nLab oplax monoidal functor

Definition

If $C$ and $D$ are monoidal categories, an oplax monoidal functor $F : C \to D$ is defined to be a lax monoidal functor $F: C^{op} \to D^{op}$. So, among other things, tensor products are preserved up to morphisms of the following sort in $D$:

$$\Delta_{c,c'} : F(c \otimes c') \to F(c) \otimes F(c')$$

which must satisfy a certain coherence law.

Properties

An oplax monoidal functor sends comonoids in $C$ to comonoids in $D$, just as a lax monoidal functor sends monoids in $C$ to monoids in $D$. For this reason an oplax monoidal functor is sometimes called a lax comonoidal functor. The other obvious terms, colax monoidal and lax opmonoidal, also exist (or at least are attested on Wikipedia).

Note that a strong opmonoidal functor –in which the morphisms $\phi$ are required to be isomorphisms— is the same thing as a strong monoidal functor.

Proposition

A functor with a right adjoint is oplax monoidal if and only if that right adjoint is a lax monoidal functor.

Proof

This is a special case of the statement of doctrinal adjunction for the case of the 2-monad whose algebras are monoidal categories,

Here is the explicit construction of the oplax monoidal structure from a lax monoidal structure on a right adjoint:

Let $(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D$ be a pair of adjoint functors and let $(C,\otimes)$ and $(D,\otimes)$ be structures of monoidal categories.

Then if $R$ is a lax monoidal functor $L$ becomes an oplax monoidal functor with oplax unit

$$L(I_D) \to I_C$$

the adjunct of the lax unit $I_D \to R(I_D)$ of $R$ and with oplax monoidal transformation

$$(L (x \otimes y) \stackrel{\Delta_{x,y}}{\to} L(x) \otimes L(y))$$

given by the adjunct of

$$x \otimes y \stackrel{\eta_x \otimes \eta_y}{\to} R L x \otimes R L y \stackrel{\nabla_{L x, L y}}{\to} R(L x \otimes L y) \,.$$

By the formula for adjuncts in terms of the adjunction counit (this prop.) this adjunct is the composite

$$L(x \otimes y) \stackrel{L(\eta_x \otimes \eta_y)}{\longrightarrow} L(R L x \otimes R L y) \stackrel{L(\nabla_{L x, L y})}{\longrightarrow} L R(L x \otimes L y) \stackrel{\epsilon_{L x \otimes L y}}{\longrightarrow} L x \otimes L y \,.$$

This appears for instance on p. 17 of (SchwedeShipley).

Related concepts

• monoidal functor

• strong monoidal functor

• lax monoidal functor

• oplax monoidal functor

• bilax monoidal functor

• Frobenius monoidal functor

References

The construction of oplax monoidal functors from right adjoint lax monoidal functors is considered for instance around page 17 of

• Stefan Schwede, Brooke Shipley, Equivalences of monoidal model categories , Algebr. Geom. Topol. 3 (2003), 287–334 (arXiv)