displaystyle f} gives a sieve on Y {\displaystyle Y} called the pullback of S {\displaystyle S} along f {\displaystyle f} , denoted by f ∗ S {\displaystyle
Grothendieck_topology
Mexican restaurant chain and food brand
Benmour, Eric (June 14, 1996). "Chi-Chi's closes oldest store here as pullback continues". Business First of Louisville. Retrieved July 2, 2010. "Salsafication
Chi-Chi's
In mathematics, collection of classes
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Conglomerate_(mathematics)
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Diagonal_functor
Tool for determining arterial narrowing
is also feasible with during manual pullback.[citation needed] Advanced computer processing can utilise iFR-pullback data to predict the hemodynamic consequences
Instantaneous_wave-free_ratio
PULLBACK
PULLBACK
PULLBACK
PULLBACK
Boy/Male
Muslim
Bahiy Udeen | بÛای یودین
The magnificent of the faith
Girl/Female
Indian, Marathi, Tamil
Beautiful Lovable Person; Lovely and Beautiful
Boy/Male
Indian, Punjabi, Sikh
Compassionate God
Boy/Male
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Oriya, Rajasthani, Sanskrit, Tamil, Telugu, Traditional
Honey Bee; Lover
Surname or Lastname
English
English : habitational name from a lost or unidentified place, probably in East Anglia.
Girl/Female
Indian, Tamil
Pure; Tree Branch; Sweet
Surname or Lastname
English
English : unexplained.Swedish : unexplained. It may have been a soldier’s name.
Boy/Male
Australian, French, Greek
Untamed; Hunter
Surname or Lastname
English
English : reduced form of Holdaway, itself a variant of Holloway.
Boy/Male
Hindu
Picture
PULLBACK
PULLBACK
PULLBACK
PULLBACK
PULLBACK
n.
The iron hook fixed to a casement to pull it shut, or to hold it party open at a fixed point.
n.
That which holds back, or causes to recede; a drawback; a hindrance.