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In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the monoidal unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian monoidal category, and any finite coproduct category can be 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 closed categories.[1]
^Cartesian monoidal category at the nLab
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In mathematics, a monoidalcategory (or tensor category) is a category C {\displaystyle \mathbf {C} } equipped with a bifunctor ⊗ : C × C → C {\displaystyle...
In category theory, a branch of mathematics, a symmetric monoidalcategory is a monoidalcategory (i.e. a category in which a "tensor product" ⊗ {\displaystyle...
example is the category of sets, Set, where the monoidal product of sets A {\displaystyle A} and B {\displaystyle B} is the usual cartesian product A × B...
monoidalcategory M. For the case that M is the category of sets and (⊗, I, α, λ, ρ) is the monoidal structure (×, {•}, ...) given by the cartesian product...
In category theory, a category is Cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with...
by the cartesian product of sets. It is also a monoidalcategory if one defines the monoidal product by the disjoint union of sets. The category Rel was...
symmetric monoidalcategory. Ab is not a topos since e.g. it has a zero object. Category of modules Abelian sheaf — many facts about the category of abelian...
one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure. The recursive...
In graph theory, the Cartesian product G □ H of graphs G and H is a graph such that: the vertex set of G □ H is the Cartesian product V(G) × V(H); and...
is Rel, the category having sets as objects and relations as morphisms, with Cartesianmonoidal structure. A symmetric monoidalcategory ( C , ⊗ , I )...
all finite biproducts exist, making C both a Cartesianmonoidalcategory and a co-Cartesianmonoidalcategory. If the product A 1 × A 2 {\textstyle A_{1}\times...
image of y {\displaystyle y} by f {\displaystyle f} . The cartesian morphisms of a fibre category F S {\displaystyle F_{S}} are precisely the isomorphisms...
preadditive category). The category of rings is a symmetric monoidalcategory with the tensor product of rings ⊗Z as the monoidal product and the ring of...
consider a 2-category with a single object; these are essentially monoidalcategories. Bicategories are a weaker notion of 2-dimensional categories in which...
category theory, the product of two categories C and D, denoted C × D and called a product category, is an extension of the concept of the Cartesian product...
More generally, any monoidal closed category is a closed category. In this case, the object I {\displaystyle I} is the monoidal unit. Eilenberg, S.;...
of a commutative monoid; a category with finite coproducts is an example of a symmetric monoidalcategory. If the category has a zero object Z {\displaystyle...
one morphism into X. symmetric monoidalcategory A symmetric monoidalcategory is a monoidalcategory (i.e., a category with ⊗) that has maximally symmetric...
In category theory, a monoidal monad ( T , η , μ , T A , B , T 0 ) {\displaystyle (T,\eta ,\mu ,T_{A,B},T_{0})} is a monad ( T , η , μ ) {\displaystyle...
John Baez and James Dolan as an instance of semistrict k-tuply monoidal n-categories, which describe general topological quantum field theories, for...
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