# Monoidal functor

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{{short description|Concept in category theory}}
In [category theory](/source/category_theory), '''monoidal functors''' are [functor](/source/functor)s between [monoidal categories](/source/monoidal_category) 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 ==
Let <math>(\mathcal C,\otimes,I_{\mathcal C})</math> and <math>(\mathcal D,\bullet,I_{\mathcal D})</math> be monoidal categories.  A '''lax monoidal functor''' from <math>\mathcal C</math> to <math>\mathcal D</math> (which may also just be called a monoidal functor) consists of a [functor](/source/functor) <math>F:\mathcal C\to\mathcal D</math> together with a [natural transformation](/source/natural_transformation)
:<math>\phi_{A,B}:FA\bullet FB\to F(A\otimes B)</math>
between functors <math>\mathcal{C}\times\mathcal{C}\to\mathcal{D}</math> and a morphism
:<math>\phi:I_{\mathcal D}\to FI_{\mathcal C}</math>,
called the '''coherence maps''' or '''structure morphisms''', which are such that for every three objects <math>A</math>, <math>B</math> and <math>C</math> of <math>\mathcal C</math> the diagrams
:332px,

:225px &nbsp;&nbsp; and &nbsp;&nbsp; 225px
commute in the category <math>\mathcal D</math>.   Above, the various natural transformations denoted using <math>\alpha, \rho, \lambda</math> are parts of the monoidal structure on <math>\mathcal C</math> and <math>\mathcal D</math>.<ref>{{harvp|Perrone|2024|pages=360-364}}</ref>

=== Variants ===

* 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 <math>\phi_{A,B}, \phi</math> 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](/source/braided_monoidal_categories) (with braidings denoted <math>\gamma</math>) such that the following diagram commutes for every pair of objects ''A'', ''B'' in <math>\mathcal C</math> :

:225px

* A '''symmetric monoidal functor''' is a braided monoidal functor whose domain and codomain are [symmetric monoidal categories](/source/symmetric_monoidal_categories).

== Examples ==
* The underlying functor <math>U\colon(\mathbf{Ab},\otimes_\mathbf{Z},\mathbf{Z}) \rightarrow (\mathbf{Set},\times,\{\ast\})</math> from the [category of abelian groups](/source/category_of_abelian_groups) to the [category of sets](/source/category_of_sets).  In this case, the map <math>\phi_{A,B}\colon U(A)\times U(B)\to U(A\otimes B)</math> sends (a, b) to <math>a\otimes b</math>; the map <math>\phi\colon \{*\}\to\mathbb Z</math> sends <math>\ast</math> to 1.
* If <math>R</math> is a (commutative) ring, then the free functor <math>\mathsf{Set},\to R\mathsf{-mod}</math> extends to a strongly monoidal functor <math>(\mathsf{Set},\sqcup,\emptyset)\to (R\mathsf{-mod},\oplus,0)</math> (and also <math>(\mathsf{Set},\times,\{\ast\})\to (R\mathsf{-mod},\otimes,R)</math> if <math>R</math> is commutative).
* If <math>R\to S</math> is a homomorphism of commutative rings, then the restriction functor <math>(S\mathsf{-mod},\otimes_S,S)\to(R\mathsf{-mod},\otimes_R,R)</math> is monoidal and the induction functor <math>(R\mathsf{-mod},\otimes_R,R)\to(S\mathsf{-mod},\otimes_S,S)</math> is strongly monoidal.
* An important example of a symmetric monoidal functor is the mathematical model of [topological quantum field theory](/source/topological_quantum_field_theory). Let <math>\mathbf{Bord}_{\langle n-1,n\rangle}</math> be the category of [cobordism](/source/cobordism)s 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 <math>F\colon(\mathbf{Bord}_{\langle n-1,n\rangle},\sqcup,\emptyset)\rightarrow(\mathbf{kVect},\otimes_k,k).</math>
* The [homology](/source/homology_(mathematics)) functor is monoidal as <math>(Ch(R\mathsf{-mod}),\otimes,R[0]) \to (grR\mathsf{-mod},\otimes,R[0])</math> via the map <math>H_\ast(C_1)\otimes H_\ast(C_2) \to H_\ast(C_1\otimes C_2), [x_1]\otimes[x_2] \mapsto [x_1\otimes x_2]</math>.

==Alternate notions==
If <math>(\mathcal C,\otimes,I_{\mathcal C})</math> and <math>(\mathcal D,\bullet,I_{\mathcal D})</math> are [closed monoidal categories](/source/closed_monoidal_categories) with internal hom-functors <math>\Rightarrow_{\mathcal C},\Rightarrow_{\mathcal D}</math> (we drop the subscripts for readability), there is an alternative formulation 
: ''ψ''<sub>''AB''</sub> : ''F''(''A'' ⇒ ''B'') → ''FA'' ⇒ ''FB''
of ''φ''<sub>''AB''</sub> commonly used in [functional programming](/source/functional_programming). The relation between ''ψ''<sub>''AB''</sub> and ''φ''<sub>''AB''</sub> 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==

* If <math>(M,\mu,\epsilon)</math> is a [monoid object](/source/monoid_object) in <math>C</math>, then <math>(FM,F\mu\circ\phi_{M,M},F\epsilon\circ\phi)</math> is a monoid object in <math>D</math>.<ref>{{harvp|Perrone|2024|pages=367-368}}</ref>

== Monoidal functors and adjunctions ==
Suppose that a functor <math>F:\mathcal C\to\mathcal D</math> is [left adjoint](/source/Adjoint_functors) to a monoidal <math>(G,n):(\mathcal D,\bullet,I_{\mathcal D})\to(\mathcal C,\otimes,I_{\mathcal C})</math>. Then <math>F</math> has a comonoidal structure <math>(F,m)</math> induced by <math>(G,n)</math>, defined by
:<math>m_{A,B}=\varepsilon_{FA\bullet FB}\circ Fn_{FA,FB}\circ F(\eta_A\otimes \eta_B):F(A\otimes B)\to FA\bullet FB</math>
and
:<math>m=\varepsilon_{I_{\mathcal D}}\circ Fn:FI_{\mathcal C}\to I_{\mathcal D}</math>.

If the induced structure on <math>F</math> is strong, then the unit and counit of the adjunction are [monoidal natural transformation](/source/monoidal_natural_transformation)s, and the adjunction is said to be a [monoidal adjunction](/source/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 ==
* [Monoidal natural transformation](/source/Monoidal_natural_transformation)

==Inline citations==
{{reflist}}

== References ==
*{{cite book |first=G. Max |last=Kelly |chapter=Doctrinal adjunction |chapter-url= |editor= |title=Category Seminar |publisher=Springer |series=Lecture Notes in Mathematics |volume=420 |date=1974 |isbn=978-3-540-37270-7 |pages=257–280 |doi=10.1007/BFb0063105}}
* {{cite book |last = Perrone |first = Paolo |title = Starting Category Theory
|date = 2024 |publisher = World Scientific|doi = 10.1142/9789811286018_0005 |isbn =  978-981-12-8600-1|url = https://www.worldscientific.com/worldscibooks/10.1142/13670}}
{{Functors}}

Category:Monoidal categories
Category:Functors

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Adapted from the Wikipedia article [Monoidal functor](https://en.wikipedia.org/wiki/Monoidal_functor) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Monoidal_functor?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.