A Syntactic Approach to Ulmer's Bialgebras

Abstract

Ulmer introduced a semantic notion of bialgebras that unifies a broad class of algebraic and coalgebraic structures. We develop a syntactic counterpart by introducing signature pairs $(\Sigma,\sigma)$ and bialgebraic theories $T$, providing a uniform language for constructing internal bialgebras in a $2$-categorical setting. For every bialgebraic theory $T$ and $\Sigma$-model $M$ within a $2$-category with PIE limits, we construct the object $M^T$ of internal $T$-bialgebras. Our approach to bialgebras admits a general Induced Functor of Algebras Theorem extending the classical lifting of lax monoidal functors to the categories of internal monoids. Since the construction of $M^T$ is expressed entirely in terms of PIE limits, accessibility, local presentability, orthogonal factorization systems, regularity, and exactness lift along the construction $M \mapsto M^T$ under suitable assumptions.