May’s conjecture on strictifying symmetric bimonoidal functors for bipermutative categories

Construct a strictification functor at the level of bipermutative categories that, for any symmetric bimonoidal functor between bipermutative categories, yields a strict symmetric bimonoidal functor with properties analogous to May’s up-to-adjunction strictification for symmetric monoidal functors between permutative categories (May’s Theorem 4.3), thereby replacing general symmetric bimonoidal functors by strict ones.

Background

May’s classical result (Theorem 4.3 referenced in the paper) provides an up-to-adjunction strictification of symmetric monoidal functors between permutative categories, enabling replacement by strict symmetric monoidal functors in the context of May’s infinite loop space machine. Extending this paradigm to bipermutative categories entails strictifying symmetric bimonoidal functors, which simultaneously respect additive and multiplicative permutative structures with distributivity.

The paper proves a weaker version that applies only to multiplicatively strong symmetric bimonoidal functors, confirming part of May’s vision but leaving the full conjecture open. The conjecture seeks a general strictification mechanism mirroring the permutative case without imposing additional strength conditions.