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[edit]

Let and be monoidal categories. A monoidal functor from to consists of a functor together with a natural isomorphism

between functors and a morphism

,

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

Lax monoidal functor associative.svg,
Lax monoidal functor right unit.svg    and    Lax monoidal functor left unit.svg

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

Variants[edit]

  • 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 such that the following diagram commutes for every pair of objects A, B in  :
Lax monoidal functor braided.svg

Examples[edit]

  • 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, which has been recently developed. 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 .

Properties[edit]

  • If is a monoid object in , then is a monoid object in .

Monoidal functors and adjunctions[edit]

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[edit]

References[edit]

  • Kelly, G. Max (1974), "Doctrinal adjunction", Lecture Notes in Mathematics, 420, 257–280