# nLab oplax monoidal functor

### Context

#### Monoidal categories

monoidal categories

## With traces

• trace

• traced monoidal category?

# Contents

## Definition

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

${\Delta }_{c,c\prime }:F\left(c\otimes c\prime \right)\to F\left(c\right)\otimes F\left(c\prime \right)$\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 $\varphi$ 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 $\left(L⊣R\right):C\stackrel{\stackrel{L}{←}}{\underset{R}{\to }}D$ be a pair of adjoint functors and let $\left(C,\otimes \right)$ and $\left(D,\otimes \right)$ be structures of monoidal categories.

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

$L\left({I}_{D}\right)\to {I}_{C}$L(I_D) \to I_C

the adjunct of the lax unit ${I}_{D}\to R\left({I}_{D}\right)$ of $R$ and with oplax monoidal transformation

$\left(L\left(x\otimes y\right)\stackrel{{\Delta }_{x,y}}{\to }L\left(x\right)\otimes L\left(y\right)\right)$(L (x \otimes y) \stackrel{\Delta_{x,y}}{\to} L(x) \otimes L(y))

given by the adjunct of

$x\otimes y\stackrel{{i}_{x}\otimes {i}_{y}}{\to }RLx\otimes RLy\stackrel{{\nabla }_{Lx,Ly}}{\to }R\left(Lx\otimes Ly\right)\phantom{\rule{thinmathspace}{0ex}}.$x \otimes y \stackrel{i_x \otimes i_y}{\to} R L x \otimes R L y \stackrel{\nabla_{L x, L y}}{\to} R(L x \otimes L y) \,.

Notice that this adjunct is the composite

$L\left(x\otimes y\right)\stackrel{L\left({i}_{x}\otimes {i}_{y}\right)}{\to }L\left(RLx\otimes RLy\right)\stackrel{L\left({\nabla }_{Lx,Ly}\right)}{\to }LR\left(Lx\otimes Ly\right)\stackrel{{ϵ}_{Lx\otimes Ly}}{\to }Lx\otimes Ly\phantom{\rule{thinmathspace}{0ex}}.$L(x \otimes y) \stackrel{L(i_x \otimes i_y)}{\to} L(R L x \otimes R L y) \stackrel{L(\nabla_{L x, L y})}{\to} L R(L x \otimes L y) \stackrel{\epsilon_{L x \otimes L y}}{\to} L x \otimes L y \,.

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

## References

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

Revised on November 3, 2010 15:16:35 by Urs Schreiber (131.211.232.76)