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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
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)
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
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)
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
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
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
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)
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
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
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
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
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
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
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
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
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)
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
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
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)
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
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
Applications of category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Applied_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
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
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
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 mathematics
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Tensor–hom_adjunction
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
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
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
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
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)
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
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)
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)
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
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
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
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
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)
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)
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
American doctor
fighters in their own words, Dec. 10, 2014 Medical Missionaries' Ebola Pullback: No More Kent Brantlys?, Deann Alford/ Nov. 21, 2014 Tryggestad, Erik (2015-02-06)
Kent_Brantly
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)
'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
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Fundamental_groupoid
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
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
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
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
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
Collection of maps which give the same result
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Commutative_diagram
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
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
Mathematical technique for vector bundles
{\displaystyle p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)} is injective, and the pullback bundle p ∗ ξ : p ∗ E → Y {\displaystyle p^{*}\xi \colon p^{*}E\rightarrow
Splitting_principle
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
Vector bundle of cotangent spaces at every point in a manifold
diagonal modulo higher order terms. The cotangent sheaf is defined as the pullback of this sheaf to M {\displaystyle M} : Γ T ∗ M = Δ ∗ ( I / I 2 ) . {\displaystyle
Cotangent_bundle
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)
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
Type of category in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Additive_category
Differential geometry construct on fiber bundles
field to γ generates a horizontal vector field in the total space of the pullback bundle γ*E. By the Picard–Lindelöf theorem, this vector field is integrable
Ehresmann_connection
Concept in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Monoidal_functor
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
Category admitting tensor products
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Monoidal_category
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
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
Quotient of a weakly contractible space by a free action
any G principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle E G → B G {\displaystyle EG\to BG} . As explained
Classifying_space
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
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
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)
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
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
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
Concept in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Point-surjective_morphism
embedding, then Z | Q {\displaystyle Z\vert _{Q}\,} is a section of the pullback bundle i Q ∗ ( T E ) → Q {\displaystyle i_{Q}^{\ast }(TE)\to Q\,} , where
Kosmann_lift
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
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Tetracategory
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
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
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
Model structure on the category of simplicial sets
_{1}^{2}\hookrightarrow \Delta ^{2}} is a weak categorical equivalence, but its pullback along the isofibration Δ 1 ≅ { 0 → 2 } ↪ Δ 2 {\displaystyle \Delta ^{1}\cong
Joyal_model_structure
Motorcycle made by Honda from 1982 to 1985
Fury design by adding larger, more highway friendly front rubber, more pullback on the handle bars and classic, cruiser styled fenders; and the Interstate
Honda_Sabre
Mathematical concept
represented as a pullback of the universal line bundle on C P ∞ {\displaystyle \mathbf {CP} ^{\infty }} , meaning there is a pullback square L → L ↓ ↓
Complex_projective_space
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
Central object of study in category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Natural_transformation
Branch of mathematics
{\mathfrak {X}}\to B} such that any deformation can be found as a unique pullback square X → X ↓ ↓ S → B {\displaystyle {\begin{matrix}X&\to &{\mathfrak
Deformation_(mathematics)
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
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
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)
Special objects used in (mathematical) category theory
Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels
Initial_and_terminal_objects
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
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
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
French fashion designer (born 1991)
Retrieved 2025-06-27. "The Best of BoF 2024: Fashion's Sustainability Pullback". The Business of Fashion. 2024-12-26. Retrieved 2025-02-12. Guilbault
Marine_Serre
2022 action role-playing game
Thai fans who were unhappy with the portrayal of the playerbase. This pullback, however, faced another backlash from a different portion of fans who were
Goddess_of_Victory:_Nikke
PULLBACK
PULLBACK
PULLBACK
PULLBACK
Girl/Female
Latin Hungarian
God's gift.
Male
English
English name derived from Latin Crispinus, CRISPIN means "curly(-headed)."
Girl/Female
Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Sindhi, Telugu
Lotus
Girl/Female
Muslim
Compassion, Sympathy, Pity
Surname or Lastname
English
English : from an Old English personal name, Lulla.German (Lüll) : from a short form of any of the Germanic personal names formed with liut- ‘people’ as the first element.Catalan (also Llull) : from the personal name Lullus, probably of Germanic origin.
Girl/Female
Biblical
Double.
Boy/Male
Muslim
Dignified
Girl/Female
Tamil
Singing bird
Boy/Male
Hindu, Indian, Traditional
Lord Krishna
Surname or Lastname
English (West Midlands)
English (West Midlands) : topographic name from West Midland Middle English rugge, a variant of rigge ‘ridge’, or a habitational name from the village of Rudge in Shropshire, which is named with this word.English (West Midlands) : from a medieval personal name, a pet form of Roger.English (West Midlands) : nickname for a person with red hair or a ruddy complexion, from Old French r(o)uge ‘red’ (Latin rubeus).
PULLBACK
PULLBACK
PULLBACK
PULLBACK
PULLBACK
n.
That which holds back, or causes to recede; a drawback; a hindrance.
n.
The iron hook fixed to a casement to pull it shut, or to hold it party open at a fixed point.