arXiv subjects

Ivan Di Liberti

Publications and source records attributed to Ivan Di Liberti.

27 records · Page 2Linked to original sources

arXiv2021

Formal Model Theory & Higher Topology

We study the $2$-categories BIon, of (generalized) bounded ionads, and $\text{Acc}_ω$, of accessible categories with directed colimits, as an abstract framework to approach formal model theory. We relate them to topoi and (lex) geometric sketches, which serve as categorical specifications of geometric theories. We provide reconstruction and completeness-like results. We relate abstract elementary classes to locally decidable topoi. We introduce the notion of categories of saturated objects and relate it to atomic topoi.

math.CT↗

arXiv2021

Towards Higher Topology

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in [Liba]. We hint that $0$-dimensional adjunction inhabits the categorified one.

math.CT↗

arXiv2020

Functorial Semantics for Partial Theories

We provide a Lawvere-style definition for partial theories, extending the classical notion of equational theory by allowing partially defined operations. As in the classical case, our definition is syntactic: we use an appropriate class of string diagrams as terms. This allows for equational reasoning about the class of models defined by a partial theory. We demonstrate the expressivity of such equational theories by considering a number of examples, including partial combinatory algebras and cartesian closed categories. Moreover, despite the increase in expressivity of the syntax we retain a well-behaved notion of semantics: we show that our categories of models are precisely locally finitely presentable categories, and that free models exist.

cs.LO↗

arXiv2020

The Scott adjunction

We introduce and study the Scott adjunction, relating accessible categories with directed colimits to topoi. Our focus is twofold, we study both its applications to formal model theory and its geometric interpretation. From the geometric point of view, we introduce the categorified Isbell duality, relating bounded (possibly large) ionads to topoi. The categorified Isbell duality interacts with the Scott adjunction offering a categorification of the Scott topology over a poset (hence the name). We show that the categorified Isbell duality is idempotent, similarly to its uncategorified version. From the logical point of view, we use this machinery to provide candidate (geometric) axiomatizations of accessible categories with directed colimits. We discuss the connection between these adjunctions and the theory of classifying topoi. We relate our framework to the more classical theory of abstract elementary classes. From a more categorical perspective, we show that the 2-category of topoi is enriched over accessible categories with directed colimits and we relate this result to the Scott adjunction.

math.CT↗

arXiv2020

Codensity: Isbell duality, pro-objects, compactness and accessibility

We study codensity monads $T$ induced by (mostly small, mostly dense) full subcategories $A \subset K$. These monads behave quite similarly, we show some connections with the Isbell duality, pro-finite objects and compact spaces. We prove that they are quite unlikely to be accessible. Finally, we introduce the notion of generically idempotent monad and comment its properties.

math.CT↗

arXiv2019

Gabriel-Ulmer duality for topoi and its relation with site presentations

Let $κ$ be a regular cardinal. We study Gabriel-Ulmer duality when one restricts the 2-category of locally $κ$-presentable categories with $κ$-accessible right adjoints to its locally full sub-2-category of $κ$-presentable Grothendieck topoi with geometric $κ$-accessible morphisms. In particular, we provide a full understanding of the locally full sub-2-category of the 2-category of $κ$-small cocomplete categories with $κ$-colimit preserving functors arising as the corresponding 2-category of presentations via the restriction. We analyse the relation of these presentations of Grothendieck topoi with site presentations and we show that the 2-category of locally $κ$-presentable Grothendieck topoi with geometric $κ$-accessible morphisms is a reflective sub-bicategory of the full sub-2-category of the 2-category of sites with morphisms of sites genearated by the weakly $κ$-ary sites in the sense of Shulman [37].

math.CT↗

arXiv2019

On the unicity of formal category theories

We prove an equivalence between cocomplete Yoneda structures and certain proarrow equipments on a 2-category $\mathcal K$. In order to do this, we recognize the presheaf construction of a cocomplete Yoneda structure as a relative, lax idempotent monad sending each admissible 1-cell $f :A \to B$ to an adjunction $\boldsymbol{P}_!f\dashv\boldsymbol{P}^*f$. Each cocomplete Yoneda structure on $\mathcal K$ arises in this way from a relative lax idempotent monad "with enough adjoint 1-cells", whose domain generates the ideal of admissibles, and the Kleisli category of such a monad equips its domain with proarrows. We call these structures "yosegi". Quite often, the presheaf construction associated to a yosegi generates an ambidextrous Yoneda structure; in such a setting there exists a fully formal version of Isbell duality.

math.CT↗

arXiv2018

Homotopical algebra is not concrete

We generalize Freyd's well-known result that "homotopy is not concrete", offering a general method to show that under certain assumptions on a model category $\mathcal M$, its homotopy category $\text{ho}(\mathcal M)$ cannot be concrete. This result is part of an attempt to understand more deeply the relation between set theory and abstract homotopy theory.

math.CT↗

arXiv2017

Weak saturation and weak amalgamation property

The two model-theoretic concepts of weak saturation and weak amalgamation property are studied in the context of accessible categories. We relate these two concepts providing sufficient conditions for existence and uniqueness of weakly saturated objects of an accessible category K. We discuss the implications of this fact in classical model theory.

math.CT↗