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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
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
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)
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)
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)
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
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
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
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
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 and earlier in
Choro-Q
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
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
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
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)
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
Concept from algebraic geometry
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
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
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)
Mathematical description of spacetime used in relativity
Lorentz group. The induced metric on this submanifold, hR1(n)=ι∗η, the pullback of the Minkowski metric η under inclusion, is a Riemannian metric. With
Minkowski_spacetime
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
Type of category in mathematics
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
Mathematical object in category theory
χ {\displaystyle \chi } in a “purely categorical way”, one can take a pullback Y → { 1 } ↓ ↓ X → χ { 0 , 1 } {\displaystyle {\begin{array}{lcl}&Y&\rightarrow
Subobject_classifier
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)
Simplicial object in the category of topological spaces
Graeme Segal in 1974. A Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More
Simplicial_space
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
a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel. A quasi-abelian
Quasi-abelian_category
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)
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
1996 film directed by Jonathan Frakes
dolly pullback required by the effects team, who needed a steady shot to blend a computer-generated version of Picard's eye with the pullback. The 135-foot
Star_Trek:_First_Contact
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
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
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
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
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)
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)
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
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)
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
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
Special kind of model structure
structure in which additionally weak equivalences are preserved under pullback (fiber product) along fibrations, called right proper, and pushouts (cofiber
Proper_model_structure
Analog of Grothendieck topology
s\subseteq w\Longrightarrow {\bar {s}}\subseteq {\bar {w}}} stability under pullback: f − 1 ( s ) ¯ ≡ f − 1 ( s ¯ ) {\displaystyle {\overline {f^{-1}(s)}}\equiv
Lawvere–Tierney_topology
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)
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
Theorem in vector calculus
v)=(𝐅(𝝍(u,v))⋅∂𝝍∂v(u,v)) then, 𝐏(u,v)=Pu(u,v)𝐞u+Pv(u,v)𝐞v. This is the pullback of F along ψ, and, by the above, it satisfies ∮∂Σ𝐅(𝐱)⋅d𝐥=∮γ𝐏(𝐲)⋅d𝐥=∮γ(Pu(u
Stokes'_theorem
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
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Fundamental_groupoid
Tensorial object depending on two points in a manifold
M\times M} , and p r i ∗ {\displaystyle \mathrm {pr} _{i}^{*}} denotes the pullback of the respective bundles. In coordinate notation, a bitensor T {\displaystyle
Bitensor
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
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)
Concept in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Monoidal_functor
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)
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Refinement_(category_theory)
the distribution of (2n−2)-planes which, at the unit vector u, is the pullback of the orthogonal complement of u in the tangent space of M. This is a
Unit_tangent_bundle
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
Method of installing underground utilities
crossings, tractor trailer mounted drills capable of 1,320,000 lbs (thrust/pullback force) and 150,000 ft-lbs (rotary force) may be used [600 tonnes- 204,000
Directional_boring
& 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
Rail yard in Vaughan, Ontario, Canada
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
Concept in mathematics
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Tensor–hom_adjunction
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)
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)
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
Projective variety that is also an algebraic group
{\displaystyle f\colon T\to A^{\vee }} so that L is isomorphic to the pullback of P along the morphism 1 A × f : A × T → A × A ∨ {\displaystyle 1_{A}\times
Abelian_variety
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
Applications of category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Applied_category_theory
Association of cohomology classes to principal bundles
functions), sending a map f : X → Y {\displaystyle f\colon X\to Y} to the pullback operation f ∗ : b G ( Y ) → b G ( X ) {\displaystyle f^{*}\colon b_{G}(Y)\to
Characteristic_class
Generalization of a foliation
general under pullbacks but Haefliger structures are. Indeed, given a continuous map f : X → Y {\displaystyle f:X\to Y} , one can take pullbacks of foliations
Haefliger_structure
Country in Central Africa
original on 22 November 2018. Retrieved 21 November 2018. "The [Ivanhoe] pullback investors have been waiting for". Mining Journal. London, UK: Aspermont
Democratic Republic of the Congo
Democratic_Republic_of_the_Congo
Israeli politician
(1944) (in Hebrew) Kibbutz Society (1954) (in Hebrew) There is No Where to Pullback To (1968) (in Hebrew) Lessons of the Six Day War (1970) (in Hebrew) Issues
Yitzhak_Tabenkin
on X×t is closed. Moreover if this set is the whole of T then L is the pullback of a line bundle on T. Mumford (2008, section 10) also gave a more precise
Seesaw_theorem
Exceptional functor
∗ ( B ) {\displaystyle H_{*}(E)\to H_{*}(B)} and a contravariant map (pullback) on cohomology H ∗ ( B ) → H ∗ ( E ) . {\displaystyle H^{*}(B)\to H^{*}(E)
Shriek_map
from T into some scheme M (better: arises from a universal family through pullback), we say M is a fine moduli space. Denote Γ ( n ) = ker [ Sp 2 g (
Moduli_of_abelian_varieties
many concepts related to random dynamical systems, it is defined in the pullback sense, which means they are understood through their long-term behavior
Absorbing set (random dynamical systems)
Absorbing_set_(random_dynamical_systems)
Relation of categories in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Isomorphism_of_categories
2024 Marvel Studios film
feel a sense of urgency in making a fourth Deadpool film amidst a wider "pullback" of the superhero genre. In January 2026, Kim Masters of Puck reported
Deadpool_&_Wolverine
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
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
Video game series
(Super Bust-a-Move 2) (2002) Ultra Puzzle Bobble (Ultra Bust-a-Move) (2004) Pullback!! Puzzle Bobble (Bust-a-Move DS) (2005) Ultra Puzzle Bobble Pocket (Bust-a-Move
Bubble_Bobble
Syrian military installment in the Rmelan base
al-Jir Base". Welat TV. 2025-10-29. "Syria takes control of Shaddadi as US pullback reshapes power balance". Middle East Online. 15 February 2026. "Syrian
Kharab_al-Jir_base
Canonical connection on a pseudo-Riemannian manifold
Formally, D {\displaystyle D} is the pullback connection γ ∗ ∇ {\displaystyle \gamma ^{*}\nabla } on the pullback bundle γ ∗ T M {\displaystyle \gamma
Levi-Civita_connection
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
Italian screenwriter
(February 6, 2013). "DreamWorks Animation Layoffs Could Reach 500 Amid Slate Pullback". Deadline Hollywood. Retrieved January 31, 2016. McNary, Dave (July 31
Alessandro_Carloni
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
Collection of objects and morphisms
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Category_(mathematics)
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Tetracategory
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
Forced return of refugees and immigrants
international law prohibition on refoulement. Pushback is contrasted with "pullback", a form of extraterritorial migration control the country seeking to repel
Pushback_(migration)
American biotechnology company
Companies". Medical Product Outsourcing. July 26, 2018. "Wait For The Pullback To Buy Edwards Lifesciences". Seeking Alpha. April 28, 2020. Whooley, Sean
Edwards_Lifesciences
Category in mathematical category theory
{\displaystyle \mathrm {Sub} (X)} have "finite" unions which are stable under pullback, then S u b ( X ) {\displaystyle \mathrm {Sub} (X)} is an upper semilattice
Coherent_category
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
American drama web television series
'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_(TV_series)
Analogs of homology groups for algebraic varieties
morphism Z → X {\displaystyle Z\to X} , and the second homomorphism is pullback with respect to the flat morphism X − Z → X {\displaystyle X-Z\to X} .
Chow_group
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Localization_of_a_category
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
For a square matrix, the transpose of the cofactor matrix
{v} \wedge \alpha .} Suppose that T : V → V is a linear transformation. Pullback by the (n − 1)th exterior power of T induces a morphism of Hom spaces.
Adjugate_matrix
Mathematical concept
satisfying ωe=id:TeG→𝔤, and ∀g∈Gωg=Ad(h)(Rh∗ωe), where h=g−1, where Rh* is the pullback of forms along the right-translation in the group and Ad(h) is the adjoint
Maurer–Cartan_form
Australian-born German mathematician (born 1949)
nonautonomous and random dynamical systems, particularly the development of pullback attractors, and for co-authoring a foundational text on the numerical solution
Peter_Kloeden
space is convex. A variety X {\displaystyle X} is called convex if the pullback of the tangent bundle to a stable rational curve f : C → X {\displaystyle
Convexity (algebraic geometry)
Convexity_(algebraic_geometry)
travel, tourism, insurance
PULLBACK
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travel, tourism, insurance