2026/07/21 by David Forsman
Mathematics · #math.CT #math.LO #math.QA
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 (Σ,σ) and bialgebraic theories T, providing a uniform language for constructing internal bialgebras in a 2-categorical setting. For every bialgebraic theory T and Σ-model M within a 2-category with PIE limits, we construct the object MT 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 MT is expressed entirely in terms of PIE limits, accessibility, local presentability, orthogonal factorization systems, regularity, and exactness lift along the construction M ↦ MT under suitable assumptions.