arXiv subjects

David Forsman

Publications and source records attributed to David Forsman.

6 recordsLinked to original sources

arXiv2026

A Syntactic Approach to Ulmer's Bialgebras

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.

math.CT↗

arXiv2026

On the Semi-Abelianness of Affine Group Schemes

We prove that the category of commutative Hopf algebras over a field $k$ is co-semi-abelian. Consequently, the category of affine group $k$-schemes is semi-abelian. We establish coregularity by identifying the orthogonal factorization system of surjections and faithfully flat injections, and we deduce coexactness from Takeuchi's correspondence between normal Hopf ideals and Hopf subalgebras of commutative Hopf $k$-algebras.

math.CT↗

arXiv2025

A Categorical Approach to Finiteness Conditions

We introduce a general categorical framework for finiteness conditions that unifies classical notions such as Noetherianness, Artinianness, and various forms of topological compactness. This is achieved through the concept of \textbf{$\tau$-compactness}, defined relative to a \textbf{coverage} $\tau$. A coverage on a category $C$ is a specified class of covering diagrams, which are functors $F\colon I \to C/c$ of a specified variance, where the indexing category $I$ is equipped with a set of 'designated small objects'. An object $c$ is $\tau$-compact if every such covering diagram over it stabilizes at some designated small object. As we permit functors of mixed variance, our framework simultaneously models ascending chain conditions (such as Noetherianness) and descending chain conditions (such as Artinianness, topological compactness via closed sets). The role of protomodularity of the ambient category emerges as a crucial property for proving strong closure results. Under suitable compatibility assumptions on the coverage, we show that in a protomodular category, the class of $\tau$-compact objects is closed under quotients and extensions. In a pointed context, this implies closure under finite products, generalizing the classical theorem that a finitely generated module over a Noetherian ring is itself Noetherian. We also leverage protomodularity to establish a categorical Hopfian property for Noetherian objects. Our main application shows that in any regular protomodular category with an initial object, the classes of Noetherian and Artinian objects are closed under subobjects, regular quotients, and extensions. As a consequence, in any abelian category, these classes of objects form exact subcategories.

math.CT↗

arXiv2025

Semi-abelian by Design: Johnstone Algebras Unifying Implication and Division

Johnstone demonstrated that Heyting semilattices form a semi-abelian category via a specific triple of terms. Inspired by this work, we introduce \emph{Johnstone algebras} or J-algebras. The algebraic $(*,\to,e)$-theory $J$ of arities $(2,2,0)$ consists of three axioms carefully chosen to ensure protomodularity in alignment with Johnstone's terms. Johnstone algebras generalize well-known structures such as groups (division) and Heyting semilattices (implication) providing a unified framework within the well-behaved setting of semi-abelian categories. We present two primary contributions. First, we identify the M-axiom, \[ (t(x,y)\to x)\to (t(x,y)\to z) \approx x\to z, \text{ where }t(x,y) = (x\to y)\to y. \] The M-axiom is satisfied by residuated Johnstone algebras, and it can be considered a weakening of the H-axiom to comparable elements. We show that $t(x,y)$ defines a \emph{relative closure term} in MBC-algebras, and it implies that MBC-algebras form a variety of algebras, thereby generalizing the corresponding theorem related to HBCK-algebras. Second, we prove several no-go results, demonstrating that balanced theories or theories admitting non-discrete monotone or inflationary algebras cannot possess Malcev terms. Together, these results establish Johnstone algebras as significant structures that achieve desirable categorical properties by carefully integrating both logical and symmetric features, while closely avoiding the constraints imposed by our no-go results.

math.LO↗

arXiv2024

On the Multicategorical Meta-Theorem and the Completeness of Restricted Algebraic Deduction Systems

Eight categorical soundness and completeness theorems are established within the framework of algebraic theories. Exactly six of the eight deduction systems exhibit complete semantics within the cartesian monoidal category of sets. The multicategorical meta-theorem via soundness and completeness enables the transference of properties of families of models from the cartesian monoidal category of sets to $\Delta$-multicategories $C$. A bijective correspondence $R \mapsto \Delta_R$ is made between context structures $R$ and structure categories $\Delta$, which are wide subcategories of $\textbf{FinOrd}$ consisting of finite ordinals and functions. Given a multisorted signature $\sigma$ with a context structure $R$, an equational deduction system $\vdash_R$ is constructed for $R$-theories. The models within $\Delta_R$-multicategories provide a natural semantic framework for the deduction system $\vdash_R$ for modelable context structures $R$. Each of the eight modelable context structures $R$ is linked with a soundness and completeness theorem for the deduction system $\vdash_R$.

math.CT↗

arXiv2023

Functors of Variance: Heuristic Naturality and Connection to Ends

The concept of a variance on a category is introduced as a two-sided strict factorization system. By employing variances, we define functors of variance in a more general setting than is usually considered, thereby eliminating the need for their domains to be product categories. Heuristic natural transformations are defined between functors of variance, whose domains are connected via a span. Heuristic naturality encompasses various known types of naturality, such as diagonal, extraordinary, and twisted naturality. We demonstrate the connection between heuristic naturality and a generalized comma category, revealing the bijective correspondence between heuristic natural transformations and sections of the forgetful functor from the generalized comma category. Moreover, we introduce the concept of limit with respect to the notion of a heuristic transformation. This is the notion of end. It is shown that ends can be calculated via products and equalizers, and the Fubini theorem still holds for ends.

math.CT↗