oplax monoidal functor in nLab

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Contents

Definition

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

Δ c,c′:F(c⊗c′)→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.

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

Let (L⊣R):C→R←LD(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 D→R(I D)I_D \to R(I_D) of RR and with oplax monoidal transformation

(L(x⊗y)→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

x⊗y→η x⊗η yRLx⊗RLy→∇ Lx,LyR(Lx⊗Ly). 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⊗y)⟶L(η x⊗η y)L(RLx⊗RLy)⟶L(∇ Lx,Ly)LR(Lx⊗Ly)⟶ϵ Lx⊗LyLx⊗Ly. 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).

• 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)