Why monoidal category?

Why monoidal category?

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 a Comonoid?

Definition A comonoid (or comonoid object) in a monoidal category M is a monoid object in the opposite category Mop (which is a monoidal category using the same operation as in M).

Is Cat Cartesian closed?

1-Categorical properties Cat is not locally Cartesian closed. Cat is locally finitely presentable.

What is a linear category?

A linear category, or algebroid, is a category whose hom-sets are all vector spaces (or modules) and whose composition operation is bilinear. This concept is a horizontal categorification of the concept of (unital associative) algebra.

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.

Are categories Monoids?

A monoid is a category with a single object. Given a monoid (M,*), one can construct a small category with only one object and whose morphisms are the elements of M. The composition of morphisms is given by the monoid operation *.

What is a Cartesian category?

A cartesian monoidal category (usually just called a cartesian category), is a monoidal category whose monoidal structure is given by the category-theoretic product (and so whose unit is a terminal object). A cartesian monoidal category which is also closed is called a cartesian closed category.

Is rel cartesian closed?

Rel is monoidal closed, with both the monoidal product A ⊗ B and the internal hom A ⇒ B given by cartesian product of sets.

What are examples categories?

: a group of people or things that are similar in some way The cars belong to the same category. Taxpayers fall into one of several categories. She competed for the award in her age category.

What is an additive functor?

Additive functors A functor F: C → D between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams.

What is an example of a monoidal category?

Such a category is sometimes called a cartesian monoidal category. For example: Set, the category of sets with the Cartesian product, any particular one-element set serving as the unit. Cat, the category of small categories with the product category, where the category with one object and only its identity map is the unit.

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 the difference between a semigroup and a semi-monoid?

A semigroup is like a monoid where there might not be an identity element. The term “semigroup” is standard, but semi-monoid would be more systematic. a set equipped with an associative binary operation.

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.