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...

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

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...

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

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

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

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

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...

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...

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...

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...

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...

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...

equivalence classes of morphisms in BordM. A TQFT on M is a symmetric monoidal functor from hBordM to the category of vector spaces. Note that cobordisms...

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

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

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...

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...

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...

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

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

Hom-functor of all morphisms in the category, while B ⇒ C {\displaystyle B\Rightarrow C} denotes the internal hom functor in the closed monoidal category...

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...