Chapter 8 - Hadamard product

Summary

We introduce the Hadamard product on the category of species (relative to a fixed hyperplane arrangement). A key property of this product is that it preserves monoids, comonoids, bimonoids. In fact, for any scalars p and q, the Hadamard product of a p-bimonoid and a q-bimonoid is a pq-bimonoid. Similarly, the Hadamard product of (co)commutative (co)monoids is again (co)commutative. These facts can be seen as formal consequences of the bilax property of the Hadamard functor. We construct the internal hom for the Hadamard product of species, and discuss its bilax property and the related constructions of the convolution monoid, coconvolution comonoid, biconvolution bimonoid. Moreover, we also construct the internal hom for the Hadamard product of monoids, comonoids and bimonoids, making critical use of the fact that these are functor categories just like the category of species. The internal hom for (co, bi)commutative bimonoids is intimately connected to the internal hom for the tensor product of modules over the Birkhoff algebra, Tits algebra, Janus algebra. We construct the universal measuring comonoid from one monoid to another monoid. It allows us to enrich the category of monoids over the category of comonoids. This enriched category possesses powers and copowers which we describe explicitly. The power is in fact the convolution monoid. The copower is a certain quotient of the free monoid on the Hadamard product of the given comonoid and monoid. We introduce the bimonoid of star families. It is constructed out of a cocommutative comonoid and a bimonoid. It builds on the internal hom for the Hadamard product of comonoids. Moreover, it has a commutative counterpart which we call the bicommutative bimonoid of star families. This one builds on the internal hom for cocommutative comonoids. There is also an analogous construction starting with a bimonoid and a commutative monoid which builds on the universal measuring comonoid. These bimonoids play an important role in the study of exp-log correspondences. We introduce the signature functor on species. It is defined by taking Hadamard product with the signed exponential species. The latter carries the structure of a signed bimonoid. This sets up an equivalence between the categories of bimonoids and signed bimonoids.