arXiv subjects

Jan Jurka

Publications and source records attributed to Jan Jurka.

4 recordsLinked to original sources

arXiv2024

Algebraic Reasoning over Relational Structures

Many important computational structures involve an intricate interplay between algebraic features (given by operations on the underlying set) and relational features (taking account of notions such as order or distance). This paper investigates algebras over relational structures axiomatized by an infinitary Horn theory, which subsume, for example, partial algebras, various incarnations of ordered algebras, quantitative algebras introduced by Mardare, Panangaden, and Plotkin, and their recent extension to generalized metric spaces and lifted algebraic signatures by Mio, Sarkis, and Vignudelli. To this end, we develop the notion of clustered equation, which is inspired by Mardare et al.'s basic conditional equations in the theory of quantitative algebras, at the level of generality of arbitrary relational structures, and we prove that it is equivalent to an abstract categorical form of equation earlier introduced by Milius and Urbat. Our main results are a family of Birkhoff-type variety theorems (classifying the expressive power of clustered equations) and an exactness theorem (classifying abstract equations by a congruence property).

cs.LO↗

arXiv2024

An enriched small object argument over a cofibrantly generated base

The small object argument is a method for transfinitely constructing weak factorization systems originally motivated by homotopy theory. We establish a variant of the small object argument that is enriched over a cofibrantly generated weak factorization system. This enriched variant of the small object argument subsumes the ordinary small object argument for categories and also certain variants of the small object argument for 2-categories, (2,1)-categories, dg-categories and simplicially enriched categories.

math.CT↗

arXiv2022

Constructing spectra using cone injectivity

We provide a generalization of the construction of a spectrum of a commutative ring as a locally ringed space, applicable to cone injectivity classes in general contexts, especially in locally finitely presentable categories. In its full generality, the spectrum functor fails to be fully faithful and we study reasonable sufficient conditions under which it is. Further, assuming the full faithfulness, we introduce a generalization of another concept from algebraic geometry -- the functor of points -- and prove equivalence of the two resulting notions of schemes.

math.CT↗

arXiv2020

Are chain-complete posets co-wellpowered?

We show that the category of chain-complete posets is nearly locally presentable, we give a characterization of strong epimorphisms, and we offer an alternative proof of co-wellpoweredness.

math.CT↗