What is Bifunctor?

A bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, the Hom functor is of the type Cop × C → Set. It can be seen as a functor in two arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other.

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.

What is a monoid Haskell?

In Haskell, the Monoid typeclass (not to be confused with Monad) is a class for types which have a single most natural operation for combining values, together with a value which doesn’t do anything when you combine it with others (this is called the identity element).

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.

What is Contravariant functor?

A functor is called contravariant if it reverses the directions of arrows, i.e., every arrow is mapped to an arrow .

Is a functor a monad?

A functor is a data type that implements the Functor typeclass. An applicative is a data type that implements the Applicative typeclass. A monad is a data type that implements the Monad typeclass. A Maybe implements all three, so it is a functor, an applicative, and a monad.

Is a functor a category?

Functor categories serve as the hom-categories in the strict 2-category Cat. In the context of enriched category theory the functor category is generalized to the enriched functor category.

What are monoidal categories in category theory?

In category theory, monoidal categories can be used to define the concept of a monoid object and an associated action on the objects of the category. They are also used in the definition of an enriched category. Monoidal categories have numerous applications outside of category theory proper.

What is a monoidal preorder?

Monoidal preorders, also known as “preordered monoids”, are special cases of monoidal categories. This sort of structure comes up in the theory of string rewriting systems, but it is plentiful in pure mathematics as well. For example, the set . We now present the general case. in C. If there happens to be a morphism from c to c’ , we could write .

What is cocartesian monoidal R-Mod?

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. As special cases one has:

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: