nLab
oplax monoidal functor

Contents

Definition

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

Δ c,c:F(cc)F(c)F(c)\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 CC to comonoids in DD, just as a lax monoidal functor sends monoids in CC to monoids in DD. 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 ϕ\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 (LR):CRLD(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D be a pair of adjoint functors and let (C,)(C,\otimes) and (D,)(D,\otimes) be structures of monoidal categories.

Then if RR is a lax monoidal functor LL becomes an oplax monoidal functor with oplax unit

L(I D)I C L(I_D) \to I_C

the adjunct of the lax unit I DR(I D)I_D \to R(I_D) of RR and with oplax monoidal transformation

(L(xy)Δ x,yL(x)L(y)) (L (x \otimes y) \stackrel{\Delta_{x,y}}{\to} L(x) \otimes L(y))

given by the adjunct of

xyi xi yRLxRLy Lx,LyR(LxLy). 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(xy)L(i xi y)L(RLxRLy)L( Lx,Ly)LR(LxLy)ϵ LxLyLxLy. 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 April 8, 2014 00:34:42 by Urs Schreiber (185.37.147.12)