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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)
Process in mathematics
Pullback (category theory) Fibred category Inverse image sheaf Fong, Brendan; Spivak, David (18 July 2019). An Invitation to Applied Category Theory:
Pullback
Most general completion of a commutative square given two morphisms with same domain
In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the
Pushout_(category_theory)
Generalization of a notion in category theory
the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as morphisms in a category of fractions. The notion
Span_(category_theory)
Overview of and topical guide to category theory
functor Yoneda lemma Product (category theory) Equaliser (mathematics) Kernel (category theory) Pullback (category theory)/fiber product Inverse limit
Outline_of_category_theory
Mathematical concept
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products
Limit_(category_theory)
Generalized object in category theory
limit – Construction in category theory Cartesian closed category – Type of category in category theory Categorical pullback – Most general completion
Product_(category_theory)
General theory of mathematical structures
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the
Category_theory
Generalization of category theory
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows
Higher_category_theory
Applications of category theory
Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer
Applied_category_theory
Contravariant functor to Set
In category theory, a branch of mathematics, a presheaf on a category C {\displaystyle C} is a functor F : C o p → S e t {\displaystyle F\colon C^{\mathrm
Presheaf_(category_theory)
properties and concepts in category theory in mathematics, including those in topos theory. (See also Outline of category theory.) Notes on foundations:
Glossary_of_category_theory
Construction in category theory
In category theory, a branch of mathematics, the cone of a functor is an abstract notion used to define the limit of that functor. Cones make other appearances
Cone_(category_theory)
Mathematical operation
^{*}\left(\nabla _{d\phi (X)}s\right).} Pushforward (differential) Pullback bundle Pullback (category theory) Jost, Jürgen (2002). Riemannian Geometry and Geometric
Pullback (differential geometry)
Pullback_(differential_geometry)
axioms and new type theories, and to use type theory as an internal language for categories, higher categories and other mathematical structures. There are
Semantics_of_type_theory
Indexed collection of objects and morphisms in a category
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in
Diagram_(category_theory)
Collection of objects and morphisms
object. A simple example is the category of sets, whose objects are sets and whose arrows are functions. Category theory is a branch of mathematics that
Category_(mathematics)
Correspondence between properties of a category and its opposite
In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite
Dual_(category_theory)
Relationship between two functors abstracting many common constructions
set theory but the general definition make for a richer range of logics. So consider an object Y {\displaystyle Y} in a category with pullbacks. Any
Adjoint_functors
Mapping between categories
terminology is contrary to the one used in category theory because it is the covectors that have pullbacks in general and are thus contravariant, whereas
Functor
Set of arguments where two or more functions have the same value
g) = Ker(f - g), where Ker denotes the category-theoretic kernel. Any category with fibre products (pullbacks) and products has equalisers. In Top where
Equaliser_(mathematics)
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
Type of category in category theory
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified
Cartesian_closed_category
In the mathematical field of category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations
Allegory_(mathematics)
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
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
Mathematical category with finite limits and coequalizers
a pullback, and if f is a regular epimorphism, then g is a regular epimorphism as well. Examples of regular categories include: Set, the category of
Regular_category
Mathematical concept
In category theory, an end of a functor S : C o p × C → X {\displaystyle S\colon \mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} } is a
End_(category_theory)
Variant of the notion of the center of a monoid, group, or ring to a category
In category theory, a branch of mathematics, the center (or Drinfeld center, after Soviet-American mathematician Vladimir Drinfeld) is a variant of the
Center_(category_theory)
Category-theoretic construction
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces
Coproduct
Map (arrow) between two objects of a category
In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures
Morphism
In category theory, a branch of mathematics, a sieve is a way of choosing arrows with a common codomain. It is a categorical analogue of a collection
Sieve_(category_theory)
In category theory and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification
Refinement_(category_theory)
Category whose hom sets have algebraic structure
In category theory, a branch of mathematics, an enriched category generalizes the idea of a locally small category by replacing hom-sets with objects
Enriched_category
Abstract mathematics relationship
In category theory, a branch of abstract mathematics, an equivalence of categories is a relation between two categories that establishes that these categories
Equivalence_of_categories
Concept in homological algebra
j ∗ {\displaystyle j^{*}} and i ∗ {\displaystyle i_{*}} are the usual pullback and pushforward functors. This works, in particular, when the sheaves in
T-structure
Generalization of a category
specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex
Quasi-category
Generalisation of a sheaf; a fibered category that admits effective descent
In a more general set-up the restrictions are replaced with pullbacks; fibred categories then make a good framework to discuss the possibility of such
Stack_(mathematics)
Topics referred to by the same term
product, in a monoidal category Product (category theory), a generalization of mathematical products Fibre product or pullback Coproduct or pushout Wick
Product
Fiber bundle induced by a map of its base space
bundles. In the language of category theory, the pullback bundle construction is an example of the more general categorical pullback. As such it satisfies the
Pullback_bundle
Concept in category theory
associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image
Fibred_category
and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel. A quasi-abelian category is an exact category.[citation needed]
Quasi-abelian_category
Mathematical concept that extends the intuitive idea of gluing in topology
projection p. The bundles on the Xij that we must control are Vi and Vj, the pullbacks to the fiber of V via the two different projection maps to X. Therefore
Descent_(mathematics)
Generalization of category
In category theory in mathematics, a 2-category is a category with "morphisms between morphisms", called 2-morphisms. A basic example is the category Cat
2-category
Concepts in algebraic topology
527 - Homotopy Theory Homotopy pullbacks A Primer on Homotopy Colimits Homotopy colimits in the category of small categories Categories and Orbispaces
Homotopy_colimit_and_limit
Concept in mathematical category theory
In category theory, a branch of mathematics, the category of elements of a presheaf is a category associated to that presheaf whose objects are the elements
Category_of_elements
Special kind of model structure
higher category theory in mathematics, a proper model structure is a model structure in which additionally weak equivalences are preserved under pullback (fiber
Proper_model_structure
Topics referred to by the same term
differential geometry Pullback (category theory), a term in category theory Pullback attractor, an aspect of a random dynamical system Pullback bundle, the fiber
Pull_back_(disambiguation)
Mathematical object in category theory
the mathematical field of category theory, a subobject classifier is a special object Ω {\displaystyle \Omega } of a category such that, informally, the
Subobject_classifier
In mathematics, invertible homomorphism
transformations, affine transformations, projective transformations. Category theory, which can be viewed as a formalization of the concept of mapping between
Isomorphism
the pullback of ( ∈ ) ↪ A × P ( A ) {\displaystyle (\in )\hookrightarrow A\times {\mathcal {P}}(A)} along χ {\displaystyle \chi } . In the category of
Power_object
Category in mathematical category theory
In category theory in mathematics, a coherent category is a regular category in which the poset of subobjects S u b ( X ) {\displaystyle \mathrm {Sub}
Coherent_category
Product of two categories, in category theory
the mathematical field of category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept
Product_category
Inclusion of one mathematical structure in another, preserving properties of interest
{\displaystyle X} and Y {\displaystyle Y} are instances. In the terminology of category theory, a structure-preserving map is called a morphism. The fact that a map
Embedding
Category admitting tensor products
the category. They are also used in the definition of an enriched category. Monoidal categories have numerous applications outside category theory proper
Monoidal_category
History of maths
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Generalization of algebraic variety
scheme theory completely subsumes the theory of commutative rings. Since Z is an initial object in the category of commutative rings, the category of schemes
Scheme_(mathematics)
Category
In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more
Pre-abelian_category
In category theory and related fields of mathematics, an envelope is a construction that generalizes the operations of "exterior completion", like completion
Envelope_(category_theory)
Category with direct sums and certain types of kernels and cokernels
it is itself an exact category and the inclusion I is an exact functor. This occurs if and only if C is closed under pullbacks of epimorphisms and pushouts
Abelian_category
Simplicial object in the category of topological spaces
is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered
Simplicial_space
Construction for categories
category theory in mathematics, the twisted diagonal of a category (also called the twisted arrow category), which makes the morphisms of a category into
Twisted diagonal (category theory)
Twisted_diagonal_(category_theory)
Embedding of categories into functor categories
The Yoneda lemma is a fundamental result in category theory, a branch of mathematics. It is an abstract result on functors of the type morphisms into
Yoneda_lemma
category, avoiding these set-theoretic issues, was one of the initial reasons for the development of the theory of model categories. A model category
Localization_of_a_category
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
Topics referred to by the same term
target measure space by a measurable function Pushout (category theory), the categorical dual of pullback Direct image sheaf, the pushforward of a sheaf by
Pushforward
In category theory, a branch of mathematics, a pulation square (also called a Doolittle diagram) is a diagram that is simultaneously a pullback square
Pulation_square
homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A
Weak equivalence (homotopy theory)
Weak_equivalence_(homotopy_theory)
Mathematical concept in category theory
In category theory, a branch of mathematics, a subobject is, roughly speaking, an object that sits inside another object in the same category. The notion
Subobject
In mathematics, specifically in category theory, an exact category is a category equipped with short exact sequences. The concept is due to Daniel Quillen
Exact_category
Mathematical structure
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C {\displaystyle {\mathcal {C}}} that makes the objects
Grothendieck_topology
Mathematical construction used in homotopy theory
quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not
Simplicial_set
Symmetric monoidal closed category equipped with a dualizing object
the existence of nontrivial *-autonomous V-categories for all symmetric monoidal categories V with pullbacks, whose objects became known a decade later
*-autonomous_category
Model structure on the category of simplicial sets
In higher category theory, the Kan–Quillen model structure is a special model structure on the category of simplicial sets. It consists of three classes
Kan–Quillen_model_structure
Category theory
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli
Kleisli_category
Category theory concept
mathematics, an overcategory (also called a slice category) is a construction from category theory used in multiple contexts, such as with covering spaces
Overcategory
In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism f ¯ : coim f → im
Semi-abelian_category
Mathematical heuristic
"pullback" therefore requires base change. The related operation in the opposite direction is descent. These ideas persist in contemporary category theory
Grothendieck's relative point of view
Grothendieck's_relative_point_of_view
Tool to track locally defined data attached to the open sets of a topological space
a sheaf on a category with respect to some Grothendieck topology, have provided applications to mathematical logic and to number theory. In many mathematical
Sheaf_(mathematics)
Category of non-empty finite ordinals and order-preserving maps
In mathematics, the simplex category (or simplicial category or nonempty finite ordinal category) is the category of non-empty finite ordinals and order-preserving
Simplex_category
In mathematics, collection of classes
In mathematics, in the framework of a one-universe foundation for category theory, the term conglomerate is applied to arbitrary sets as a contraposition
Conglomerate_(mathematics)
Concept in mathematical category theory
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ⊗ {\displaystyle
Symmetric_monoidal_category
Central object of study in category theory
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal
Natural_transformation
Relation of categories in category theory
In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C that are mutually inverse to each other, i.e
Isomorphism_of_categories
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
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
Theory of stochastic partial differential equations
differential forms and dynamical systems theory defines their dynamics by the generalized transfer operator (GTO)—the pullback averaged over noise. GTO commutes
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Generalization of vector bundles
coherent), but pullbacks of coherent sheaves are coherent if X {\displaystyle X} is locally Noetherian. An important special case is the pullback of a vector
Coherent_sheaf
Injective homomorphism
generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks. The categorical dual of a monomorphism
Monomorphism
Axiomatic approach to quantum field theory
(isotony). The Poincaré group acts continuously on Mink. There exists a pullback of this action, which is continuous in the norm topology of A ( M ) {\displaystyle
Algebraic quantum field theory
Algebraic_quantum_field_theory
In category theory, a branch of mathematics, a stable ∞-category is an ∞-category such that (i) It has a zero object. (ii) Every morphism in it admits
Stable_∞-category
Unified field theory
In physics, Kaluza–Klein theory (KK theory) is an attempt at creating a unified field theory of gravitation and electromagnetism based on the idea of
Kaluza–Klein_theory
Mathematics construct
comma category is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to
Comma_category
Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category C, the construction creates
Q-construction
Special case of colimit in category theory
concept of colimit in category theory. Direct limits are dual to inverse limits, which are a special case of limits in category theory. We will first give
Direct_limit
Quotient space of a codomain of a linear map by the map's image
cokernel is called the corank of f. Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to
Cokernel
higher category theory in mathematics, co- and contravariant model structures are special model structures on slice categories of the category of simplicial
Co- and contravariant model structure
Co-_and_contravariant_model_structure
Mathematics glossary
glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, timeline of manifolds
Glossary of algebraic topology
Glossary_of_algebraic_topology
Construction in algebraic geometry
field, or the pullback of a family of varieties, or a fiber of a family of varieties. Base change is a closely related notion. The category of schemes is
Fiber_product_of_schemes
More precisely, an adhesive category is one where any of the following equivalent conditions hold: C has all pullbacks, it has pushouts along monomorphisms
Adhesive_category
PULLBACK CATEGORY-THEORY
PULLBACK CATEGORY-THEORY
Surname or Lastname
English, Scottish, and Irish (of Norman origin)
English, Scottish, and Irish (of Norman origin) : of disputed origin. It may be from a Celtic personal name derived from the element cam ‘bent’, ‘crooked’ (compare Cameron and Campbell). This was relatively frequent in Norfolk, Lincolnshire, and Yorkshire in the 12th and 13th centuries, perhaps as a result of Breton immigration. According to another theory it is a habitational name from Comines near Lille, but there is no evidence for this (no early forms with de have been found). In southern Ireland this Anglo-Norman name has been confused with 2.Irish : Anglicized form of Gaelic Mac CuimÃn (or Ó CuimÃn) ‘son (or ‘descendant’) of CuimÃn’, a personal name formed from a diminutive of cam ‘crooked’.Americanized form of French Canadian Vien, Viens, based on the misconception that these derive from French venire ‘to come’.
Surname or Lastname
English
English : according to Reaney this is a nickname from an unattested Old English word cybbe meaning ‘clumsy’ or ‘thickset’. Reaney’s speculation is apparently based on taking the Middle English word kibble ‘cudgel’ as a diminutive of an unattested Old English word. Corresponding personal names have been postulated for the place names Kibworth (‘enclosure of a man called Cybba’) and Kibblesworth (‘enclosure of a man called Cybbel’); so, in theory, the surname could be a reflex of these Old English personal names.North German : nickname for a cantankerous person, from Middle Low German, Middle High German kiven ‘to quarrel’.
Surname or Lastname
English
English : from a short form of the personal names Giles, Julian, or William. In theory the name would have a soft initial when derived from the first two of these, and a hard one when from William or from the other possibilities discussed in 2–4 below. However, there has been much confusion over the centuries.Northern English : topographic name for someone who lived by a ravine or deep glen, Middle English gil(l), Old Norse gil ‘ravine’.Scottish and Irish : reduced Anglicized form of Gaelic Mac Gille (Scottish), Mac Giolla (Irish), patronymics from an occupational name for a servant or a short form of the various personal names formed by attaching this element to the name of a saint. See McGill. The Old Norse personal name Gilli is probably of this origin, and may lie behind some examples of the name in northern England.Scottish and Irish : reduced Anglicized form of Gaelic Mac An Ghoill (see Gall 1).Norwegian : habitational name from any of three farmsteads in western Norway named Gil, from Old Norse gil ‘ravine’.Dutch : cognate of Giles.Jewish (Israeli) : ornamental name from Hebrew gil ‘joy’.German : from a vernacular short form of the medieval personal name Aegidius (see Gilger).Indian (Panjab) : Sikh name, probably from Panjabi gil ‘moisture’, also meaning ‘prosperity’. There is a Jat tribe that bears this name; the Ramgarhia Sikhs also have a clan called Gill.
Surname or Lastname
English (mainly Gloucestershire), Dutch, and German (also Türk)
English (mainly Gloucestershire), Dutch, and German (also Türk) : from Middle English, Old French turc, Middle High and Low German Turc ‘Turk’, from Turkish türk. In theory this could be an ethnic name but, both in England and northwest Europe, it is generally a nickname for a person with black hair and a swarthy complexion or a cruel, rowdy, or unruly person. The Dutch and German surname also represents a house name, derived from the use of a picture of a Turk as a house sign. It is also found as a nickname for someone who had taken part in the wars against the Turks.English : from a medieval personal name, a back-formation from Turkel, misanalyzed as containing the Old French diminutive suffix -el.Scottish : reduced Anglicized form of Gaelic Mac Tuirc, a patronymic from the byname Torc ‘boar’.Jewish (Ashkenazic) : ethnic name denoting someone from Turkey or anywhere in the Ottoman Empire, or a nickname for someone thought to resemble a Turk.Americanized form of the Greek ethnic name Tourkos ‘Turk’. See also Turco.
Boy/Male
British, English
Crown
Surname or Lastname
English and Scottish
English and Scottish : topographic name for someone who lived by a patch of wet ground overgrown with brushwood, northern Middle English kerr (Old Norse kjarr). A legend grew up that the Kerrs were left-handed, on theory that the name is derived from Gaelic cearr ‘wrong-handed’, ‘left-handed’.Irish : see Carr.This surname has also absorbed examples of German Kehr.
Surname or Lastname
English
English : unexplained. It may be a variant of a medieval name, Preville, a habitational name from a Norman place named with the elements pré ‘meadow’ + ville ‘settlement’. However, this theory is not supported by evidence of early forms.
PULLBACK CATEGORY-THEORY
PULLBACK CATEGORY-THEORY
Boy/Male
German, Greek
Strong; Masculine
Girl/Female
Hindu, Indian, Traditional
Wise
Girl/Female
Basque, Gujarati, Hindu, Indian, Kannada, Malayalam, Marathi, Punjabi, Sanskrit, Sikh, Sindhi, Telugu
Name of a Learned Woman of the Past
Male
English
English variant of French Pépin, PIPPIN means "seed of a fruit."
Girl/Female
Tamil
Little black one, Dusky
Girl/Female
Tamil
Poorvika | பூரà¯à®µà®¿à®•à®¾Â
Orient, Formerly
Boy/Male
Indian, Punjabi, Sikh
God's Love
Boy/Male
English
Blond.
Girl/Female
Irish
Has been used mainly in Northern Ireland as a female form ofUltach “an Ulsterman.†There have been eighteen saints named Ultan. St. Ultan of Ardbraccan, c. 650 AD, noted for his care of the poor, orphans and the sick is considered the patron saint of children and a hospital for sick children in Dublin is named after him.
Girl/Female
Tamil
Smiritha | ஸà¯à®®à¯€à®°à¯€à®¤à®¾
Remembered
PULLBACK CATEGORY-THEORY
PULLBACK CATEGORY-THEORY
PULLBACK CATEGORY-THEORY
PULLBACK CATEGORY-THEORY
PULLBACK CATEGORY-THEORY
n.
See Category.
n.
The curve formed by a rope or chain of uniform density and perfect flexibility, hanging freely between two points of suspension, not in the same vertical line.
n.
The American pollock; the coalfish.
n.
A marine gadoid food fish of Europe (Pollachius virens). Called also greenfish, greenling, lait, leet, lob, lythe, and whiting pollack.
pl.
of Category
n.
The pollack.
n.
One of the highest classes to which the objects of knowledge or thought can be reduced, and by which they can be arranged in a system; an ultimate or undecomposable conception; a predicament.
a.
Of or pertaining to a category.
n.
The quillback.
a.
Belonging to the same category of individuality; -- a morphological term applied to organisms so related.
n.
The quillback.
n.
The iron hook fixed to a casement to pull it shut, or to hold it party open at a fixed point.
n.
One who inserts in a category or list; one who classifies.
a.
Alt. of Catenarian
v. t.
To insert in a category or list; to class; to catalogue.
n.
A porpoise.
n.
That which holds back, or causes to recede; a drawback; a hindrance.
n.
An American fresh-water fish (Ictiobus, / Carpiodes, cyprinus); -- called also carp sucker, sailfish, spearfish, and skimback.
n.
The European pollack; -- called also laith, and leet.
n.
Class; also, state, condition, or predicament; as, we are both in the same category.