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Simona Paoli

Publications and source records attributed to Simona Paoli.

At least 19 recordsLinked to original sources
arXiv2025

A study of Kock's fat Delta

Motivated by the study of weak identity structures in higher category theory we explore the fat Delta category, a modification of the simplex category introduced by J. Kock and provide a comprehensive study via the theory of monads with arities. Specifically, by proving that the free relative semicategory monad is strongly cartesian and identifying a dense generator, the theory of monads with arities immediately gives rise to the nerve theorem. We characterise the essential image of the nerve via the Segal condition, and show that fat Delta possesses an active-inert factorisation system. Building on these results, we also establish an isomorphism between two presentations of fat Delta.

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arXiv2020

Weakly globular double categories and weak units

Weakly globular double categories are a model of weak $2$-categories based on the notion of weak globularity, and they are known to be suitably equivalent to Tamsamani $2$-categories. Fair $2$-categories, introduced by J. Kock, model weak $2$-categories with strictly associative compositions and weak unit laws. In this paper we establish a direct comparison between weakly globular double categories and fair $2$-categories and prove they are equivalent after localisation with respect to the $2$-equivalences. This comparison sheds new light on weakly globular double categories as encoding a strictly associative, though not strictly unital, composition, as well as the category of weak units via the weak globularity condition.

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arXiv2020

Stable homotopy hypothesis in the Tamsamani model

We prove that symmetric monoidal weak n-groupoids in the Tamsamani model provide a model for stable n-types. Moreover, we recover the classical statement that Picard categories model stable 1-types.

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arXiv2019

Comonad Cohomology of Track Categories

We define a comonad cohomology of track categories and we show it is linked by a long exact sequence to its Dwyer-Kan-Smith cohomology . Under mild hypothesis on the track category, we show that its comonad cohomology coincides, up to dimension shift, with its Dwyer-Kan-Smith cohomology, therefore obtaining an algebraic formulation of the latter. We also specialize our results to the case where the track category is a $2$-groupoid.

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arXiv2017

Segal-type models of higher categories

Higher category theory is an exceedingly active area of research, whose rapid growth has been driven by its penetration into a diverse range of scientific fields. Its influence extends through key mathematical disciplines, notably homotopy theory, algebraic geometry and algebra, mathematical physics, to encompass important applications in logic, computer science and beyond. Higher categories provide a unifying language whose greatest strength lies in its ability to bridge between diverse areas and uncover novel applications. In this foundational work we introduce a new approach to higher categories. It builds upon the theory of iterated internal categories, one of the simplest possible higher categorical structures available, by adopting a novel and remarkably simple "weak globularity" postulate and demonstrating that the resulting model provides a fully general theory of weak n-categories. The latter are among the most complex of the higher structures, and are crucial for applications. We show that this new model of "weakly globular n-fold categories" is suitably equivalent to the well studied model of weak n-categories due to Tamsamani and Simpson.

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arXiv2016

Weakly globular Tamsamani N-categories and their rigidification

We introduce a new class of higher categorical structures called weakly globular Tamsamani n-categories. These generalize the Tamsamani-Simpson model of higher categories by using the new paradigm of weak globularity to weaken higher categorical structures. We prove this new structure is suitably equivalent to a simpler one previously introduced by the author, called weakly globular n-fold categories.

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arXiv2016

Weakly globular N-fold categories as a model of weak N-categories

We study a new type of higher categorical structure, called weakly globular n-fold category, previously introduced by the author. We show that this structure is a model of weak n-categories by proving that it is suitably equivalent to the Tamsamani-Simpson model. We also introduce groupoidal weakly globular n-fold categories and show that they are algebraic models of n-types.

math.CT↗
arXiv2016

Pseudo-functors modelling higher structures

We introduce a new higher categorical structure called a weakly globular n-fold category. This structure is based on iterated internal categories and on the notion of weak globularity. We identify a suitable class of pseudo-functors whose strictification produces weakly globular n-fold categories.

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arXiv2016

Homotopically discrete higher categorical structures

We introduce the notion of homotopically discrete n-fold category as an n-fold generalization of a groupoid with no non-trivial loops. We give two equivalent descriptions of this structure: in terms of a Segal-type model and in terms of iterated internal equivalence relations. We also show that homotopically discrete n-fold categories form an n-fold categorical model of 0-types.

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arXiv2014

Segal-type algebraic models of n-types

For each n\geq 1 we introduce two new Segal-type models of n-types of topological spaces: weakly globular n-fold groupoids, and a lax version of these. We show that any n-type can be represented up to homotopy by such models via an explicit algebraic fundamental n-fold groupoid functor. We compare these models to Tamsamani's weak n-groupoids, and extract from them a model for (k-1)connected n-types

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arXiv2013

Bicategories and Weakly Globular Double Categories

This paper introduces the notion of weakly globular double categories, a particular class of strict double categories, as a way to model weak 2-categories; it explores its use in defining a double category of fractions, and shows that the sub-2-category of groupoidal weakly globular double categories forms an algebraic model of homotopy 2-types.

math.CT↗
arXiv2010

A Thomason Model Structure on the Category of Small n-fold Categories

We construct a cofibrantly generated Quillen model structure on the category of small n-fold categories and prove that it is Quillen equivalent to the standard model structure on the category of simplicial sets. An n-fold functor is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak equivalence of simplicial sets. This is an n-fold analogue to Thomason's Quillen model structure on Cat. We introduce an n-fold Grothendieck construction for multisimplicial sets, and prove that it is a homotopy inverse to the n-fold nerve. As a consequence, we completely prove that the unit and counit of the adjunction between simplicial sets and n-fold categories are natural weak equivalences.

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arXiv2010

Two-track categories

We describe a 2-dimensional analogue of track categories, called two-track categories, and show that it can be used to model categories enriched in 2-type mapping spaces. We also define a Baues-Wirsching type cohomology theory for track categories, and explain how it can be used to classify two-track extensions of a track category D by a module over D.

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arXiv2008

Model Structures on the Category of Small Double Categories

In this paper we obtain several model structures on {\bf DblCat}, the category of small double categories. Our model structures have three sources. We first transfer across a categorification-nerve adjunction. Secondly, we view double categories as internal categories in {\bf Cat} and take as our weak equivalences various internal equivalences defined via Grothendieck topologies. Thirdly, {\bf DblCat} inherits a model structure as a category of algebras over a 2-monad. Some of these model structures coincide and the different points of view give us further results about cofibrant replacements and cofibrant objects. As part of this program we give explicit descriptions and discuss properties of free double categories, quotient double categories, colimits of double categories, several nerves, and horizontal categorification.

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arXiv2007

Semistrict Tamsamani n-groupoids and connected n-types

Tamsamani's weak n-groupoids are known to model n-types. In this paper we show that every Tamsamani weak n-groupoid representing a connected n-type is equivalent in a suitable way to a semistrict one. We obtain this result by comparing Tasmamani's weak n-groupoids and cat^(n-1)-groups as models of connected n-types.

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arXiv2006

Semistrict models of connected 3-types and Tamsamani's weak 3-groupoids

Homotopy 3-types can be modelled algebraically by Tamsamani's weak 3-groupoids as well as, in the path-connected case, by cat^2-groups. This paper gives a comparison between the two models in the path-connected case. This leads to two different semistrict algebraic models of connected 3-types using the Tamsamani's model. Both are then related to Gray groupoids.

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