Endofunctors as a monoidal category. #
We give the monoidal category structure on
C ⥤ C,
and show that when
C itself is monoidal, it embeds via a monoidal functor into
C ⥤ C.
Can we use this to show coherence results, e.g. a cheap proof that
λ_ (𝟙_ C) = ρ_ (𝟙_ C)?
I suspect this is harder than is usually made out.
The category of endofunctors of any category is a monoidal category, with tensor product given by composition of functors (and horizontal composition of natural transformations).
m ⊗ n ≅ 𝟙_M, then
F.obj m is a left inverse of
m ⊗ n ≅ 𝟙_M and
n ⊗ m ≅ 𝟙_M (subject to some commuting constraints),
F.obj m and
F.obj n forms a self-equivalence of