arXiv subjects

Graham Manuell

Publications and source records attributed to Graham Manuell.

At least 19 recordsLinked to original sources

arXiv2026

Clifford semigroups and the monoidal Grothendieck construction

We show that the monoidal Grothendieck construction can be applied to recover the known structure theorem for Clifford monoids, which states that they correspond to functors from a semilattice into the category of groups. Furthermore, we capture the category of Clifford monoids itself as a Grothendieck construction of the functor sending a semilattice L to the functor category [L, Grp] and use this to construct a number of factorisation systems on this category. Finally, we prove a general result on taking monoids in a monoidal fibration and apply it to establish a correspondence between inverse semirings and lax monoidal functors from an idempotent semiring into the category of abelian groups.

math.CT↗

arXiv2026

Generic bundles over a localic category

In this paper we construct classifying localic categories and groupoids for various bundles equipped with logical structure. When these bundles are local homeomorphisms, we recover the localic groupoids that classify geometric theories, demonstrating that these groupoids satisfy a stronger universal property than that of their corresponding classifying toposes. We also prove a dual result that there exist classifying localic categories and groupoids for proper separated bundles satisfying a dual geometric theory. Thus, localic groupoids classify strictly more kinds of logical theories than toposes. Our approach provides a concrete construction of the localic categories and the generic bundles involved in terms of generalised frame presentations. To accommodate our approach, we prove en passant a constructive, pointfree version of the Alexandroff--Hausdorff theorem and that internal functors that are fully faithful and effective descent morphisms on objects induce equivalences between the categories of discrete opfibrations over the source and target categories.

math.CT↗

arXiv2026

Machine Space I: Weak exponentials and quantification over compact spaces

Topology may be interpreted as the study of verifiability, where opens correspond to semi-decidable properties. In this paper we make a distinction between verifiable properties themselves and processes which carry out the verification procedure. The former are simply opens, while we call the latter \emph{machines}. Given a frame presentation $\mathcal{O} X = \langle G \mid R\rangle$ we construct a space of machines $Σ^{Σ^G}$ whose points are given by formal combinations of basic machines corresponding to generators in $G$. This comes equipped with an `evaluation' map making it a weak exponential with base $Σ$ and exponent $X$. When it exists, the true exponential $Σ^X$ occurs as a retract of machine space. We argue this helps explain why some spaces are exponentiable and others not. We then use machine space to study compactness by giving a purely topological version of Escardó's algorithm for universal quantification over compact spaces in finite time. Finally, we relate our study of machine space to domain theory and domain embeddings.

math.GN↗

arXiv2026

Bicategories of Lax Fractions

The well-known calculus of fractions of Gabriel and Zisman provides a convenient way to formally invert morphisms in a category. This was generalised to bicategories by Pronk. We extend these constructions by presenting a calculus of lax fractions for 2-categories that formally turns given morphisms into left adjoint right inverses and given pseudo-commutative squares into Beck-Chevalley squares.

math.CT↗

arXiv2025

Sequences suffice for pointfree uniform completions

Completions of metric spaces are usually constructed using Cauchy sequences. However, this does not work for general uniform spaces, where Cauchy filters or nets must be used instead. The situation in pointfree topology is more straightforward: the correct completion of uniform locales can indeed be obtained as a quotient of a locale of Cauchy sequences.

math.GN↗

arXiv2025

Frobenius reciprocity, modular connections, lattice isomorphism theorem and abstract principal ideals

The purpose of this short note is to fill a gap in the literature: Frobenius reciprocity in the theory of doctrines is closely related to modular connections in projective homological algebra and the notion of a principal element in abstract commutative ideal theory. These concepts are based on particular properties of Galois connections which play an important role also in the abstract study of group-like structures from the perspective of categorical/universal algebra; such role stems from a classical and basic result in group theory: the lattice isomorphism theorem.

math.RA↗

arXiv2024

Pointfree topology and constructive mathematics

The constructive approach to mathematics has the advantage that witnesses can be extracted from statements of existence and theorems can be unwound to give algorithms. Even better, constructive theorems can be interpreted in any topos, giving many different results for the price of one. On the other hand, you might have heard that fundamental results from topology such as Tychonoff's theorem or even the intermediate value theorem do not hold constructively, which can make the price of constructive theorems seem rather steep. However, almost all of these pathologies disappear if we take the pointfree approach to topology, in which spaces are studied algebraically and logically through their lattices of opens without reference to a predefined underlying set of points. In fact, this perspective also sheds light on aspects of constructive mathematics that might at first appear to have little to do with topology. These notes provide a gentle introduction to the main aspects of constructive pointfree topology and some of its applications.

math.GN↗

arXiv2024

The Case for Inverse Semirings

A semiring generalises the notion of a ring, replacing the additive abelian group structure with that of a commutative monoid. In this paper, we study a notion positioned between a ring and a semiring -- a semiring whose additive monoid is a commutative inverse semigroup. These inverse semirings include some important classes of semirings, as well as some new motivating examples. We devote particular attention to the inverse semiring of bounded polynomials and argue for their computational significance. We then prove a number of fundamental results about inverse semirings, their modules and their ideals. Parts of the theory show strong similarities with rings, while other parts are akin to the theory of idempotent semirings or distributive lattices. We note in particular that downward-closed submodules are precisely kernels. We end by exploring a connection to the theory of E-unitary inverse semigroups.

math.RA↗

arXiv2024

The representing localic groupoid for a geometric theory

We give an expository, and hopefully approachable, account of the Joyal-Tierney result that every topos can be represented as a topos of sheaves on a localic groupoid. We give an explicit presentation of a representing localic groupoid for the classifying topos of a given geometric theory and discuss links with the topological groupoids of Forssell.

math.CT↗

arXiv2024

Uniform locales and their constructive aspects

Much work has been done on generalising results about uniform spaces to the pointfree context. However, this has almost exclusively been done using classical logic, whereas much of the utility of the pointfree approach lies in its constructive theory, which can be interpreted in many different toposes. Johnstone has advocated for the development of a constructive theory of uniform locales and wrote a short paper on the basic constructive theory via covering uniformities, but he never followed this up with a discussion of entourage uniformities or completions. We present a more extensive constructive development of uniform locales, including both entourage and covering approaches, their equivalence, completions and some applications to metric locales and localic algebra. Some aspects of our presentation might also be of interest even to classically minded pointfree topologists. These include the definition and manipulation of entourage uniformities using the internal logic of the geometric hyperdoctrine of open sublocales and the emphasis on pre-uniform locales. The latter leads to a description of the completion as the uniform reflection of the pre-uniform locale of Cauchy filters and a new result concerning the completion of pre-uniform localic rings, which can be used to easily lift addition and multiplication on $\mathbb{Q}$ to $\mathbb{R}$ (or $\mathbb{Q}_p$) in the pointfree setting.

math.GN↗

arXiv2023

Presenting quotient locales

It is often useful to be able to deal with locales in terms of presentations of their underlying frames, or equivalently, the geometric theories which they classify. Given a presentation for a locale, presentations for its sublocales can be obtained by simply appending additional relations, but the case of quotient locales is more subtle. We provide simple procedures for obtaining presentations of open quotients, proper quotients or general triquotients from presentations of the parent locale. The results are proved with the help of the suplattice, preframe and dcpo coverage theorems and applied to obtain presentations of the circle from ones for the reals and the closed unit interval.

math.AG↗

arXiv2023

Weakly Schreier extensions for general algebras

Weakly Schreier split extensions are a reasonably large, yet well-understood class of monoid extensions, which generalise some aspects of split extensions of groups. This short note provides a way to define and study similar classes of split extensions in general algebraic structures (parameterised by a term $θ$). These generalise weakly Schreier extensions of monoids, as well as general extensions of semi-abelian varieties (using the $θ$ appearing in their syntactical characterisation). Restricting again to the case of monoids, a different choice of $θ$ leads to a new class of monoid extensions, more general than the weakly Schreier split extensions.

math.RA↗

arXiv2022

Artin glueings of toposes as adjoint split extensions

Artin glueings of frames correspond to adjoint split extensions in the category of frames and finite-meet-preserving maps. We extend these ideas to the setting of toposes and show that Artin glueings of toposes correspond to a 2-categorical notion of adjoint split extensions in the 2-category of toposes, finite-limit-preserving functors and natural transformations. A notion of morphism between these split extensions is introduced, which allows the category Ext(H,N) to be constructed. We show that Ext(H,N) is contravariantly equivalent to Hom(H,N), and moreover, that this can be extended to a 2-natural contravariant equivalence between the Hom 2-functor and a naturally defined Ext 2-functor.

math.CT↗

arXiv2022

Monoid extensions and the Grothendieck construction

In category theory circles it is well-known that the Schreier theory of group extensions can be understood in terms of the Grothendieck construction on indexed categories. However, it is seldom discussed how this relates to extensions of monoids. We provide an introduction to the generalised Grothendieck construction and apply it to recover classifications of certain classes of monoid extensions (including Schreier and weakly Schreier extensions in particular).

math.CT↗

arXiv2022

The spectrum of a localic semiring

A number of spectrum constructions have been devised to extract topological spaces from algebraic data. Prominent examples include the Zariski spectrum of a commutative ring, the Stone spectrum of a bounded distributive lattice, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann-Lawson spectrum of a continuous frame. Inspired by the examples above, we define a spectrum for localic semirings. We use arguments in the symmetric monoidal category of suplattices to prove that, under conditions satisfied by the aforementioned examples, the spectrum can be constructed as the frame of overt weakly closed radical ideals and that it reduces to the usual constructions in those cases. Our proofs are constructive. Our approach actually gives 'quantalic' spectrum from which the more familiar localic spectrum can then be derived. For a discrete ring this yields the quantale of ideals and in general should contain additional 'differential' information about the semiring.

math.RA↗

arXiv2022

Quantalic spectra of semirings

Spectrum constructions appear throughout mathematics as a way of constructing topological spaces from algebraic data. Given a commutative localic semiring R (the pointfree analogue of a topological semiring), we define a spectrum of R which generalises the Stone spectrum of a distributive lattice, the Zariski spectrum of a commutative ring, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann-Lawson spectrum of a continuous frame. We then provide an explicit construction of this spectrum under conditions on R which are satisfied by our main examples. Our results are constructively valid and hence admit interpretation in any elementary topos with natural number object. For this reason the spectrum we construct should actually be a locale instead of a topological space. A simple modification to our construction gives rise to a quantic spectrum in the form of a commutative quantale. Such a quantale contains `differential' information in addition to the purely topological information of the localic spectrum. In the case of a discrete ring, our construction produces the quantale of ideals. This prompts us to study the quantale of ideals in more detail. We discuss some results from abstract ideal theory in the setting of quantales and provide a tentative definition for what it might mean for a quantale to be nonsingular by analogy to commutative ring theory.

math.RA↗

arXiv2022

Quotients of monoid extensions and their interplay with Baer sums

Cosetal extensions of monoids generalise extensions of groups, special Schreier extensions of monoids and Leech's normal extensions of groups by monoids. They share a number of properties with group extensions, including a notion of Baer sum when the kernel is abelian. However, unlike group extensions (with fixed kernel and cokernel) there may be nontrivial morphisms between them. We explore the structure of the category of cosetal extensions and relate it to an analogue of second cohomology groups. Finally, the order structure and additive structures are combined to give an indexed family of inverse semigroups of extensions. These in turn can be combined into an inverse category.

math.RA↗

arXiv2022

Congruence frames of frames and $κ$-frames

We describe the congruence lattices of frames and $κ$-frames. We look at the role that congruence biframes play in the category of strictly zero-dimensional biframes and discuss some reflections and coreflections of congruence frames.

math.RA↗