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
.
TODO #
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).
Instances For
Tensoring on the right gives a monoidal functor from C
into endofunctors of C
.
Instances For
If m ⊗ n ≅ 𝟙_M
, then F.obj m
is a left inverse of F.obj n
.
Instances For
If m ⊗ n ≅ 𝟙_M
and n ⊗ m ≅ 𝟙_M
(subject to some commuting constraints),
then F.obj m
and F.obj n
forms a self-equivalence of C
.