nLab monoidal functor



Monoidal categories

monoidal categories

With braiding

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 a functor:

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

together with coherence maps:

  1. a morphism

    Ο΅:1 π’ŸβŸΆF(1 π’ž) \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}})
  2. 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.

More generally, lax functors send enriched categories to enriched categories, an operation known as change of enriching category. See there for more details.



(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.


Exposition of basics of monoidal categories and categorical algebra:

Last revised on September 2, 2023 at 12:15:47. See the history of this page for a list of all contributions to it.