In mathematics, a monoidal category (or tensor category) is a category equipped with a bifunctor. that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.

What is a monoidal in category theory?

In mathematics, a monoidal category (or tensor category) is a category equipped with a bifunctor. that is associative up to a natural isomorphism, and an object I that is both a left and right identity for ⊗, again up to a natural isomorphism.

Is set a monoidal category?

For example the category Set can be made into a monoidal category with cartesian product or disjoint union (i.e. coproduct) as the ‘tensor product’.

Is Cat Cartesian closed?

Cartesian closed structure The category Cat, at least in its traditional version comprising small categories only, is cartesian closed: the exponential objects are functor categories.

What is semigroup and monoid?

A semigroup may have one or more left identities but no right identity, and vice versa. A two-sided identity (or just identity) is an element that is both a left and right identity. Semigroups with a two-sided identity are called monoids.

Is a Monad a monoid?

@AlexanderBelopolsky, technically, a monad is a monoid in the monoidal category of endofunctors equipped with functor composition as its product. In contrast, classical “algebraic monoids” are monoids in the monoidal category of sets equipped with the cartesian product as its product.

What is an Endofunctor?

Endofunctor. A functor that maps a category to that same category; e.g., polynomial functor. Identity functor. in category C, written 1C or idC, maps an object to itself and a morphism to itself. The identity functor is an endofunctor.

What is a Monad in mathematics?

In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations required to fulfill certain coherence conditions.

Is there a category of all categories?

No, there is no category of all categories.

Is Cat locally small?

Cat has all small limits and colimits. Cat is not locally Cartesian closed. Cat is locally finitely presentable.

What do you mean by semi group?

A mathematical object defined for a set and a binary operator in which the multiplication operation is associative. No other restrictions are placed on a semigroup; thus a semigroup need not have an identity element and its elements need not have inverses within the semigroup.

What is semi group example?

Example: Consider a semigroup (N, +), where N is the set of all natural numbers and + is an addition operation. The algebraic system (E, +) is a subsemigroup of (N, +), where E is a set of +ve even integers.

Is a monad a category?

Monads are often considered in the 2-category Cat where they are given by endofunctors with a monoid structure on them. In particular, monads in Cat on Set are equivalent to the equational theories studied in universal algebra.

What is the difference between strict monoidal and monoidal categories?

Every monoidal category is monoidally equivalent to a strict monoidal category. Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit.

What are the applications of monoidal categories?

Monoidal categories have numerous applications outside of category theory proper. They are used to define models for the multiplicative fragment of intuitionistic linear logic. They also form the mathematical foundation for the topological order in condensed matter physics.

What is the difference between strict and cartesian monoidal categories?

Every monoidal category is monoidally equivalent to a strict monoidal category. Any category with finite products can be regarded as monoidal with the product as the monoidal product and the terminal object as the unit. Such a category is sometimes called a cartesian monoidal category. For example:

What is cocartesian monoidal?

Such a monoidal category is called cocartesian monoidal. R-Mod, the category of modules over a commutative ring R, is a monoidal category with the tensor product of modules ⊗R serving as the monoidal product and the ring R (thought of as a module over itself) serving as the unit.