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...

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...

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...

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....

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...

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

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...

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...

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...

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

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...

called "operands". In category theory, semiautomata essentially are functors. A transformation semigroup or transformation monoid is a pair ( M , Q ) {\displaystyle...

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

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...

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...

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 opposite category or dual category C op {\displaystyle C^{\text{op}}} of a given category C {\displaystyle...

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...

If it includes the identity function, it is a monoid, called a transformation (or composition) monoid. This is the semigroup analogue of a permutation...

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...

n × n {\displaystyle n\times n} matrices, and form a monoid, canonically isomorphic to the monoid of linear endomorphisms of R n {\displaystyle \mathbb...

In category theory, an end of a functor S : C o p × C → X {\displaystyle S:\mathbf {C} ^{\mathrm {op} }\times \mathbf {C} \to \mathbf {X} } is a universal...

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...

endomorphisms of X forms a monoid, the full transformation monoid, and denoted End(X) (or EndC(X) to emphasize the category C). An invertible endomorphism...

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