arXiv subjects

Gabriele Lobbia

Publications and source records attributed to Gabriele Lobbia.

9 recordsLinked to original sources

arXiv2025

Canonical differential calculi via functorial geometrization

Given a category $\mathcal{E}$, we establish sufficient conditions on a faithful isofibration $\mathcal{E}\rightarrow\operatorname{Mon}(\mathcal{V})$ valued in the category of monoids internal to a monoidal additive category $\mathcal{V}$ such that $\mathcal{E}$ admits a canonical functor to the category of first order differential calculi in $\mathcal{V}$. Generalizing the procedure of extending a first order differential calculus to its maximal prolongation to this setting, we obtain a canonical functor from $\mathcal{E}$ to the category of differential calculi in $\mathcal{V}$. This yields a simultaneous generalization of the de Rham complex on $C^{\infty}$-rings, the K\"{a}hler differentials on commutative algebras, and the universal differential calculus on associative algebras. As a consequence, such categories $\mathcal{E}$ admit natural analogues of the notions of smooth map and diffeomorphism, as well as a functorial de Rham theory. Moreover, whenever two such faithful isofibrations to $\operatorname{Mon}(\mathcal{V})$ factor suitably, their corresponding de Rham functors are related via a comparison map. Developing this theory requires first extending the noncommutative geometry formalism of differential calculi from associative algebras to the setting of monoids internal to monoidal additive categories.

math.CT↗

arXiv2025

Quantitative Monoidal Algebra: Axiomatising Distance with String Diagrams

String diagrammatic calculi have become increasingly popular in fields such as quantum theory, circuit theory, probabilistic programming, and machine learning, where they enable resource-sensitive and compositional algebraic analysis. Traditionally, the equations of diagrammatic calculi only axiomatise exact semantic equality. However, reasoning in these domains often involves approximations rather than strict equivalences. In this work, we develop a quantitative framework for diagrammatic calculi, where one may axiomatise notions of distance between string diagrams. Unlike similar approaches, such as the quantitative theories introduced by Mardare et al., this requires us to work in a monoidal rather than a cartesian setting. We define a suitable notion of monoidal theory, the syntactic category it freely generates, and its models, where the concept of distance is established via enrichment over a quantale. To illustrate the framework, we provide examples from probabilistic and linear systems analysis.

cs.LO↗

arXiv2025

A finite approach to representable multicategories and related structures

It is known that monoidal categories have a finite definition, whereas multicategories have an infinite (albeit finitary) definition. Since monoidal categories correspond to representable multicategories, it goes without saying that representable multicategories should also admit a finite description. With this in mind, we give a new finite definition of a structure called a short multicategory, which only has multimaps of dimension at most four, and show that under certain representability conditions short multicategories correspond to various flavours of representable multicategories. This is done in both the classical and skew settings.

math.CT↗

arXiv2024

Sketches and Classifying Logoi

Inspired by the theory of classifying topoi for geometric theories, we define rounded sketches and logoi and provide the notion of classifying logos for a rounded sketch. Rounded sketches can be used to axiomatise all the known fragments of infinitary first order logic in $\mathbf{L}_{\infty,\infty}$, in a spectrum ranging from weaker than finitary algebraic to stronger than $λ$-geometric for $λ$ a regular cardinal. We show that every rounded sketch has an associated classifying logos, having similar properties to the classifying topos of a geometric theory. This amounts to a Diaconescu-type result for rounded sketches and (Morita small) logoi, which generalises the one for classifying topoi.

math.CT↗

arXiv2023

A skew approach to enrichment for Gray-categories

It is well known that the category of Gray-categories does not admit a monoidal biclosed structure that models weak higher-dimensional transformations. In this paper, the first of a series on the topic, we describe several skew monoidal closed structures on the category of Gray-categories, one of which captures higher lax transformations, and another which models higher pseudo-transformations.

math.CT↗

arXiv2023

Some Remarks on the Interchange in Gray-categories

We prove a generalised interchange equality for 3-cells in a Gray-category, i.e. we show that it still holds modulo the unique isomorphism given by the Gray-categorical pasting theorem of Di Vittorio. This significantly simplifies many calculations in Gray-categories.

math.CT↗

arXiv2023

KZ-pseudomonads and Kan Injectivity

We introduce the notion of Kan injectivity in 2-categories and study its properties. For an adequate 2-category $\mathcal{K}$, we show that every set of morphisms $\mathcal{H}$ induces a KZ-pseudomonad on $\mathcal{K}$ whose 2-category of pseudoalgebras is the locally full sub-2-category of all objects (left) Kan injective with respect to $\mathcal{H}$ and morphisms preserving Kan extensions. The main ingredient is the construction of a (pseudo)chain whose appropriate ``convergence" is ensured by a small object argument.

math.CT↗

arXiv2023

Distributive laws for relative monads

We introduce the notion of a distributive law between a relative monad and a monad. We call this a relative distributive law and define it in any 2-category $\mathcal{K}$. In order to do that, we introduce the 2-category of relative monads in a 2-category $\mathcal{K}$ with relative monad morphisms and relative monad transformations as 1- and 2-cells, respectively. We relate our definition to the 2-category of monads in $\mathcal{K}$ defined by Street. Thanks to this view we prove two Beck-type theorems regarding relative distributive laws. We also describe what does it mean to have Eilenberg-Moore and Kleisli objects in this context and give examples in the 2-category of locally small categories.

math.CT↗

arXiv2021

On the formal theory of pseudomonads and pseudodistributive laws

We contribute to the formal theory of pseudomonads, i.e. the analogue for pseudomonads of the formal theory of monads. In particular, we solve a problem posed by Steve Lack by proving that, for every Gray-category K, there is a Gray-category Psm(K) of pseudomonads, pseudomonad morphisms, pseudomonad transformations and pseudomonad modifications in K. We then establish a triequivalence between Psm(K) and the Gray-category of pseudomonads introduced by Marmolejo. Finally, these results are applied to give a clear account of the coherence conditions for pseudodistributive laws. 41 pages. Comments welcome.

math.CT↗