Units in lax and oplax monoidal functors

Units in lax and oplax monoidal functors

[David Yetter]

I seem to recall that there was some difficulty for the coherence of
lax and oplax monoidal functors (properly so called, meaning with
unit maps in addition to the natural transformation).  Epstein's 1966
paper handles what I would call symmetric semigroupal functors (symmetry,
but no units), and his proof goes through without the symmetry.  I have
also once as an exercise prove the coherence theorem for strong monoidal
functors.  

This query arises tangentially in regard to a paper I am writing on
the deformation theory of monoidal categories and functors.  An answer
is not strictly needed, but I'd be greatful for a reference to the
difficulty, or a corrective to a mis-recollection on my part. (It would
be nice to include in the paper.)

Best Thoughts to all,
David Yetter

Re: Units in lax and oplax monoidal functors

[Max Kelly]

David Yetter asks about coherence results for monoidal functors. These were
studied in the PhD thesis of my then-student Geoff Lewis round about 1971, and
he has an article about them in that Springer Lecture Notes volume - was it
number 129 ? - on coherence in categories, edited by Saunders Mac Lane in the
early 1970s. It is one of the cases covered by the "club" idea, where the free
structure on 1 tells you all about the free structure on any category. Moreover
the case of two monoidal categories and a monoidal functor (lax, of course) is
interesting in that Lewis finds the club COMPLETELY, even though it is false
that "every diagram commutes". What is true, if f is the monoidal functor, is
that a diagram commutes if its codomain has the form f(x), in contrast to say
f(x)of(y) where o is the tensor product. Lewis also studies there the case of
a monoidal f between monoidal CLOSED categories (everything symmetric), getting
for these a PARTIAL determination of the club, like that of Kelly and Mac Lane
for a single symmetric monoidal category.

Max Kelly.

Re: Units in lax and oplax monoidal functors

[Ross Street]

>I seem to recall that there was some difficulty for the coherence of
>lax and oplax monoidal functors (properly so called, meaning with
>unit maps in addition to the natural transformation).

For the lax case, see

Geoffrey Lewis, Coherence for a closed functor, LNM 281 (Springer, 1972)
148-195.

For coherence of strong monoidal (= tensor preserving) functors, see

A. Joyal and R. Street, Braided tensor categories, Advances in Math  102
(1993) 20-78; MR94m:18008.

Best wishes,
Ross