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Monoidal functor

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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:

Commutative diagram demonstrating how a monoidal coherence map gives rise to its applicative formulation
Commutative diagram demonstrating how a monoidal coherence map can be recovered from its applicative formulation

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.