In category theory, monoidal functors are functors between monoidal categories which preserve the monoidal structure. More specifically, a monoidal functor between two monoidal categories consists of a functor between the categories, along with two coherence maps—a natural transformation and a morphism that preserve monoidal multiplication and unit, respectively. Mathematicians require these coherence maps to satisfy additional properties depending on how strictly they want to preserve the monoidal structure; each of these properties gives rise to a slightly different definition of monoidal functors

  • The coherence maps of lax monoidal functors satisfy no additional properties; they are not necessarily invertible.
  • The coherence maps of strong monoidal functors are invertible.
  • The coherence maps of strict monoidal functors are identity maps.

Although we distinguish between these different definitions here, authors may call any one of these simply monoidal functors.

Definition

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Let   and   be monoidal categories. A lax monoidal functor from   to   (which may also just be called a monoidal functor) consists of a functor   together with a natural transformation

 

between functors   and a morphism

 ,

called the coherence maps or structure morphisms, which are such that for every three objects  ,   and   of   the diagrams

 ,
     and     

commute in the category  . Above, the various natural transformations denoted using   are parts of the monoidal structure on   and  .[1]

Variants

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  • The dual of a monoidal functor is a comonoidal functor; it is a monoidal functor whose coherence maps are reversed. Comonoidal functors may also be called opmonoidal, colax monoidal, or oplax monoidal functors.
  • A strong monoidal functor is a monoidal functor whose coherence maps   are invertible.
  • A strict monoidal functor is a monoidal functor whose coherence maps are identities.
  • A braided monoidal functor is a monoidal functor between braided monoidal categories (with braidings denoted  ) such that the following diagram commutes for every pair of objects A, B in   :
 

Examples

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  • The underlying functor   from the category of abelian groups to the category of sets. In this case, the map   sends (a, b) to  ; the map   sends   to 1.
  • If   is a (commutative) ring, then the free functor   extends to a strongly monoidal functor   (and also   if   is commutative).
  • If   is a homomorphism of commutative rings, then the restriction functor   is monoidal and the induction functor   is strongly monoidal.
  • An important example of a symmetric monoidal functor is the mathematical model of topological quantum field theory. Let   be the category of cobordisms of n-1,n-dimensional manifolds with tensor product given by disjoint union, and unit the empty manifold. A topological quantum field theory in dimension n is a symmetric monoidal functor  
  • The homology functor is monoidal as   via the map  .

Alternate notions

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If   and   are closed monoidal categories with internal hom-functors   (we drop the subscripts for readability), there is an alternative formulation

ψAB : F(AB) → FAFB

of φAB commonly used in functional programming. The relation between ψAB and φAB is illustrated in the following commutative diagrams:

 
 

Properties

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  • If   is a monoid object in  , then   is a monoid object in  .[2]

Monoidal functors and adjunctions

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Suppose that a functor   is left adjoint to a monoidal  . Then   has a comonoidal structure   induced by  , defined by

 

and

 .

If the induced structure on   is strong, then the unit and counit of the adjunction are monoidal natural transformations, and the adjunction is said to be a monoidal adjunction; conversely, the left adjoint of a monoidal adjunction is always a strong monoidal functor.

Similarly, a right adjoint to a comonoidal functor is monoidal, and the right adjoint of a comonoidal adjunction is a strong monoidal functor.

See also

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Inline citations

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  1. ^ Perrone (2024), pp. 360–364
  2. ^ Perrone (2024), pp. 367–368

References

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  • Kelly, G. Max (1974). "Doctrinal adjunction". Category Seminar. Lecture Notes in Mathematics. Vol. 420. Springer. pp. 257–280. doi:10.1007/BFb0063105. ISBN 978-3-540-37270-7.
  • Perrone, Paolo (2024). Starting Category Theory. World Scientific. doi:10.1142/9789811286018_0005. ISBN 978-981-12-8600-1.