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Mathematical structure
specifically in category theory, an F {\displaystyle F} -coalgebra is a structure defined according to a functor F {\displaystyle F} , with specific properties
F-coalgebra
Proof method in mathematical logic
A × ν F {\displaystyle \mathrm {out} :\nu F\rightarrow F(\nu F)=A\times \nu F} This induces another coalgebra F ( ν F ) {\displaystyle F(\nu F)} with
Coinduction
Structure dual to a unital associative algebra
schemes). There are also F-coalgebras, with important applications in computer science. One frequently recurring example of coalgebras occurs in representation
Coalgebra
Function type in category theory
concepts are initial F-algebras which may serve to encapsulate the induction principle, and the dual construction F-coalgebras. If C {\displaystyle C}
F-algebra
Mathematical object
list object, respectively. Dually, a final coalgebra is a terminal object in the category of F-coalgebras. The finality provides a general framework for
Initial_algebra
Programming function applied recursively to its previous result
. F X of a functor F. By the universal property of final coalgebras, there is a unique coalgebra morphism A → ν X . F X for any other F-coalgebra a :
Anamorphism
Topics referred to by the same term
general structures: In category theory and computer science: F-algebra and F-coalgebra T-algebra Algebra Blessett, singer from the U.S. whose stage name
Algebra_(disambiguation)
algebra, the cofree coalgebra of a vector space or module is a coalgebra analog of the free algebra of a vector space. The cofree coalgebra of any vector space
Cofree_coalgebra
Ring that is also a vector space or a module
structure of a coalgebra. There is also an abstract notion of F-coalgebra, where F is a functor. This is vaguely related to the notion of coalgebra discussed
Associative_algebra
In mathematics a Lie coalgebra is the dual structure to a Lie algebra. In finite dimensions, these are dual objects: the dual vector space to a Lie algebra
Lie_coalgebra
Universal construction in multilinear algebra
also has two coalgebra structures; one simple one, which does not make it a bi-algebra, but does lead to the concept of a cofree coalgebra, and a more
Tensor_algebra
Portuguese computer scientist
publications indexed by Google Scholar Silva, Alexandra (2010). Kleene coalgebra (PDF). ru.nl (PhD thesis). 694079062. hdl:2066/83205. OCLC 694079062.
Alexandra_Silva
comodule over a coalgebra is formed by dualizing the definition of a module over an associative algebra. Let K be a field, and C be a coalgebra over K. A (right)
Comodule
Type of algorithm in computer science
of codata. Given a final coalgebra A of a polynomial functor F—i.e. F ( X ) = ∑ a : α X β ( a ) {\displaystyle \textstyle F(X)=\sum _{a:\alpha }X^{\beta
Corecursion
Lie-theoretic case of a bialgebra: it is a set with a Lie algebra and a Lie coalgebra structure which are compatible. It is a bialgebra where the multiplication
Lie_bialgebra
Construction in algebra
simultaneously a (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that
Hopf_algebra
Mathematical concept in category theory
product), is a unital associative K-algebra, and a comonoid object is a K-coalgebra. For any category C, the category [C, C] of its endofunctors has a monoidal
Monoid_(category_theory)
z ≤ y f ( x , z ) g ( z , y ) . {\displaystyle (f*g)(x,y):=\sum _{x\leq z\leq y}f(x,z)g(z,y).} There is also a definition of incidence coalgebra. In theoretical
Locally_finite_poset
Poisson manifold that is also a Lie group
{\mathfrak {g}}} of a Poisson–Lie group has a natural structure of Lie coalgebra given by linearising the Poisson tensor P : G → T G ∧ T G {\displaystyle
Poisson–Lie_group
abstract algebra, a dialgebra is the generalization of both algebra and coalgebra. The notion was originally introduced by Lambek as "subequalizers", and
Dialgebra
as Morphism Morphisms of F-algebras From an initial algebra to an algebra: Catamorphism From a coalgebra to a final coalgebra: Anamorphism An anamorphism
Paramorphism
Homomorphism from an initial algebra into another algebra
"YXX") (end "YXY")) (end "YY")) Morphism Morphisms of F-algebras From a coalgebra to a final coalgebra: Anamorphism An anamorphism followed by an catamorphism:
Catamorphism
the space of all maps from the morphisms of C to R, denoted F(C), and has a natural coalgebra structure. Thus for a locally finite category, the dual of
Category_algebra
Recursive function
nodes. Morphism Morphisms of F-algebras From an initial algebra to an algebra: Catamorphism From a coalgebra to a final coalgebra: Anamorphism Extension of
Hylomorphism (computer science)
Hylomorphism_(computer_science)
cocomplete if C = ⋃ k F k C {\textstyle C=\bigcup _{k}F_{k}C} . The reduced tensor coalgebra is the universal cocomplete coalgebra over V {\displaystyle
Homotopy_associative_algebra
Concept in mathematics
coefficient of xiyj in F(x,y). Conversely, given a Hopf algebra whose coalgebra structure is given above, we can recover a formal group law F from it. So 1-dimensional
Formal_group_law
British philosopher
Hamilton (2007). Corfield, David (2011). "Understanding the Infinite II: Coalgebra". Studies in History and Philosophy of Science. 42 (4): 571–579. Bibcode:2011SHPSA
David_Corfield
State machine that may have infinite states
{\alpha } q\,\}} . In other words, a labelled state transition system is a coalgebra for the functor P ( Λ × − ) {\displaystyle P(\Lambda \times {-})} . There
Transition_system
Algebraic structure with "nice" duality properties
Hopf modules and integrals. Note: the coalgebra structure for the Hopf algebra is different from the coalgebra structure for the Frobenius algebra. The
Frobenius_algebra
Overview of and topical guide to algebraic structures
product space: an F-vector space V with a definite bilinear form V × V → F. Bialgebra: an associative algebra with a compatible coalgebra structure. Lie
Outline of algebraic structures
Outline_of_algebraic_structures
Algebraic structure used in analysis
generators are [ F 1 , F 2 ] = F 3 , {\displaystyle [F_{1},F_{2}]=F_{3},} [ F 2 , F 3 ] = F 1 , {\displaystyle [F_{2},F_{3}]=F_{1},} [ F 3 , F 1 ] = F 2 . {\displaystyle
Lie_algebra
{\displaystyle ({\mathfrak {g}},\triangleleft )} is a pre-Lie algebra. Lie coalgebra Lie bialgebra Lie algebra cohomology Frobenius algebra Quasi-Frobenius
Quasi-Frobenius_Lie_algebra
Subset of a preorder that contains all larger elements
point in it. Let F {\displaystyle F} be the set of all (not-necessarily-open) neighborhoods of x {\displaystyle x} . Then F {\displaystyle F} is an upper
Upper_and_lower_sets
Type of binary relation
Set Theory, 3rd edition, "Well-founded relations", pages 251–5, Marcel Dekker ISBN 0-8247-7915-0 https://ncatlab.org/nlab/show/well-founded+coalgebra
Well-founded_relation
Generalization of bialgebra
by F 1 {\displaystyle F_{1}} and then F 2 {\displaystyle F_{2}} is equivalent to twisting by F 2 F 1 {\displaystyle F_{2}F_{1}} , and twisting by F {\displaystyle
Quasi-bialgebra
(Corecursion). Morphism Morphisms of F-algebras From an initial algebra to an algebra: Catamorphism From a coalgebra to a final coalgebra: Anamorphism An anamorphism
Apomorphism
Operation in algebra and mathematics
usual tensor product are important and widely studied under the name of coalgebras. The notion of monad was invented by Roger Godement in 1958 under the
Monad_(category_theory)
R_{21}R_{12}=1} . The twisting of H A {\displaystyle {\mathcal {H_{A}}}} by F ∈ A ⊗ A {\displaystyle F\in {\mathcal {A\otimes A}}} is the same as for a quasi-Hopf algebra
Quasi-triangular quasi-Hopf algebra
Quasi-triangular_quasi-Hopf_algebra
Algebra associated to any vector space
{\displaystyle \textstyle \bigwedge (V)} , giving the structure of a coalgebra. The coproduct is a linear function Δ : ⋀ ( V ) → ⋀ ( V ) ⊗ ⋀ ( V )
Exterior_algebra
the coalgebra. To see this, let Δ ( a ) = ∑ a i ⊗ b i {\displaystyle \Delta (a)=\textstyle \sum a_{i}\otimes b_{i}} and, given linear functionals f {\displaystyle
Schur_algebra
Pasquali, Fabio; Rosolini, Giuseppe (2020). "Elementary doctrines as coalgebras". Journal of Pure and Applied Algebra. 224 (12). doi:10.1016/j.jpaa.2020
Doctrine_(mathematics)
Associative algebra used in combinatorics
p_{k}^{-s}} , we obtain the usual Euler product. Graph algebra Incidence coalgebra Path algebra Incidence algebras of locally finite posets were treated
Incidence_algebra
One of five systems of modal logic
"Steve Awodey. Category Theory. Chapter 10. Monads. 10.4 Comonads and Coalgebras" (PDF). Archived from the original (PDF) on 2022-10-07. Retrieved 2022-03-08
S5_(modal_logic)
Branch of mathematics that studies abstract algebraic structures
\phi _{2}(X)} . This product can be recognized as the coproduct on a coalgebra. In general, the tensor product of irreducible representations is not
Representation_theory
representations in terms of F-matrices associated with finite-dimensional irreducible representations of quantum affine algebra. F-matrices can be used to
Quasi-Hopf_algebra
ISBN 978-0-387-90192-3. Oliveira, J.N. (2013). "Weighted Automata as Coalgebras in Categories of Matrices". Int. J. Found. Comput. Sci. 24 (6): 709–728
Probabilistic_bisimulation
Links the homology groups of a product space with those of the individual spaces
graded coalgebras. The composite C ∗ ( X ) → C ∗ ( X ) ⊗ C ∗ ( X ) {\displaystyle C_{*}(X)\to C_{*}(X)\otimes C_{*}(X)} itself is not a map of coalgebras. The
Eilenberg–Zilber_theorem
Theorem in category theory
condition that B is faithfully flat implies that T reflects isomorphisms. A coalgebra over T turns out to be essentially a B-module with descent data, so the
Beck's_monadicity_theorem
Left adjoint to a forgetful functor to sets
the right adjoint to the forgetful functor. See, for example, Cofree coalgebra. There are general existence theorems that apply; the most basic of them
Free_object
Mathematical constructs and creation rules
M-types) can be defined up to isomorphism as initial algebras (resp. final coalgebras) for polynomial functors. In this case, the property of initiality (res
Inductive_type
Computer science professor
Coinduction: A Proof Theoretical Foundation In Proceedings of Algebra and Coalgebra in Computer Science (CALCO), pp. 127-144. J. Endrullis, D. Hendriks, M
Grigore_Roșu
Differential form in commutative algebra
complex into a commutative differential graded algebra. It also has a coalgebra structure inherited from the one on the exterior algebra. The hypercohomology
Kähler_differential
Algebraic construct of interest in theoretical physics
of the Hopf *-algebra (a corepresentation of a counital coassociative coalgebra A is a square matrix v = ( v i j ) i , j = 1 , … , n {\displaystyle v=(v_{ij})_{i
Quantum_group
Mathematical structure in differential geometry
\mu } endows g {\displaystyle {\mathfrak {g}}} with a structure of Lie coalgebra, which is moreover compatible with the original Lie algebra structure
Poisson_manifold
Simplicial Lie algebra Homotopy Lie algebra Hinich, Vladimir (2001). "DG coalgebras as formal stacks". Journal of Pure and Applied Algebra. 162 (2–3): 209–250
Differential graded Lie algebra
Differential_graded_Lie_algebra
Mathematical theory of topological spaces
differential graded cocommutative coalgebras. (The associated coalgebra is the rational homology of X as a coalgebra; the dual vector space is the rational
Rational_homotopy_theory
W-types (resp. M-types) are (isomorphic to) initial algebras (resp. final coalgebras) of such functors. Polynomial functors have been studied in the more general
Polynomial functor (type theory)
Polynomial_functor_(type_theory)
{C}}} . For P {\displaystyle P} an algebra and C {\displaystyle C} a coalgebra over a shared field k {\displaystyle k} , an entwining between them is
Distributive law between monads
Distributive_law_between_monads
the maps f {\displaystyle f} and p {\displaystyle p} induce maps of differential graded coalgebras f ∗ : S ∗ ( X ) → S ∗ ( B ) {\displaystyle f_{\ast }\colon
Eilenberg–Moore spectral sequence
Eilenberg–Moore_spectral_sequence
Generalization of associativity properties
chain complexes, groupoids (or even the category of categories itself), coalgebras, etc. Given a commutative ring R we consider the category R - M o d {\displaystyle
Operad
Result of partitioning the elements of an algebraic structure using a congruence relation
ISBN 978-0-8218-8323-5. Klaus Denecke; Shelly L. Wismath (2009). Universal algebra and coalgebra. World Scientific. pp. 14–17. ISBN 978-981-283-745-5. Purna Chandra Biswal
Quotient_(universal_algebra)
Technique from algebraic geometry
{\displaystyle T} -coactions a : F → T ( F ) {\displaystyle a:F\to T(F)} (despite the name, they are more like comodules than coalgebras). Then the key point here
Faithfully_flat_descent
Freely generated algebraic structure over a given signature
ISBN 0-521-77920-0. Klaus Denecke; Shelly L. Wismath (2009). Universal Algebra and Coalgebra. World Scientific. pp. 21–23. ISBN 978-981-283-745-5. T.H. Tse (2010)
Term_algebra
Dutch mathematician
Komenda, J.H. van Schuppen, Modular control of discrete-event systems with coalgebra, IEEE Transactions on Automatic Control 53 (2008), 447–460. Jan Komenda
Jan_H._van_Schuppen
American mathematician (1936–2024)
ISBN 0-521-24870-1 Steven Shnider and Shlomo Sternberg (1993) Quantum Groups. From Coalgebras to Drinfeld Algebras: A Guided Tour (Mathematical Physics Ser.) International
Shlomo_Sternberg
Armenian theoretical physicist (born 1936)
in n-Li - Nambu algebras, he introduced the fundamental concepts of 3-coalgebras, 3-algebras and 3-algebras of Hopf. In 1990 Rudolf Muradyan proposed the
Rudolf_Muradyan
and coideal in H, and the quotient module Q = H/R°H is a right H-module coalgebra. For example, if H is a group algebra, then R is a subgroup algebra of
Depth of noncommutative subrings
Depth_of_noncommutative_subrings
homology coalgebra E ∗ ( E ) {\displaystyle E_{*}(E)} (of co-operations). Note for the case E = H F p {\displaystyle E=H\mathbb {F} _{p}} , H F p ∗ ( H F p )
Adams_resolution
H → R that make (H, R, Δ, ε) an R-coring (with axioms like that of a coalgebra such that all mappings are R-R-bimodule homomorphisms and all tensors
Hopf_algebroid
Abstract structure in mathematics
groups", arXiv:math/9803122 a corepresentation of a counital coassiative coalgebra A is a square matrix v = ( v i j ) i , j = 1 , … , n {\displaystyle v=(v_{ij})_{i
Compact_quantum_group
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