arXiv subjects

Giacomo Tendas

Publications and source records attributed to Giacomo Tendas.

16 recordsLinked to original sources

arXiv2026

Isoregular theories, accessible 2-categories, and free constructions

We introduce isoregular theories, in which it is possible to express existential quantification up to unique isomorphism, as typically used to characterise category-theoretic universal constructions, such as limits. We then develop a functorial semantics for isoregular theories and prove that their 2-categories of models are accessible with flexible limits. We apply these results by showing that a number of 2-categories of interest in general category theory, categorical algebra, and categorical logic are models of isoregular theories, thereby establishing that they are accessible 2-categories with flexible limits and obtaining a number of new free constructions.

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arXiv2026

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 Kähler 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.

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arXiv2026

Towards enriched universal algebra

Following the classical approach of Birkhoff, we suggest an enriched version of enriched universal algebra. Given a suitable base of enrichment $\mathcal V$, we define a language $\mathbb L$ to be a collection of $(X,Y)$-ary function symbols whose arities are taken among the objects of $\mathcal V$. The class of $\mathbb L$-terms is constructed recursively from the symbols of $\mathbb L$, the morphisms in $\mathcal V$, and by incorporating the monoidal structure of $\mathcal V$. Then, $\mathbb L$-structures and interpretations of terms are defined, leading to enriched equational theories. In this framework we characterize algebras for finitary monads on $\mathcal V$ as models of an equational theories.

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arXiv2025

More on soundness in the enriched context

Working within enriched category theory, we further develop the use of soundness, introduced by Adámek, Borceux, Lack, and Rosický for ordinary categories. In particular we investigate: (1) the theory of locally $Φ$-presentable $\mathcal V$-categories for a sound class $Φ$, (2) the problem of whether every $Φ$-accessible $\mathcal V$-category is $Ψ$-accessible, for given sound classes $Φ\subseteqΨ$, and (3) a notion of $Φ$-ary equational theory whose $\mathcal V$-categories of models characterize algebras for $Φ$-ary monads on $\mathcal V$.

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arXiv2025

Enriched positive logic

Building on our previous work on enriched regular logic, we introduce an enriched version of positive logic and relate it to enriched cone-injectivity classes and enriched accessible categories. To do this, we need a factorization system on the base of enrichment in order to interpret existential quantification and disjunctions. We will also show how to treat unique existence in enriched logic, and how to relate it to local presentability.

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arXiv2025

On enriched terms and 2-categorical universal algebra

We introduce a new notion of recursively generated enriched term which generalizes the one studied in joint work with Rosický. These new terms come together with a notion of term-interpretability, which recovers the same type of interpretability that has been considered for enrichment over posets, metric spaces, and $ω$-complete posets. As an application of this, we specialize to the 2-categorical case by considering 2-dimensional terms and 2-dimensional equational theories. In this context we also give an explicit description of free structures and prove a 2-dimensional Birkhoff variety theorem.

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arXiv2025

Dualities in the theory of accessible categories

Through the notion of weakly sound class of weights, we recover many known dualities involving accessible categories with a chosen class of limits, as instances of a general duality theorem. These include the Gabriel-Ulmer duality for locally finitely presentable categories, Diers duality for locally finitely multipresentable categories, and the Makkai-Paré duality for finitely accessible categories. In doing so, we extend these to the enriched setting, provide a more formal and unifying approach to the theory, and also discuss new dualities that arise as a consequence of our main theorem.

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arXiv2025

Enriched concepts of regular logic

Building on our previous work on enriched universal algebra, we define a notion of enriched language consisting of function and relation symbols whose arities are objects of the base of enrichment. In this context, we construct atomic formulas and define the regular fragment of enriched logic by taking conjunctions and existential quantifications of those. We then characterize enriched categories of models of regular theories as enriched injectivity classes in the enriched category of structures. These notions rely on the choice of a factorization system on the base of enrichment which will be used to interpret relation symbols and existential quantifications.

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arXiv2024

Notions of enriched purity

We introduce enriched notions of purity depending on the left class $\mathcal E$ of a factorization system on the base $\mathcal V$ of enrichment. Ordinary purity is given by the class of surjective mappings in the category of sets. Under specific assumptions, covering enrichment over quantale-valued metric spaces, $ω$-complete posets, and quasivarieties, we characterize the $(λ,\mathcal E)$-injectivity classes of locally presentable $\mathcal V$-categories in terms of closure under a class of limits, $λ$-filtered colimits, and $(λ,\mathcal E)$-pure subobjects.

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arXiv2023

Flatness, weakly lex colimits, and free exact completions

We capture in the context of lex colimits, introduced by Garner and Lack, the universal property of the free regular and Barr-exact completions of a weakly lex category. This is done by introducing a notion of flatness for functors $F\colon\mathcal C\to\mathcal E$ with lex codomain, and using this to describe the universal property of free $Φ$-exact completions in the absence of finite limits, for any given class $Φ$ of lex weights. In particular, we shall give necessary and sufficient conditions for the existence of free lextensive and free pretopos completions in the non-lex world, and prove that the ultraproducts, in the categories of models of such completions, satisfy an universal property.

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arXiv2023

Accessible categories with a class of limits

In this paper we characterize those accessible $\mathcal V$-categories that have limits of a specified class. We do this by introducing the notion of companion $\mathfrak C$ for a class of weights $Ψ$, as a collection of special types of colimit diagrams that are compatible with $Ψ$. We then characterize the accessible $\mathcal V$-categories with $Ψ$-limits as those accessibly embedded and $\mathfrak C$-virtually reflective in a presheaf $\mathcal V$-category, and as the $\mathcal V$-categories of $\mathfrak C$-models of sketches. This allows us to recover the standard theorems for locally presentable, locally multipresentable, and locally polypresentable categories as instances of the same general framework. In addition, our theorem covers the case of any weakly sound class $Ψ$, and provides a new perspective on the case of weakly locally presentable categories.

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arXiv2022

Virtual concepts in the theory of accessible categories

We provide a new characterization of enriched accessible categories by introducing the two new notions of virtual reflectivity and virtual orthogonality as a generalization of the usual reflectivity and orthogonality conditions for locally presentable categories. The word virtual refers to the fact that the reflectivity and orthogonality conditions are given in the free completion of the $\mathcal V$-category involved under small limits, instead of the $\mathcal V$-category itself. In this way we hope to provide a clearer understanding of the theory as well as a useful way of recognizing accessible $\mathcal V$-categories. In the last section we prove that the 2-category of accessible $\mathcal V$-categories, accessible $\mathcal V$-functors, and $\mathcal V$-natural transformations has all flexible limits.

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arXiv2022

On continuity of accessible functors

We prove that for each locally $α$-presentable category $\mathcal K$ there exists a regular cardinal $γ$ such that any $α$-accessible functor out of $\mathcal K$ (into another locally $α$-presentable category) is continuous if and only if it preserves $γ$-small limits; as a consequence we obtain a new adjoint functor theorem specific to the $α$-accessible functors out of $\mathcal K$. Afterwards we generalize these results to the enriched setting and deduce, among other things, that a small $\mathcal V$-category is accessible if and only if it is Cauchy complete.

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arXiv2022

Flat vs. filtered colimits in the enriched context

The importance of accessible categories has been widely recognized; they can be described as those freely generated in some precise sense by a small set of objects and, because of that, satisfy many good properties. More specifically finitely accessible categories can be characterized as: (a) free cocompletions of small categories under filtered colimits, and (b) categories of flat presheaves on some small category. The equivalence between (a) and (b) is what makes the theory so general and fruitful. Notions of enriched accessibility have also been considered in the literature for various bases of enrichment, such as $\mathbf{Ab},\mathbf{SSet},\mathbf{Cat}$ and $\mathbf{Met}$. The problem in this context is that the equivalence between (a) and (b) is no longer true in general. The aim of this paper is then to: (1) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow$ (b) holds; (2) give sufficient conditions on $\mathcal V$ so that (a) $\Leftrightarrow $ (b) holds up to Cauchy completion; (3) explore some examples not covered by (1) or (2).

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arXiv2021

Cauchy completeness for DG-categories

We go back to the roots of enriched category theory and study categories enriched in chain complexes; that is, we deal with differential graded categories (DG-categories for short). In particular, we recall weighted colimits and provide examples. We solve the 50 year old question of how to characterize Cauchy complete DG-categories in terms of existence of some specific finite absolute colimits. As well as the interactions between absolute weighted colimits, we also examine the total complex of a chain complex in a DG-category as a non-absolute weighted colimit.

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arXiv2019

Enriched Regular Theories

Regular and exact categories were first introduced by Michael Barr in 1971; since then, the theory has developed and found many applications in algebra, geometry, and logic. In particular, a small regular category determines a certain theory, in the sense of logic, whose models are the regular functors into Set. Barr further showed that each small and regular category can be embedded in a particular category of presheaves; then in 1990 Makkai gave a simple explicit characterization of the essential image of the embedding, in the case where the original regular category is moreover exact. More recently Prest and Rajani, in the additive context, and Kuber and Rosický, in the ordinary one, described a duality which connects an exact category with its (definable) category of models. Considering a suitable base for enrichment, we define an enriched notion of regularity and exactness, and prove a corresponding version of the theorems of Barr, of Makkai, and of Prest-Rajani/Kuber-Rosický.

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