specifically in category theory, an F {\displaystyle F} -coalgebra is a structure defined according to a functor F {\displaystyle F} , with specific properties...

schemes). There are also F-coalgebras, with important applications in computer science. One frequently recurring example of coalgebras occurs in representation...

{out} :\nu F\rightarrow F(\nu F)=A\times \nu F} This induces another coalgebra F ( ν F ) {\displaystyle F(\nu F)} with associated morphism F ( o u t )...

concepts are initial F-algebras which may serve to encapsulate the induction principle, and the dual construction F-coalgebras. If C {\displaystyle C}...

. 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 :...

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...

list object, respectively. Dually, a final coalgebra is a terminal object in the category of F-coalgebras. The finality provides a general framework for...

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...

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...

general structures: In category theory and computer science: F-algebra and F-coalgebra T-algebra Algebra Blessett, singer from the U.S, goes by the stage...

also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated...

publications indexed by Google Scholar Silva, Alexandra (2010). Kleene coalgebra (PDF). ru.nl (PhD thesis). 694079062. hdl:2066/83205. OCLC 694079062....

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)...

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...

as Morphism Morphisms of F-algebras From an initial algebra to an algebra: Catamorphism From a coalgebra to a final coalgebra: Anamorphism An anamorphism...

{\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...

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...

simultaneously an (unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that...

"YXX") (end "YXY")) (end "YY")) Morphism Morphisms of F-algebras From a coalgebra to a final coalgebra: Anamorphism An anamorphism followed by an catamorphism:...

{\alpha } q\,\}} . In other words, a labelled state transition system is a coalgebra for the functor P ( Λ × − ) {\displaystyle P(\Lambda \times {-})} . There...

(Corecursion). Morphism Morphisms of F-algebras From an initial algebra to an algebra: Catamorphism From a coalgebra to a final coalgebra: Anamorphism An anamorphism...

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...

Hamilton (2007). Corfield, David (2011). "Understanding the Infinite II: Coalgebra". Studies in History and Philosophy of Science. 42 (4): 571–579. Bibcode:2011SHPSA...

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...

fixpoint of a type equation; the isomorphism is then given by a final coalgebra. If the domain of discourse is the category of sets and total functions...

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...

\phi _{2}(X)} . This product can be recognized as the coproduct on a coalgebra. In general, the tensor product of irreducible representations is not...

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...

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...

{\displaystyle {\textstyle \bigwedge }(V)} ⁠, giving the structure of a coalgebra. The coproduct is a linear function ⁠ Δ : ⋀ ( V ) → ⋀ ( V ) ⊗ ⋀ ( V )...