theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between...

enriched functor, etc... reduce to the original definitions from ordinary category theory. An enriched category with hom-objects from monoidal category...

Cat (equipped with the monoidal structure induced by the cartesian product). Monoidal functors are the functors between monoidal categories that preserve...

internal product functor defining a monoidal category. The isomorphism is natural in both X and Z. In other words, in a closed monoidal category, the internal...

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

Combinatorial species Exact functor Derived functor Dominant functor Enriched functor Kan extension of a functor Hom functor Product (category theory) Equaliser...

applicative functor, or an applicative for short, is an intermediate structure between functors and monads. In category theory they are called closed monoidal functors...

{\displaystyle (T,\eta ,\mu )} on a monoidal category ( C , ⊗ , I ) {\displaystyle (C,\otimes ,I)} such that the functor T : ( C , ⊗ , I ) → ( C , ⊗ , I )...

Monoidal may refer to: Monoidal category, concept in category theory Monoidal functor, between monoidal categories Monoidal natural transformation, between...

A closed monoidal category is a monoidal category C {\displaystyle {\mathcal {C}}} such that for every object B {\displaystyle B} the functor given by...

a category as a functor category. A vertical composition is the composition of natural transformations. Similarly, given a monoidal category V, the category...

Formally, a diagram of shape J {\displaystyle J} in C {\displaystyle C} is a functor from J {\displaystyle J} to C {\displaystyle C} : F : J → C . {\displaystyle...

relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in...

preadditive category is the category Ab itself. More precisely, Ab is a closed monoidal category. Note that commutativity is crucial here; it ensures that the...

symmetric monoidal, and moreover, (Ste, ⊛ {\displaystyle \circledast } ) is a closed symmetric monoidal category with the internal hom-functor ⊘ {\displaystyle...

In algebra, an action of a monoidal category ( S , ⊗ , e ) {\displaystyle (S,\otimes ,e)} on a category X is a functor ⋅ : S × X → X {\displaystyle \cdot...

monoidal category C Σ {\displaystyle C_{\Sigma }} . The interpretation in a monoidal category D {\displaystyle D} is a defined by a monoidal functor F...

is a fundamental result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation...

thought of as a cocartesian monoidal category. Cartesian categories with an internal Hom functor that is an adjoint functor to the product are called Cartesian...

a branch of mathematics, a functor category D C {\displaystyle D^{C}} is a category where the objects are the functors F : C → D {\displaystyle F:C\to...

adjunction is that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ⁡ ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form an adjoint...

composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product...

that there exists a K-linear exact and faithful tensor functor (i.e., a strong monoidal functor) F from C to the category of finite dimensional L-vector...

comonoid object in a monoidal category gives rise to a simplicial object since it can then be viewed as the image of a functor from Δ + op {\displaystyle...

mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition...

in 1970 in the general context of enriched functor categories. Day convolution gives a symmetric monoidal structure on H o m ( C , D ) {\displaystyle...

the category of contravariant functors from D {\displaystyle D} to the category of sets; such a contravariant functor is frequently called a presheaf...

and Automata in Monoidal Categories. ASL North American Annual Meeting, 17 March 2010 Aguiar, M. and Mahajan, S.2010. "Monoidal Functors, Species, and Hopf...

({\mathcal {D}},\bullet ,J)} are two monoidal categories. A monoidal adjunction between two lax monoidal functors ( F , m ) : ( C , ⊗ , I ) → ( D , ∙ ...

for the monad). 2.  A functor is said to be monadic if it is a constituent of a monadic adjunction. monoidal category A monoidal category, also called...