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A monoid object in K-Vect, the category of K-vector spaces (again, with the tensor 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 structure induced by the composition and the identity functor I C. A monoid object in [C, C] is a monad ...
Conversely, the set of isomorphism classes (if such a thing makes sense) of a monoidal category is a monoid w.r.t. the tensor product. Any commutative monoid (,,) can be realized as a monoidal category with a single object. Recall that a category with a single object is the same thing as an ordinary monoid.
More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming , the set of strings built from a given set of characters is a free monoid .
Category theory is a general theory of mathematical structures and their relations. ... For example, a monoid may be viewed as a category with a single object, ...
Cartesian monoidal category; Categorical quantum mechanics; Category of finite-dimensional Hilbert spaces; Category of relations; Center (category theory) Closed monoidal category; Compact closed category
A category C is called small if both ob(C) and hom(C) are actually sets and not proper classes, and large otherwise. A locally small category is a category such that for all objects a and b, the hom-class hom(a, b) is a set, called a homset. Many important categories in mathematics (such as the category of sets), although not small, are at ...
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In concise terms, a monad is a monoid in the category of endofunctors of some fixed category (an endofunctor is a functor mapping a category to itself). According to John Baez, a monad can be considered at least in two ways: [1] A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category,