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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
Mathematical term
(also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the
Disjoint_union_(topology)
Topics referred to by the same term
a generalization of mathematical products Fibre product or pullback Coproduct or pushout Wick product of random variables Graph product Product (Brand
Product
Most general completion of a commutative square given two morphisms with same domain
category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit
Pushout_(category_theory)
In mathematics, operation on sets
disjoint sets is their union. In category theory, the disjoint union is the coproduct of the category of sets, and thus defined up to a bijection. In this context
Disjoint_union
Object that is both a product and coproduct
zero objects, is both a product and a coproduct. In a preadditive category the notions of product and coproduct coincide for finite collections of objects
Biproduct
Type of category in category theory
monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a
Cartesian_monoidal_category
Universal construction in multilinear algebra
unital and associative. The unit is explicitly required to define the coproduct. Let V be a vector space over a field K. For any nonnegative integer k
Tensor_algebra
Structure dual to a unital associative algebra
(and for representation theory in general), the coproduct must be linear. As a general rule, the coproduct in representation theory is reducible; the factors
Coalgebra
Category with direct sums and certain types of kernels and cokernels
family (Ai) of objects of A, the coproduct *Ai exists in A (i.e. A is cocomplete). AB4) A satisfies AB3), and the coproduct of a family of monomorphisms is
Abelian_category
Chemical compound
beverages. tert-Butyl alcohol is derived commercially from isobutane as a coproduct of propylene oxide production. It can also be produced by the catalytic
Tert-Butyl_alcohol
Algebra associated to any vector space
is dual to a coproduct defined on ⋀ ( V ) {\displaystyle \textstyle \bigwedge (V)} , giving the structure of a coalgebra. The coproduct is a linear
Exterior_algebra
Category whose objects are rings and whose morphisms are ring homomorphisms
component-wise. The coproduct of a family of rings exists and is given by a construction analogous to the free product of groups. The coproduct of nonzero rings
Category_of_rings
Octahedric silicon compound
produced naturally on a large scale in volcanoes. It is manufactured as a coproduct in the production of phosphate fertilizers. The resulting hexafluorosilicic
Hexafluorosilicic_acid
V\in {}_{H}^{H}{\mathcal {YD}}} is always a braided Hopf algebra. The coproduct Δ {\displaystyle \Delta } of T V {\displaystyle TV} is defined in such
Braided_Hopf_algebra
Algebraic structure formed from a collection of algebraic structures
the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, the direct sum is a coproduct. That
Direct_sum
Links the homology groups of a product space with those of the individual spaces
F} yields a coproduct C ∗ ( X ) → C ∗ ( X ) ⊗ C ∗ ( X ) {\displaystyle C_{*}(X)\to C_{*}(X)\otimes C_{*}(X)} inducing the standard coproduct on H ∗ ( X
Eilenberg–Zilber_theorem
In algebra, the free product (coproduct) of a family of associative algebras A i , i ∈ I {\displaystyle A_{i},i\in I} over a commutative ring R is the
Free product of associative algebras
Free_product_of_associative_algebras
Example of a non-commutative and non-cocommutative Hopf algebra
describes a slight variant of this Hopf algebra using the opposite coproduct, i.e. the coproduct described above composed with the tensor flip on H4⊗H4. This
Sweedler's_Hopf_algebra
Operation that combines groups
universal group with a given set of generators). The free product is the coproduct in the category of groups. That is, the free product plays the same role
Free_product
Mathematical category
fiber products distribute over coproducts; that is, given a set I {\displaystyle I} , an I {\displaystyle I} -indexed coproduct mapping to A {\displaystyle
Topos
Algebraic structure with "nice" duality properties
circles) gives a product map V ⊗ V → V {\displaystyle V\otimes V\to V} or a coproduct map V → V ⊗ V {\displaystyle V\to V\otimes V} , depending on how the boundary
Frobenius_algebra
Investment in the precious metal silver
reserves at 610,000 metric tons. Silver is often recovered as a byproduct or coproduct from lead-zinc, copper, and gold mines. Silver demand comes from both
Silver_as_an_investment
Type of category in category theory
every finitary product is necessarily a coproduct, and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence
Additive_category
Generalized object in category theory
where the plus sign here denotes the coproduct. To see this, note that the universal property of the coproduct X × Y + X × Z {\displaystyle X\times Y+X\times
Product_(category_theory)
Mathematical concept
colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related
Limit_(category_theory)
Special objects used in (mathematical) category theory
colimit of the empty diagram 0 → C and can be thought of as an empty coproduct or categorical sum. It follows that any functor which preserves limits
Initial_and_terminal_objects
Mathematical category whose hom sets form Abelian groups
in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised
Preadditive_category
American chairman & CEO (born 1965)
1965) is the founder of POET, LLC, a leading producer of biofuels and coproducts. He currently serves as chairman and CEO. Broin's family ran a farm in
Jeff_Broin
Operation in abstract algebra
submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion. The most
Direct_sum_of_modules
Generalization of the Cartesian product
example of a product in a category, whereas a direct sum is an example of a coproduct. If R {\displaystyle \mathbb {R} } is thought of as the set of real numbers
Direct_product
Theorem in algebra
_{R}\to \mathbf {Mod} _{S}} is additive, is right-exact and preserves coproducts if and only if it is of the form F ≃ − ⊗ R F ( R ) {\displaystyle F\simeq
Eilenberg–Watts_theorem
Union data structure with enforced cases
record, choice type, discriminated union, disjoint union, sum type, or coproduct, is a data structure used to hold a value that could take on several different
Tagged_union
In mathematics, invertible homomorphism
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Isomorphism
Algebraic construct of interest in theoretical physics
coproduct T o Δ, where T is given by T(x ⊗ y) = y ⊗ x, giving three more possible versions. The counit on Uq(A) is the same for all these coproducts:
Quantum_group
Category admitting tensor products
identity map is the unit. Dually, any category with finite coproducts is monoidal with the coproduct as the monoidal product and the initial object as the
Monoidal_category
Tensor product of algebras over a field; itself another algebra
coproduct in the category of commutative R-algebras. The tensor product is not the coproduct in the category of all R-algebras; there the coproduct is
Tensor_product_of_algebras
Chemical compound
1-(trimethylsilyl)imidazole requires more preparative effort with the advantage that the coproduct trimethylsilyl chloride is volatile. CDI hydrolyzes readily to give back
Carbonyldiimidazole
Chemical compound
converts to silicon dioxide: Si(OCH2CH3)4 → SiO2 + 2 (CH3CH2)2O The volatile coproduct is diethyl ether. Inhalation of TEOS induces eye and nose irritation,
Tetraethyl_orthosilicate
Type of category in category theory
Cartesian closed categories with binary coproducts and an initial object, with products distributing over coproducts. Their equational theory is extended
Cartesian_closed_category
Chemical compound
isopropanol in presence of ammonia. Hydrogen chloride is formed as a coproduct: TiCl4 + 4 (CH3)2CHOH → Ti{OCH(CH3)2}4 + 4 HCl Titanium isopropoxide reacts
Titanium_isopropoxide
Category
(equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts;
Pre-abelian_category
Chemical compound
derived from oxygenation of isobutane, which affords t-butanol. This coproduct can be dehydrated to isobutene, converted to MTBE, an additive for gasoline
Propylene_oxide
Mapping between categories
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Functor
Category theory concept
example, if C {\displaystyle {\mathcal {C}}} has finite products and coproducts, it is immediate the categories C / X {\displaystyle {\mathcal {C}}/X}
Overcategory
Field theory involving topological effects in physics
manifolds), the map associated with a pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative
Topological quantum field theory
Topological_quantum_field_theory
Applications of category theory
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Applied_category_theory
Homological algebra statement
built up inductively with the nth item in the resolution equal to the coproduct of the nth items in the resolutions of A ′ {\displaystyle A'} and A ″
Horseshoe_lemma
Mathematical concept
filtered homotopy colimits). For a triangulated category C which admits all coproducts, Neeman (2001a) defines an object to be compact if Hom C ( X , ⋅ ) :
Compact_object_(mathematics)
in terms of projections from coproducts, having a family of projective generators along with the existence of coproducts guarantees the category has "enough
Generator_(category_theory)
Chemical compound
chloride gas is dissolved in water to produce hydrochloric acid as a useful coproduct of the reaction. Sodium bisulfate can be generated as a byproduct of the
Sodium_bisulfate
Ring built from other rings (mathematics)
homomorphism. (A finite coproduct in the category of commutative algebras over a commutative ring is a tensor product of algebras. A coproduct in the category
Product_of_rings
Chemical compound
give both nitrous acid and nitric acid: N2O4 + H2O → HNO2 + HNO3 The coproduct HNO2 upon heating disproportionates to NO and more nitric acid. When exposed
Dinitrogen_tetroxide
Chemical compound
amides into nitriles: P4O10 + RC(O)NH2 → P4O9(OH)2 + RCN The indicated coproduct P4O9(OH)2 is an idealized formula for undefined products resulting from
Phosphorus_pentoxide
Concept in mathematics
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Tensor–hom_adjunction
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Smooth_functor
Function type in category theory
commutative diagrams: Now use the coproduct (the disjoint union of sets) to glue the three morphisms in one: α = e
F-algebra
American biofuel manufacturer
Among its coproducts in the process are feed ingredients such as distillers grains, syrup and corn oil, along with additional coproducts including an
POET
Chemical compound
preparing certain nylons, is produced by cleavage of ricinoleic acid. The coproduct is 2-octanol. The mechanism of the base-induced cleavage is proposed to
Ricinoleic_acid
Debate concerning diversion of food supply for biofuels
Food versus fuel is the dilemma regarding the risk of diverting farmland or crops for biofuel production to the detriment of the food supply. The biofuel
Food_vs._fuel
Concept in Hopf algebra
x , y ∈ X {\displaystyle x,y\in X} . Here the Sweedler's notation of coproduct of Hopf algebra is used. For matched pair of Hopf algebras A {\displaystyle
Bicrossed product of Hopf algebra
Bicrossed_product_of_Hopf_algebra
Species of grass cultivated as a food crop
including exports of corn silage and dried distiller grains (ethanol coproducts). As a grain crop, the dried kernels are used as feed. They are often
Maize
Result from multiplying no factors
of finite sets. Dually, the coproduct of an empty family is an initial object. Nullary categorical products or coproducts may not exist in a given category;
Empty_product
Category whose only morphisms are the identity morphisms
into another category is called a product, while the colimit is called a coproduct. Thus, for example, the discrete category with just two objects can be
Discrete_category
Indexed collection of objects and morphisms in a category
of the limit, the result is the product; for the colimit, one gets the coproduct. So, for example, when J is the discrete category with two objects, the
Diagram_(category_theory)
Chemical compound
dimerization of butadiene in the presence of a nickel or iron catalyst, a coproduct being vinylcyclohexene. Approximately 10,000 tons were produced in 2005
1,5-Cyclooctadiene
Generalizations of '"`UNIQ--math-00000000-QINU`"' in algebraic structures
union An empty sum or empty coproduct An initial object in a category (an empty coproduct, and so an identity under coproducts) An absorbing element in a
Zero_element
Category whose objects are sets and whose morphisms are functions
product of sets. The coproduct is given by the disjoint union: given sets Ai where i ranges over some index set I, we construct the coproduct as the union of
Category_of_sets
of the symmetric group. The ring of symmetric functions can be given a coproduct and a bilinear form making it into a positive selfadjoint graded Hopf
Ring_of_symmetric_functions
General theory of mathematical structures
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Category_theory
Construction in algebra
is usually taken to be a finite-dimensional algebra and coalgebra with coproduct Δ: H → H ⊗ H and counit ε: H → k satisfying all the axioms of Hopf algebra
Hopf_algebra
Map (arrow) between two objects of a category
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Morphism
) {\displaystyle D(x,x^{\prime })=(dx,d^{\prime }x^{\prime })} . The coproduct of two differential graded Lie algebras, L ∗ L ′ {\displaystyle L*L^{\prime
Differential graded Lie algebra
Differential_graded_Lie_algebra
In type theory, a type with no terms
} is a type with no terms. Such a type may be defined as the nullary coproduct (i.e. disjoint sum of no types). It may also be defined as the polymorphic
Empty_type
Repeated application of an operation to a sequence
of all the terms gcd {\displaystyle \gcd } Category theory Coproduct Disjoint union Coproduct of objects ∐ {\displaystyle \coprod } Product Cartesian product
Iterated_binary_operation
Object in category theory
not cartesian closed. In a category with a terminal object 1 and binary coproducts (denoted by +), an NNO can be defined as the initial algebra of the endofunctor
Natural_numbers_object
Theorem in category theory
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Lawvere's_fixed-point_theorem
realizer of y {\displaystyle y} . The coproduct of X {\displaystyle X} and Y {\displaystyle Y} is the coproduct of sets X + Y {\displaystyle X+Y} where
Effective_topos
Category in which all small limits exist
and only if it has coequalizers and all (small) coproducts, or, equivalently, pushouts and coproducts. Finite completeness can be characterized in several
Complete_category
requires a strong Brønsted base and a weak Lewis acid, and gives a methanol coproduct: (H2CO)2n + nB + nLA + nHArOH → nHC(=O)ArOH + n[HB][LA:OMe] Formally,
Casiraghi_formylation
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Traced_monoidal_category
Space in topology mathematics
sum of the underlying spaces. The wedge sum can be understood as the coproduct in the category of pointed spaces. Alternatively, the wedge sum can be
Wedge_sum
Concept in category theory
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Point-surjective_morphism
made into a commutative and cocommutative Hopf algebra as follows. The coproduct of Exp(G) is defined so that all the elements exp(gt) are group-like.
Exp_algebra
Commercially important migratory fish
compared to fish-fed salmon. Another possible alternative is a yeast-based coproduct of bioethanol production, proteinaceous fermentation biomass. Substituting
Salmon
In mathematics, collection of classes
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Conglomerate_(mathematics)
Most general completion of a commutative square given two morphisms with same codomain
also the pushout. Since the pushout in the category of rings, i.e, the coproduct in a category of algebras over a ring R, is given by the tensor product
Pullback_(category_theory)
Chemical compound
same stereoselectivity as metal-catalysed syn addition of H2. The only coproduct released is nitrogen gas. Although the method is cumbersome, the use of
Diimide
∇ Del operator Gradient Divergence Curl ∈ Element (mathematics) ƛ Reduced wavelength ∐ Coproduct
List of letters used in mathematics, science, and engineering
List_of_letters_used_in_mathematics,_science,_and_engineering
Cleavage of C=C, C≡C, or N=N bonds with ozone
Ozonolysis of oleic acid is an important route to azelaic acid. The coproduct is nonanoic acid: CH3(CH2)7CH=CH(CH2)7CO2H} + 4 O3 → HO2C(CH2)7CO2H} +
Ozonolysis
Monoidal category
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Tannakian_formalism
Construction in category theory
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Cone_(category_theory)
Branch of mathematics that studies abstract algebraic structures
_{1}(X)\otimes I+I\otimes \phi _{2}(X)} . This product can be recognized as the coproduct on a coalgebra. In general, the tensor product of irreducible representations
Representation_theory
Topics referred to by the same term
theory) (also called an amalgamated sum or a cocartesian square, fibered coproduct, or fibered sum), the colimit of a diagram consisting of two morphisms
Sum
Arithmetic operation
case of the coproduct operation, and general coproducts are perhaps the most abstract of all the generalizations of addition. The coproduct such as direct
Addition
Measure in functional analysis
X_{i}} contain a nonzero element, the Lp sum is neither a product nor a coproduct. Helemskii, A. Ya. (2006). Lectures and Exercises on Functional Analysis
Lp_sum
unit/ source map η R : A → Γ right unit/ target map Δ : Γ → Γ ⊗ A Γ coproduct/ composition map ε : Γ → A counit/ identity map c : Γ → Γ conjugation/
Hopf_algebroid
Category theory constructs
Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories
Kan_extension
Overview of and topical guide to category theory
(category theory)/fiber product Inverse limit Pro-finite group Colimit Coproduct Coequalizer Cokernel Pushout (category theory) Direct limit Biproduct
Outline_of_category_theory
Formal system in mathematical logic
of the simply typed lambda calculus with constructs such as products, coproducts or natural numbers (System T) or even full recursion (like PCF). In contrast
Simply_typed_lambda_calculus
COPRODUCT
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Girl/Female
Irish
Ancient.
Girl/Female
Tamil
Tasteful
Girl/Female
Indian, Sikh
Pure Like Water
Boy/Male
Hindu
Girl/Female
Arabic, Muslim
Crusader; Warrior
Girl/Female
American, Australian
Wisdom; Wise
Girl/Female
Tamil
Dedicated to the gods
Boy/Male
Arabic, Muslim
Another Name for Prophet Muhammad
Boy/Male
Hindu
Flower, Blossom
Surname or Lastname
English
English : habitational name from Tarleton in Lancashire, near Croston, named with the Old Norse personal name þóraldr (composed of the elements þórr, name of the Norse god of thunder (see Thor) + valdr ‘rule’) + Old English tūn ‘enclosure’, ‘settlement’.English : habitational name from Tarlton in Gloucestershire, recorded in Domesday Book as Torentune and in 1204 as Torleton, probably from Old English thorn ‘thorn tree’ + lēah ‘(forest) clearing’ + tūn ‘enclosure’, ‘settlement’.
COPRODUCT
COPRODUCT
COPRODUCT
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COPRODUCT