monoidal functor



Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



A monoidal functor is a functor between monoidal categories that preserves the monoidal structure: a homomorphism of monoidal categories.



Let (π’ž,βŠ— π’ž,1 π’ž)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (π’Ÿ,βŠ— π’Ÿ,1 π’Ÿ)(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} ) be two monoidal categories. A lax monoidal functor between them is

  1. a functor

    F:π’žβŸΆπ’Ÿ, F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,,
  2. a morphism

    Ο΅:1 π’ŸβŸΆF(1 π’ž) \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
  3. a natural transformation

    ΞΌ x,y:F(x)βŠ— π’ŸF(y)⟢F(xβŠ— π’žy) \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y)

    for all x,yβˆˆπ’žx,y \in \mathcal{C}

satisfying the following conditions:

  1. (associativity) For all objects x,y,zβˆˆπ’žx,y,z \in \mathcal{C} the following diagram commutes

    (F(x)βŠ— π’ŸF(y))βŠ— π’ŸF(z) βŸΆβ‰ƒa F(x),F(y),F(z) π’Ÿ F(x)βŠ— π’Ÿ(F(y)βŠ— π’ŸF(z)) ΞΌ x,yβŠ—id↓ ↓ idβŠ—ΞΌ y,z F(xβŠ— π’žy)βŠ— π’ŸF(z) F(x)βŠ— π’ŸF(yβŠ— π’žz) ΞΌ xβŠ— π’žy,z↓ ↓ ΞΌ x,yβŠ— π’žz F((xβŠ— π’žy)βŠ— π’žz) ⟢F(a x,y,z π’ž) F(xβŠ— π’ž(yβŠ— π’žz)) \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} F(y \otimes_{\mathcal{C}} z) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) }

    where a π’ža^{\mathcal{C}} and a π’Ÿa^{\mathcal{D}} denote the associators of the monoidal categories;

  2. (unitality) For all xβˆˆπ’žx \in \mathcal{C} the following diagrams commute

    1 π’ŸβŠ— π’ŸF(x) βŸΆΟ΅βŠ—id F(1 π’ž)βŠ— π’ŸF(x) β„“ F(x) π’Ÿβ†“ ↓ ΞΌ 1 π’ž,x F(x) ⟡F(β„“ x π’ž) F(1βŠ— π’žx) \array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} x ) }


    F(x)βŠ— π’Ÿ1 π’Ÿ ⟢idβŠ—Ο΅ F(x)βŠ— π’ŸF(1 π’ž) r F(x) π’Ÿβ†“ ↓ ΞΌ x,1 π’ž F(x) ⟡F(r x π’ž) F(xβŠ— π’ž1) \array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) }

    where β„“ π’ž\ell^{\mathcal{C}}, β„“ π’Ÿ\ell^{\mathcal{D}}, r π’žr^{\mathcal{C}}, r π’Ÿr^{\mathcal{D}} denote the left and right unitors of the two monoidal categories, respectively.

If Ο΅\epsilon and all ΞΌ x,y\mu_{x,y} are isomorphisms, then FF is called a strong monoidal functor. (Note that β€˜strong’ is also sometimes applied to β€˜monoidal functor’ to indicate possession of a tensorial strength.) If they are even identity morphisms, then FF is called a strict monoidal functor.


In the literature often the term β€œmonoidal functor” refers by default to what in def. is called a strong monoidal functor. With that convention then what def. calls a lax monoidal functor is called a weak monoidal functor.


Lax monoidal functors are the lax morphisms for an appropriate 2-monad.


An oplax monoidal functor (with various alternative names including comonoidal), is a monoidal functor from the opposite categories C opC^{op} to D opD^{op}.


A monoidal transformation between monoidal functors is a natural transformation that respects the extra structure in an obvious way.



(Lax monoidal functors send monoids to monoids)

If F:(C,βŠ—)β†’(D,βŠ—)F : (C,\otimes) \to (D,\otimes) is a lax monoidal functor and

(A∈C,ΞΌ A:AβŠ—Aβ†’A,i A:Iβ†’A) (A \in C,\;\; \mu_A : A \otimes A \to A, \; i_A : I \to A)

is a monoid object in CC, then the object F(A)F(A) is naturally equipped with the structure of a monoid in DD by setting

i F(A):I D→F(I C)→F(i A)F(A) i_{F(A)} : I_D \stackrel{}{\to} F(I_C) \stackrel{F(i_A)}{\to} F(A)


ΞΌ F(A):F(A)βŠ—F(A)β†’βˆ‡ F(A),F(A)F(AβŠ—A)β†’F(ΞΌ A)F(A). \mu_{F(A)} : F(A) \otimes F(A) \stackrel{\nabla_{F(A), F(A)}}{\to} F(A \otimes A) \stackrel{F(\mu_A)}{\to} F(A) \,.

This construction defines a functor

Mon(f):Mon(C)β†’Mon(D) Mon(f) : Mon(C) \to Mon(D)

between the categories of monoids in CC and DD, respectively.



(oplax monoidal functors sends comonoids to comonoids)

For (C,βŠ—)(C,\otimes) a monoidal category write BC\mathbf{B}C for the corresponding delooping 2-category.

Lax monoidal functor f:C→Df : C \to D correspond to lax 2-functor

BF:BC→BD. \mathbf{B}F : \mathbf{B}C \to \mathbf{B}D \,.

If FF is strong monoidal then this is an ordinary 2-functor. If it is strict monoidal, then this is a strict 2-functor.

String diagrams

Just like monoidal categories, monoidal functors have a string diagram calculus; see these slides for some examples.


  • Marcelo Aguiar and Swapneel Mahajan, Monoidal functors, species and Hopf algebras. (pdf)

Exposition of basics of monoidal categories and categorical algebra:

Last revised on September 19, 2021 at 14:17:55. See the history of this page for a list of all contributions to it.