arXiv subjects

Ülo Reimaa

Publications and source records attributed to Ülo Reimaa.

4 recordsLinked to original sources

arXiv2026

Exactness of the 2-categories of abelian and triangulated categories

We introduce a notion of $2$-homological category modelled on pointed homological categories in the sense of Grandis, using bizero objects whose null $1$-cells are zero objects in the hom-categories for formulating $2$-dimensional pointedness. We first prove directly that the $2$-category of di-exact homological categories, which generalize Puppe exact categories, functors preserving all kernels and cokernels, and arbitrary natural transformations is $2$-homological. Its normal subcategories are saturated thick subcategories, and its exact quotients are constructed by a complete three-arrow fraction calculus similar to the one known for Puppe exact categories. We then adapt this proof to prove that the $2$-category of triangulated categories is also $2$-homological; here, instead of the ternary fractions we use the well-known Verdier fractions. Abstracting the common quotient structure of these proofs yields a general criterion, using which we further establish that the $2$-categories of pointed, additive and abelian categories are also $2$-homological. The criterion also applies to categories enriched in semimodules over a fixed commutative rig, with a zero object, and to their full sub-$2$-category with finite biproducts. Their quotients are linear congruence quotients. Combining enrichment with Puppe exactness gives further examples whose quotients are exact linear localizations, including linear abelian categories as the finite-biproduct case. In the abelian and triangulated cases the normal subcategories and quotients are, respectively, Serre subcategories and Serre quotients, and thick triangulated subcategories and Verdier localizations.

math.CT↗

arXiv2025

A Structural Account of Combinatory Completeness

We give a general notion of combinatory completeness with respect to a faithful cartesian club and use it systematically to obtain characterisations of a number of different kinds of applicative system. Moreover, we characterise combinatory completeness of a given applicative system in terms of multicategory structure on its computable maps.

math.CT↗

arXiv2024

Fibrations of algebras

We study fibrations arising from indexed categories of the following form: fix two categories $\mathcal{A},\mathcal{X}$ and a functor $F : \mathcal{A} \times \mathcal{X} \longrightarrow\mathcal{X} $, so that to each $F_A=F(A,-)$ one can associate a category of algebras $\mathbf{Alg}_\mathcal{X}(F_A)$ (or an Eilenberg-Moore, or a Kleisli category if each $F_A$ is a monad). We call the functor $\int^{\mathcal{A}}\mathbf{Alg} \to \mathcal{A}$, whose typical fibre over $A$ is the category $\mathbf{Alg}_\mathcal{X}(F_A)$, the "fibration of algebras" obtained from $F$. Examples of such constructions arise in disparate areas of mathematics, and are unified by the intuition that $\int^\mathcal{A}\mathbf{Alg} $ is a form of semidirect product of the category $\mathcal{A}$, acting on $\mathcal{X}$, via the `representation' given by the functor $F : \mathcal{A} \times \mathcal{X} \longrightarrow\mathcal{X}$. After presenting a range of examples and motivating said intuition, the present work focuses on comparing a generic fibration with a fibration of algebras: we prove that if $\mathcal{A}$ has an initial object, under very mild assumptions on a fibration $p : \mathcal{E}\longrightarrow \mathcal{A}$, we can define a canonical action of $\mathcal{A}$ letting it act on the fibre $\mathcal{E}_\varnothing$ over the initial object. This result bears some resemblance to the well-known fact that the fundamental group $π_1(B)$ of a base space acts naturally on the fibers $F_b = p^{-1}b$ of a fibration $p : E \to B$.

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↗