 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2016/iii/paper-122/2/a/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 122 / 2 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-06

An opmonoidal functor

F : (C, \otimes, I) \to (D, \otimes, J)

consists of a functor, a natural transformation

F_{2} (X, Y) : F (X \otimes Y) ⟶ FX \otimes F Y,

F_{0} : F I \to J

. These maps need not be invertible. If

α, λ, ρ

are the respective associators and unitors, its axioms are

α^{D} (F_{2} (X, Y) \otimes 1) F_{2} (X \otimes Y, Z) = (1 \otimes F_{2} (Y, Z)) F_{2} (X, Y \otimes Z) F (α^{C}),

λ^{D} (F_{0} \otimes 1) F_{2} (I, X) = F (λ^{C}), ρ^{D} (1 \otimes F_{0}) F_{2} (X, I) = F (ρ^{C}) .

The first equation has domain

F ((X \otimes Y) \otimes Z)

FX \otimes (F Y \otimes FZ)

, which fixes the direction of every arrow. This is also called a colax monoidal functor.

An opmonoidal natural transformation

τ : F \Rightarrow G

between opmonoidal functors is a natural transformation satisfying

G_{2} (X, Y) τ_{X \otimes Y} = (τ_{X} \otimes τ_{Y}) F_{2} (X, Y), G_{0} τ_{I} = F_{0} .

It respects both the tensor and unit comparison maps. These are commutative diagrams and a unit commutative diagram; the arrow directions are opposite to those for a lax monoidal functor.

 Read the full article

 New to topics? Read the docs here!