Units in lax and oplax monoidal functors
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
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
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.
participants (3)
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David Yetter -
Max Kelly -
street＠mpce.mq.edu.au