nLab
monoidal functor

Idea

Just as a functor is a morphism between categories, so a monoidal functor is a morphism between monoidal categories. Monoidal functors come in varying levels of strictness and strength.

Definition

A monoidal functor is a functor $F:C\to D$ between monoidal categories $(C,{\otimes }_C,I_C)$ and $(D,{\otimes }_D,I_D)$ that respects the tensor product and the unit up to coherent but not necessarily invertible morphisms

$$ϵ:I_D\stackrel{}{\to }F(I_c)$$\epsilon : I_D \stackrel{}{\to} F(I_c)

and

$${\mu }_{x,y}:F(x){\otimes }_{D}F(y)\to F(x{\otimes }_{C}y).$$\mu_{x,y} : F(x) \otimes_D F(y) \to F(x \otimes_C y) \,.

If these structure morphisms are isomorphisms then $F$ is called a strong monoidal functor. If they are even identities it is called a strict monoidal functor. In contrast to this, a strong monoidal functor may also be called a weak monoidal functor. Sometimes the plain term “monoidal functor” is used to mean a strong monoidal functor, in which case the general situation is called a lax monoidal functor.

Lax monoidal functors send monoids to monoids.

Colax monoidal functors

A colax monoidal functor (with various alternative names including comonoidal), is a monoidal functor from the opposite categories $C^{\mathrm{op}}$ to $D^{\mathrm{op}}$.

Colax monoidal functors send comonoids to comonoids.

Am I the only one who objects to the term ‘comonoidal functor’? Surely there are people who study (at least enriched over a symmetric —but non-cartesian— monoidal category) categories equipped with comonoidal structures; I would want ‘comonoidal functor’ to describe structure-preserving functors between them. For functors between bimonoidal categories, we would have a conflict. —Toby

You’re not the only one! The Australians call these ‘comonoidal functors’ oplax monoidal functors. —John Baez

I know that term, which is why I included it in the list of terms (^_^). It's nice that you made that page, but you should also link to it in the text here and move the redirects from the bottom of this page to that one. (I did that now.) —Toby

Mike Shulman: I think some Australians, at least, call them “opmonoidal functors.” I agree that “comonoidal functor” has nothing to recommend it. I myself prefer always to add the adjective “lax” when it applies, and to put the “co” or the “op” on the “lax.”

Zoran: I say colax monoidal in my articles, though in informal conversations I am used to geometers who mostly like “comonoidal” as they have dual roles when applied to categories of sheaves and transporting various properties on the tensor categories of sheaves for example; but the main reson is probably because most examples are not “strong” monoidal so usage of words lax and weak would be superfluous for most applications they have in mind.

2-Category perspective

If we pass to the delooping 2-categories $BC$ and $BD$ then a lax monoidal functor corresponds to a lax 2-functor

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

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