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An algebra for this endofunctor is a set X together with a function 1 + N × X → X. To define such a function, we need a point x ∈ X and a function N × X → X. The set of finite lists of natural numbers is an initial algebra for this functor.
The algebra (, [,]) in the above example is an initial algebra. Various finite data structures used in programming , such as lists and trees , can be obtained as initial algebras of specific endofunctors.
Initial and terminal objects are not required to exist in a given category. However, if they do exist, they are essentially unique. Specifically, if I 1 and I 2 are two different initial objects, then there is a unique isomorphism between them. Moreover, if I is an initial object then any object isomorphic to I is also an initial object. The ...
def – define or definition. deg – degree of a polynomial, or other recursively-defined objects such as well-formed formulas. (Also written as ∂.) del – del, a differential operator. (Also written as.) det – determinant of a matrix or linear transformation. DFT – discrete Fourier transform.
In mathematics, specifically category theory, a functor is a mapping between categories.Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces.
The definitions of categories and functors provide only the very basics of categorical algebra; additional important topics are listed below. Although there are strong interrelations between all of these topics, the given order can be considered as a guideline for further reading.
“Any linear map from to an algebra can be uniquely extended to an algebra homomorphism from () to .” This statement is an initial property of the tensor algebra since it expresses the fact that the pair ( T ( V ) , i ) {\displaystyle (T(V),i)} , where i : V → U ( T ( V ) ) {\displaystyle i:V\to U(T(V))} is the inclusion map, is a ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
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