A bialgebra axiom and the Dold-Kan correspondence

TL;DR

This paper introduces a new bialgebra axiom for functors between symmetric monoidal categories, demonstrating its validity for the normalized chain complex functor in the Dold-Kan correspondence, with implications for future applications.

Contribution

It defines a bialgebra axiom for pairs of monoidal structures on functors and proves its validity for the normalized chain complex functor in the Dold-Kan correspondence.

Findings

The bialgebra axiom weakens the requirement for a functor to be strict symmetric monoidal.

The axiom is preserved under passing to adjoint functors and categories of monoids.

The axiom holds for the Alexander-Whitney and Eilenberg-MacLane structures in the Dold-Kan correspondence.

Abstract

We introduce a bialgebra axiom for a pair $(c,\ell)$ of a colax-monoidal and a lax-monoidal structures on a functor $F\colon \mathscr{M}_1\to \mathscr{M}_2$ between two (strict) symmetric monoidal categories. This axiom can be regarded as a weakening of the property of $F$ to be a strict symmetric monoidal functor. We show that this axiom transforms well when passing to the adjoint functor or to the categories of monoids. Rather unexpectedly, this axiom holds for the Alexander-Whitney colax-monoidal and the Eilenberg-MacLane lax-monoidal structures on the normalized chain complex functor in the Dold-Kan correspondence. This fact, proven in Section 2, opens up a way for many applications, which we will consider in our sequel paper(s).