arXiv subjects

Nathanael Arkor

Publications and source records attributed to Nathanael Arkor.

12 recordsLinked to original sources

arXiv2026

Presheaves and cocompletions in formal category theory

We study the relationship between presheaf constructions and free cocompletions in the context of formal category theory, elucidating the coincidence between the two concepts in familiar settings. We show that, in a virtual equipment satisfying mild assumptions, free cocompletions under classes of weights are exhibited by presheaf constructions. We furthermore extend the theory of weighted colimits from enriched category theory to this setting, developing the concepts of atomicity and rank, and providing recognition theorems for presheaf objects, free cocompletions, and cocomplete objects. As an application of our methods, we construct free cocompletions, under arbitrary classes of colimit-small weights, of (possibly large) categories enriched in (not necessarily symmetric) monoidal categories and bicategories; this resolves a longstanding omission in the literature on enriched category theory.

math.CT↗

arXiv2026

Exponentiable virtual double categories and presheaves for double categories

Given a pair of pseudo double categories $\mathbb A$ and $\mathbb B$, the lax functors from $\mathbb A$ to $\mathbb B$, along with their transformations, modules, and multimodulations, assemble into a virtual double category $\mathbf{\mathbb Lax}(\mathbb A, \mathbb B)$. We exhibit a universal property of this construction by observing that it arises naturally from the consideration of exponentiability for virtual double categories. In particular, we show that every pseudo double category is exponentiable as a virtual double category, whereby the virtual double category $\mathbf{\mathbb Lax}(\mathbb A, \mathbb B)$ of lax functors arises as the virtual double category $\mathbf{\mathbb Mod}(\mathbb B^{\mathbb A})$ of monads and modules in the exponential $\mathbb B^{\mathbb A}$. We explore some consequences of this characterisation, demonstrating that it leads to simple proofs of statements that heretofore required unwieldy computations. For instance, we deduce that the 2-category of pseudo double categories and lax functors is enriched in the 2-category of normal virtual double categories, and demonstrate that several aspects of the Yoneda theory of pseudo double categories - such as the correspondence between presheaves and discrete fibrations - are substantially simplified by this perspective.

math.CT↗

arXiv2025

The formal theory of relative monads

We develop the theory of relative monads and relative adjunctions in a virtual equipment, extending the theory of monads and adjunctions in a 2-category. The theory of relative comonads and relative coadjunctions follows by duality. While some aspects of the theory behave analogously to the non-relative setting, others require new insights. In particular, the universal properties that define the algebra object and the opalgebra object for a monad in a virtual equipment are stronger than the classical notions of algebra object and opalgebra object for a monad in a 2-category. Inter alia, we prove a number of representation theorems for relative monads, establishing the unity of several concepts in the literature, including the devices of Walters, the $j$-monads of Diers, and the relative monads of Altenkirch, Chapman, and Uustalu. A motivating setting is the virtual equipment $\mathbb{V}\text{-}\mathbf{\mathbb{C}at}$ of categories enriched in a monoidal category $\mathbb{V}$, though many of our results are new even for $\mathbb{V} = \mathbf{Set}$.

math.CT↗

arXiv2025

Idempotence for relative monads

We study the concept of idempotence for relative monads, which exhibits several subtleties not present for non-relative monads. In particular, there is a bifurcation of notions of idempotence in the relative setting, which are indistinguishable for idempotent monads. As a special case, we obtain several characterisations of idempotence for monads in extension form.

math.CT↗

arXiv2025

Magmal characterisations of cocartesian categories

We present a survey of characterisations of cocartesian categories in terms of monoidal categories - and, more generally, magmal categories - satisfying additional properties. In particular, we show that the following are equivalent for a unital magmal category $(\mathcal M, \otimes)$, sharpening several classical characterisations. * $(\mathcal M, \otimes)$ is cocartesian monoidal. * Every object of $\mathcal M$ admits the structure of a unital magma with respect to $\otimes$, such that every morphism is a homomorphism, and a single compatibility condition holds between the magma structures and $\otimes$. * The tensor product functor ${\otimes} \colon \mathcal M \times \mathcal M \to \mathcal M$ admits a right adjoint.

math.CT↗

arXiv2025

Bicategories of algebras for relative pseudomonads

We introduce pseudoalgebras for relative pseudomonads and develop their theory. For each relative pseudomonad $T$, we construct a free--forgetful relative pseudoadjunction that exhibits the bicategory of $T$-pseudoalgebras as terminal among resolutions of $T$. The Kleisli bicategory for $T$ thus embeds into the bicategory of pseudoalgebras as the sub-bicategory of free pseudoalgebras. We consequently obtain a coherence theorem that implies, for instance, that the bicategory of distributors is biequivalent to the 2-category of presheaf categories. In doing so, we extend several aspects of the theory of pseudomonads to relative pseudomonads, including doctrinal adjunction, transport of structure, and lax-idempotence. As an application of our general theory, we prove that, for each class of colimits $Φ$, there is a correspondence between monads relative to free $Φ$-cocompletions, and $Φ$-cocontinuous monads on free $Φ$-cocompletions.

math.CT↗

arXiv2024

Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation

Many structures of interest in two-dimensional category theory have aspects that are inherently strict. This strictness is not a limitation, but rather plays a fundamental role in the theory of such structures. For instance, a monoidal fibration is - crucially - a strict monoidal functor, rather than a pseudo or lax monoidal functor. Other examples include monoidal double categories, double fibrations, and intercategories. We provide an explanation for this phenomenon from the perspective of enhanced 2-categories, which are 2-categories having a distinguished subclass of 1-cells representing the strict morphisms. As part of our development, we introduce enhanced 2-categorical limit sketches and explain how this setting addresses shortcomings in the theory of 2-categorical limit sketches. In particular, we establish the symmetry of internalisation for such structures, entailing, for instance, that a monoidal double category is equivalently a pseudomonoid in an enhanced 2-category of double categories, or a pseudocategory in an enhanced 2-category of monoidal categories.

math.CT↗

arXiv2024

The nerve theorem for relative monads

A fundamental result in the theory of monads is the characterisation of the category of algebras for a monad in terms of a pullback of the category of presheaves on the category of free algebras: intuitively, this expresses that every algebra is a colimit of free algebras. We establish an analogous result for enriched relative monads with dense roots, and explain how it generalises the nerve theorems for monads with arities and nervous monads. As an application, we derive sufficient conditions for the existence of algebraic colimits of relative monads. More generally, we establish such a characterisation of the category of algebras in the context of an exact virtual equipment. In doing so, we are led to study the relationship between a $j$-relative monad $T$ and its associated loose-monad $E(j, T)$, and consequently show that the opalgebra object and the algebra object for $T$ may be constructed from certain double categorical limits and colimits associated to $E(j, T)$.

math.CT↗

arXiv2024

Relative monadicity

We establish a relative monadicity theorem for relative monads with dense roots in a virtual equipment, specialising to a relative monadicity theorem for enriched relative monads. In particular, for a dense $\mathbb V$-functor $j \colon A \to E$, a $\mathbb V$-functor $r \colon D \to E$ is $j$-monadic if and only if $r$ admits a left $j$-relative adjoint and creates $j$-absolute colimits. This provides a refinement of the classical monadicity theorem -- characterising those categories whose objects are given by those of $E$ equipped with algebraic structure -- in which the arities of the algebraic operations are valued in $A$. In particular, when $j = 1$, we recover a formal monadicity theorem. Furthermore, we examine the interaction between the pasting law for relative adjunctions and relative monadicity. As a consequence, we derive necessary and sufficient conditions for the ($j$-relative) monadicity of the composite of a $\mathbb V$-functor with a ($j$-relatively) monadic $\mathbb V$-functor.

math.CT↗

arXiv2024

Adjoint functor theorems for lax-idempotent pseudomonads

For each pair of lax-idempotent pseudomonads $R$ and $I$, for which $I$ is locally fully faithful and $R$ distributes over $I$, we establish an adjoint functor theorem, relating $R$-cocontinuity to adjointness relative to $I$. This provides a new perspective on the nature of adjoint functor theorems, which may be seen as methods to decompose adjointness into cocontinuity and relative adjointness. As special cases, we recover variants of the adjoint functor theorem of Freyd, the multiadjoint functor theorem of Diers, and the pluriadjoint functor theorem of Solian--Viswanathan, as well as the adjoint functor theorems for locally presentable categories. More generally, we recover enriched $Φ$-adjoint functor theorems for weakly sound classes of weight $Φ$.

math.CT↗

arXiv2021

Abstract clones for abstract syntax

We give a formal treatment of simple type theories, such as the simply-typed $λ$-calculus, using the framework of abstract clones. Abstract clones traditionally describe first-order structures, but by equipping them with additional algebraic structure, one can further axiomatize second-order, variable-binding operators. This provides a syntax-independent representation of simple type theories. We describe multisorted second-order presentations, such as the presentation of the simply-typed $λ$-calculus, and their clone-theoretic algebras; free algebras on clones abstractly describe the syntax of simple type theories quotiented by equations such as $β$- and $η$-equality. We give a construction of free algebras and derive a corresponding induction principle, which facilitates syntax-independent proofs of properties such as adequacy and normalization for simple type theories. Working only with clones avoids some of the complexities inherent in presheaf-based frameworks for abstract syntax.

cs.LO↗

arXiv2020

Algebraic models of simple type theories: a polynomial approach

We develop algebraic models of simple type theories, laying out a framework that extends universal algebra to incorporate both algebraic sorting and variable binding. Examples of simple type theories include the unityped and simply-typed $λ$-calculi, the computational $λ$-calculus, and predicate logic. Simple type theories are given models in presheaf categories, with structure specified by algebras of polynomial endofunctors that correspond to natural deduction rules. Initial models, which we construct, abstractly describe the syntax of simple type theories. Taking substitution structure into consideration, we further provide sound and complete semantics in structured cartesian multicategories. This development generalises Lambek's correspondence between the simply-typed $λ$-calculus and cartesian-closed categories, to arbitrary simple type theories.

cs.LO↗