Published online by Cambridge University Press: 28 February 2020
We discuss some important rigidity theorems related to universal constructions. They usually take the form of an adjoint equivalence between suitable categories. The Loday-Ronco theorem says that the category of 0-bimonoids is equivalent to the category of species. In particular, 0-bimonoids are both free and cofree. This theorem is a special case of a more general result in which 0-bimonoids are replaced by q-bimonoids with q not a root of unity. We refer to this result as the rigidity of q-bimonoids. Invertibility of the Varchenko matrix associated to the q-distance function on faces plays a critical role here. The Leray-Samelson theorem says that the category of bicommutative bimonoids is equivalent to the category of species. In particular, bicommutative bimonoids are both free commutative and cofree cocommutative. There is also a signed analogue of Lerayâ€“Samelson which applies to signed bicommutative signed bimonoids. The Borelâ€“Hopf theorem says that any cocommutative bimonoid is cofree on its primitive part, and dually, any commutative bimonoid is free on its indecomposable part. This result also has a signed analogue which applies to signed (co)commutative signed bimonoids. We present three broad approaches to these rigidity theorems. The first approach is elementary and proceeds by an induction on the primitive filtration of the bimonoid. Here a key role is played by how the bimonoid axiom works on the primitive part. The second approach is more direct and proceeds by constructing an explicit inverse to the appropriate universal map. The universal map is defined using a zeta function and the inverse using a Möbius function. These maps have connections to the exponential and logarithm operators. The third approach is also constructive and employs (commutative, usual or two-sided) characteristic operations by suitable families of idempotents in the (Birkhoff, Tits or q-Janus) algebra, respectively, to decompose the given bimonoid. All results in this chapter are independent of the characteristic of the base field.