arXiv subjects

Tim Van der Linden

Publications and source records attributed to Tim Van der Linden.

At least 19 recordsLinked to original sources

arXiv2026

A two-categorical Snake Lemma

We prove a Snake Lemma for 2-categories. Working in a 2-category with a strong bizero object, we develop 2-kernels and 2-cokernels, 2-monomorphisms and 2-epimorphisms as fully faithful and cofully faithful 1-cells, short 2-exact sequences and normal image factorisations, and we prove a two-dimensional Normal Short Five Lemma. We then introduce the dinversion of an antinormal pair and the notion of a homologically self-dual 2-category, characterised equally by the self-duality of homology, by a Pure Snake Lemma and by a Third Isomorphism Property. Two-dimensional di-exactness implies homological self-duality. Our main result is the Snake Lemma in a 2-di-exact 2-category: a ladder of 2-exact rows with normal verticals induces a 2-exact six-term sequence, with a connecting 1-cell that is 2-natural in the ladder. Di-exactness can be traded for two hypotheses that are not self-dual: that dinversion preserve normality, and that normal 2-epimorphisms compose. Everything specialises, on passing to a locally discrete 2-category, to its classical counterpart. We close by exhibiting three models: a 2-di-exact 2-category of abelian categories containing $\mathsf{Coh}(X)$ for every noetherian scheme $X$; the locally ordered 2-category of complete modular lattices, in which 2-di-exactness amounts to Dedekind's transposition principle; and the 2-category of Hilbert lattices, which is not 2-di-exact but satisfies the non-self-dual hypotheses, by a theorem of Mackey on pairs of closed subspaces.

math.CT↗

arXiv2026

Intrinsic tensor products and a Ganea-type extension of the five-term exact sequence

We define an intrinsic symmetric bi-right-exact (and for varieties, bi-cocontinuous) bilinear product on objects of a semi-abelian category, constructed as the cosmash product in the two-nilpotent reflection. When applied to abelian objects, this recovers classical tensor products in many cases. A recognition theorem states that any symmetric bi-cocontinuous bifunctor on an abelian variety of algebras is realised as the bilinear product in the variety of algebras over a suitable 2-nilpotent symmetric operad in the monoidal category of abelian groups. For abelian groups replaced with any commutative ring, the bilinear product of algebras over such an operad is associative as long as the only unary operations are given by multiplication with scalars, but not in general. This relies on a right-exactness theorem for cross-effects of bifunctors, and consequently for cosmash products. We develop basic properties, compare the bilinear product to the Brown-Loday non-abelian tensor product, and prove a categorical version of Ganea's six-term exact homology sequence. We further characterise abelian extensions via internal action cores, obtaining explicit descriptions of bilinear products in categories of representations; in particular, the bilinear product of the associated Beck modules generalises the classical tensor product of representations for groups and Lie algebras.

math.CT↗

arXiv2026

Weak action representability of 2-nilpotent groups

In this article, we investigate the representability of actions of the category $\mathsf{Nil}_2(\mathsf{Grp})$ of $2$-nilpotent groups. We first provide an algebraic characterisation of derived actions in $\mathsf{Nil}_2(\mathsf{Grp})$ by determining a universal strict general actor of an object $X$, which turns out to be the group $\operatorname{Aut}_c(X)$ of central automorphisms of $X$. We also characterise the morphisms $B \to \operatorname{Aut}_c(X)$ that define an action of $B$ on $X$ in $\mathsf{Nil}_2(\mathsf{Grp})$. We then show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not action representable, and that the existence of a weak representation is related to the amalgamation property. Using the construction of an amalgam of a suitable family of abelian subgroups of $\operatorname{Aut}_c(X)$, we prove that the category $\mathsf{Nil}_2(\mathsf{Grp})$ is weakly action representable, and that a weak representing object can be chosen to be an abelian group. Finally, we show that $\mathsf{Nil}_2(\mathsf{Grp})$ is not locally algebraically cartesian closed.

math.CT↗

arXiv2025

Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context

Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper, we show how the context of a `noetherian form', recently introduced by the second and third authors, allows a self-dual treatment of these lemmas even in the case of non-abelian categories of group-like structures. This context covers a wide range of examples: module categories, the category of groups, of graded abelian groups, the categories of Lie algebras, of cocommutative Hopf algebras, the category of Heyting semilattices, of loops, the dual of the category of pointed sets, the category of modular/distributive lattices and modular connections, the category of sets and partial bijections, and many others. More generally, it includes all semi-abelian and Grandis exact categories.

math.CT↗

arXiv2025

A comparison between weakly protomodular and protomodular objects in unital categories

We compare the concepts of protomodular and weakly protomodular objects within the context of unital categories. Our analysis demonstrates that these two notions are generally distinct. To establish this, we introduce left pseudocancellative unital magmas and characterise weakly protomodular objects within the variety of algebras they constitute. Subsequently, we present an example of a weakly protomodular object that is not protomodular in this category.

math.CT↗

arXiv2025

From Abelianization to Tangent Categories

A tangent category is a category with an endofunctor, called the tangent bundle functor, which is equipped with various natural transformations that capture essential properties of the classical tangent bundle of smooth manifolds. In this paper, we show that, surprisingly, the category of groups is a tangent category whose tangent bundle functor is induced by abelianization and whose differential bundles correspond to abelian groups. We generalize this construction by introducing the concept of linear assignments, which are endofunctors assigning to every object a commutative monoid in a natural and idempotent manner. We then show that a linear assignment induces a tangent bundle functor, whose differential bundles correspond to a notion of linear algebras. We show that any finitely cocomplete regular unital category is a tangent category whose tangent bundle functor is induced by the canonical abelianization functor, which is a monadic linear assignment. This allows us to provide multiple new examples of tangent categories including monoids, pointed magmas, loops, non-unital rings, Jónsson--Tarski varieties, and pointed Mal'tsev varieties.

math.CT↗

arXiv2025

Categorical-algebraic aspects of Heyting semilattices

This article gives an overview of some key categorical-algebraic properties of the variety of Heyting semilattices, with the aim of correcting a misconception in the literature. We confirm that the category of Heyting semilattices is not algebraically coherent, even though it satisfies a strong version of the so-called Smith is Huq condition (on the equivalence of two types of commutators). We also prove that Higgins commutators of normal subobjects are normal, as a consequence of the fact that Heyting semilattices form an arithmetical category. We provide an elementary characterisation of when a pair of subobjects commutes, and use this in the construction of two counterexamples. We further show that centralisers exist, centralisers of normal monomorphisms are normal monomorphisms, and normal monomorphisms are closed under composition. We study the latter condition in detail. On the other hand, we show that the category of Heyting semilattices does not satisfy normality of unions. Hence, it is not action accessible and so it does not admit all normalisers. In particular, this means that the known implication between action accessibility and the condition requiring the existence of centralisers of normal monomorphisms which are themselves normal, is strict.

math.CT↗

arXiv2025

The cohomology objects of a semi-abelian variety are small

A well-known, but often ignored issue in Yoneda-style definitions of cohomology objects via collections of $n$-step extensions (i.e., equivalence classes of exact sequences of a given length $n$ between two given objects, usually subject to further criteria, and equipped with some algebraic structure) is, whether such a collection of extensions forms a set. We explain that in the context of a semi-abelian variety of algebras, the answer to this question is, essentially, yes: for the collection of all $n$-step extensions between any two objects, a set of representing extensions can be chosen, so that the collection of extensions is "small" in the sense that a bijection to a set exists. We further consider some variations on this result, involving double extensions and crossed extensions (in the context of a semi-abelian variety), and Schreier extensions (in the category of monoids).

math.CT↗

arXiv2025

Weak representability of actions of non-associative algebras

We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field. Our first aim is to give a complete characterization of action accessible, operadic quadratic varieties of non-associative algebras which satisfy an identity of degree two and to study the representability of actions for them. Here we prove that the varieties of two-step nilpotent (anti-)commutative algebras and that of commutative associative algebras are weakly action representable, and we explain that the condition (WRA) is closely connected to the existence of a so-called amalgam. Our second aim is to work towards the construction, still within the context of algebras over a field, of a weakly representing object $E(X)$ for the actions on (or split extensions of) an object $X$. We actually obtain a partial algebra $E(X)$, which we call external weak actor of $X$, together with a monomorphism of functors ${\operatorname{SplExt}(-,X) \rightarrowtail \operatorname{Hom}(U(-),E(X))}$, which we study in detail in the case of quadratic varieties. Furthermore, the relations between the construction of the universal strict general actor $\operatorname{USGA}(X)$ and that of $E(X)$ are described in detail. We end with some open questions.

math.CT↗

arXiv2025

Non-additive derived functors via chain resolutions

Let $F\colon \mathcal{C} \to \mathcal{E}$ be a functor from a category $\mathcal{C}$ to a homological (Borceux-Bourn) or semi-abelian (Janelidze-Márki-Tholen) category $\mathcal{E}$. We investigate conditions under which the homology of an object $X$ in $\mathcal{C}$ with coefficients in the functor $F$, defined via projective resolutions in $\mathcal{C}$, remains independent of the chosen resolution. Consequently, the left derived functors of $F$ can be constructed analogously to the classical abelian case. Our approach extends the concept of chain homotopy to a non-additive setting using the technique of imaginary morphisms. Specifically, we utilize the approximate subtractions of Bourn-Janelidze, originally introduced in the context of subtractive categories. This method is applicable when $\mathcal{C}$ is a pointed regular category with finite coproducts and enough projectives, provided the class of projectives is closed under protosplit subobjects, a new condition introduced in this article and naturally satisfied in the abelian context. We further assume that the functor $F$ meets certain exactness conditions: for instance, it may be protoadditive and preserve proper morphisms and binary coproducts - conditions that amount to additivity when $\mathcal{C}$ and $\mathcal{E}$ are abelian categories. Within this framework, we develop a basic theory of derived functors, compare it with the simplicial approach, and provide several examples.

math.CT↗

arXiv2024

A universal Kaluzhnin--Krasner embedding theorem

Given two groups $A$ and $B$, the Kaluzhnin--Krasner universal embedding theorem states that the wreath product $A\wr B$ acts as a universal receptacle for extensions from $A$ to $B$. For a split extension, this embedding is compatible with the canonical splitting of the wreath product, which is further universal in a precise sense. This result was recently extended to Lie algebras and to cocommutative Hopf algebras. The aim of the present article is to explore the feasibility of adapting the theorem to other types of algebraic structures. By explaining the underlying unity of the three known cases, our analysis gives necessary and sufficient conditions for this to happen. From those we may for instance conclude that a version for crossed modules can indeed be attained, while the theorem cannot be adapted to, say, associative algebras, Jordan algebras or Leibniz algebras, when working over an infinite field: we prove that then, amongst non-associative algebras, only Lie algebras admit a universal Kaluzhnin--Krasner embedding theorem.

math.CT↗

arXiv2024

A Homological View of Categorical Algebra

We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic selection of suitable axiomatic frameworks, some well known and some new, and (b) the development of categorical tools for effective reasoning and computing within those frameworks. The selection of axiomatic frameworks begins `from the ground up' with z-exact categories. These are pointed categories in which every morphism has a kernel and cokernel. Then we progress all the way to abelian categories, en route meeting contexts such as Borceux--Bourn homological categories and Janelidze--Márki--Tholen semiabelian categories. We clarify the relationship between these axiomatic frameworks by direct comparison, but also by explaining how concrete examples fit into the selection. The outcome is a fine-grained set of criteria by which one can map varieties of algebras (in the sense of Universal Algebra) and topological models of algebraic theories into the various frameworks. The categorical tools for effective computation deal mostly with situations involving universal factorizations of morphisms, with exact sequences, and with the homology of chain complexes. Further, the categorical tools for computation include the `basic diagram lemmas' of homological algebra, that is the (Short) $5$-Lemma, the $(3\times 3)$-Lemma, the Snake Lemma, and applications thereof. We find that these tools are even available in some surprisingly weak categorical environments, such as the category of pointed sets. In discovering such features, we make systematic use of what we call the self-dual axis of a category.

math.CT↗

arXiv2023

Schur- and Baer-type theorems for Lie and Leibniz algebras

The aim of this article is to obtain variations on the classical theorems of Schur and Baer on finiteness of commutator subgroups, valid in the contexts of Lie algebras and Leibniz algebras over a field. Using non-abelian tensor products and exterior products, we prove Schur's Theorem for finitely generated Leibniz algebras, both Schur's Theorem and Baer's Theorem for finitely generated Lie algebras, and a version of these theorems for finitely presented Lie algebras.

math.RA↗

arXiv2023

Twisted Commutators and Internal Crossed Modules

We introduce a notion of relative commutator -- an important special case being commutators twisted by an action -- as a straightforward modification of the definition of the Higgins commutator, establish its relation with a new notion of commutativity -- also obtained as a modification of the usual notion -- and show how we can use it to characterise internal crossed modules in the context of a semi-abelian category.

math.CT↗

arXiv2023

Associativity and the cosmash product in operadic varieties of algebras

In this article, we characterise the operadic variety of commutative associative algebras over a field via a (categorical) condition: the associativity of the so-called cosmash product. This condition, which is closely related to commutator theory, is quite strong: for example, groups do not satisfy it. However, in the case of commutative associative algebras, the cosmash product is nothing more than the tensor product; which explains why in this case it is associative. We prove that in the setting of operadic varieties of algebras over a field, it is the only example. Further examples in the non-operadic case are also discussed.

math.RA↗

arXiv2022

A symmetric approach to higher coverings in categorical Galois theory

In the context of a tower of (strongly Birkhoff) Galois structures in the sense of categorical Galois theory, we show that the concept of a higher covering admits a characterisation which is at the same time absolute (with respect to the base level in the tower), rather than inductively defined relative to extensions of a lower order; and symmetric, rather than depending on a perspective in terms of arrows pointing in a certain chosen direction. This result applies to the Galois theory of quandles, for instance, where it helps us characterising the higher coverings in purely algebraic terms.

math.CT↗

arXiv2021

Intrinsic Schreier special objects

Motivated by the categorical-algebraic analysis of split epimorphisms of monoids, we study the concept of a special object induced by the intrinsic Schreier split epimorphisms in the context of a regular unital category with binary coproducts, comonadic covers and a natural imaginary splitting in the sense of our article [Intrinsic Schreier split extensions, Appl. Categ. Structures 28 (2020), 517--538]. In this context, each object comes naturally equipped with an imaginary magma structure. We analyse the intrinsic Schreier split epimorphisms in this setting, showing that their properties improve when the imaginary magma structures happen to be associative. We compare the intrinsic Schreier special objects with the protomodular objects, and characterise them in terms of the imaginary magma structure. We furthermore relate them to the Engel property in the case of groups and Lie algebras.

math.CT↗

arXiv2021

Algebras with representable representations

Just like group actions are represented by group automorphisms, Lie algebra actions are represented by derivations: up to isomorphism, a split extension of a Lie algebra $B$ by a Lie algebra $X$ corresponds to a Lie algebra morphism $B\to \mathit{Der}(X)$ from $B$ to the Lie algebra $\mathit{Der}(X)$ of derivations on $X$. In this article, we study the question whether the concept of a derivation can be extended to other types of non-associative algebras over a field $\mathbb{K}$, in such a way that these generalised derivations characterise the $\mathbb{K}$-algebra actions. We prove that the answer is no, as soon as the field $\mathbb{K}$ is infinite. In fact, we prove a stronger result: already the representability of all abelian actions -- which are usually called representations or Beck modules -- suffices for this to be true. Thus we characterise the variety of Lie algebras over an infinite field of characteristic different from $2$ as the only variety of non-associative algebras which is a non-abelian category with representable representations. This emphasises the unique role played by the Lie algebra of linear endomorphisms $\mathfrak{gl}(V)$ as a representing object for the representations on a vector space $V$.

math.RA↗