nLab
monoidal functor

Contents

Context

Monoidal categories

monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

Contents

Idea

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

Definition

Definition

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 ) }

    and

    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.)

Remark

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.

Remark

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

Definition

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}.

Definition

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

Properties

Proposition

(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)

and

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.

Similarly:

Proposition

(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.

References

Exposition of basics of monoidal categories and categorical algebra:

Last revised on July 12, 2019 at 00:43:17. See the history of this page for a list of all contributions to it.