In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) (M, μ, η) in a monoidal category (C, ⊗, I) is...

itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely...

least in two ways: A monad as a generalized monoid; this is clear since a monad is a monoid in a certain category, A monad as a tool for studying algebraic...

category-theoretic concept of kernel pair. In particular, kernel pairs can be used to interpret kernels in monoid theory or ring theory in category-theoretic...

classical and quantum information theory. In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action...

Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and...

From a category theoretic point of view, a monoid is a category with one object, and an act is a functor from that category to the category of sets....

monoid may be viewed as a category with a single object, whose morphisms are the elements of the monoid. The second fundamental concept of category theory...

turned into a monoid by just adding an identity element. Consequently, monoids are studied in the theory of semigroups rather than in group theory. Semigroups...

respective categories of monoids and semigroups. It follows that every monoid (or semigroup) arises as a homomorphic image of a free monoid (or semigroup)...

In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas...

In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows...

the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford...

In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general...

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic...

In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In...

In category theory, a branch of mathematics, the opposite category or dual category C op {\displaystyle C^{\text{op}}} of a given category C {\displaystyle...

variant of the notion of the center of a monoid, group, or ring to a category. The center of a monoidal category C = ( C , ⊗ , I ) {\displaystyle {\mathcal...

In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the...

a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.) Notes on foundations: In many expositions...

In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products...

theory Semigroup algebra Transformation semigroup Monoid Aperiodic monoid Free monoid Monoid (category theory) Monoid factorisation Syntactic monoid Structure...

objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids. Let A = ( Σ 1 , Σ 2 , … , Σ n ) {\displaystyle...

The study of this category is known as group theory. There are two forgetful functors from Grp, M: Grp → Mon from groups to monoids and U: Grp → Set from...

group. In K-theory, the point of departure is to observe that the category of vector bundles on a topological space has a commutative monoid structure under...

and normalizer Center (category theory) Kilp, Mati; Knauer, Ulrich; Mikhalev, Aleksandr V. (2000). Monoids, Acts and Categories. De Gruyter Expositions...

In category theory, a branch of mathematics, duality is a correspondence between the properties of a category C and the dual properties of the opposite...

functors) since monads can be viewed as monoid objects in endofunctor categories. Simplicial category PROP (category theory) Abstract simplicial complex Goerss...

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit...

Applied category theory Category of sets Concrete category Category of vector spaces Category of graded vector spaces Category of chain complexes Category of...