arXiv subjects

Muriel Livernet

Publications and source records attributed to Muriel Livernet.

At least 19 recordsLinked to original sources

arXiv2026

Homotopy theories via the magnitude-path spectral sequence

We introduce a family of homotopy theories for generalized metric spaces with natural number distances, via the magnitude-path spectral sequence (MPSS). The first page of the MPSS is known as magnitude homology; the second page is known as bigraded path homology, and contains GLMY path homology as its top row. For each natural number r, we define a class of maps of metric spaces called r-quasi-isomorphisms: those maps that induce a quasi-isomorphism at page r of the MPSS. We show that every page of the spectral sequence satisfies a suitable metric analogue of each of the Eilenberg-Steenrod axioms. In particular, we introduce the notion of r-cofibration and prove a Mayer-Vietoris theorem for page r with respect to r-cofibrations. We establish a family of Brown category structures on generalized metric spaces which allow us to explicitly compute homotopy colimits. We apply this to describe r-suspension and r-spheres of dimension n, and compute their spectral sequences. Finally we prove that for r = 1 the entire theory restricts to directed graphs.

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arXiv2026

Spectral sequences via linear presheaves

We study homotopy theory of the category of spectral sequences with respect to the class of weak equivalences given by maps which are quasi-isomorphisms on a fixed page. We introduce the category of extended spectral sequences and show that this is bicomplete by analysis of a certain linear presheaf category modelled on discs. We endow the category of extended spectral sequences with various model category structures, restricting to give the almost Brown category structures on spectral sequences of our earlier work. One of these has the property that spectral sequences is a homotopically full subcategory. By results of Meier, this exhibits the category of spectral sequences as a fibrant object in the Barwick-Kan model structure on relative categories, that is, it gives a model for an infinity category of spectral sequences. We also use the presheaf approach to define two décalage functors on spectral sequences, left and right adjoint to a shift functor, thereby clarifying prior use of the term décalage in connection with spectral sequences.

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arXiv2025

Model category structures on truncated multicomplexes for complex geometry

To a bicomplex one can associate two natural filtrations, the column and row filtrations, and then two associated spectral sequences. This can be generalized to $N$-multicomplexes. We present a family of model category structures on the category of $N$-multicomplexes where the weak equivalences are the morphisms inducing a quasi-isomorphism at a fixed page $r$ of the first spectral sequence and at a fixed page $s$ of the second spectral sequence. Such weak equivalences arise naturally in complex geometry. In particular, the model structures presented here establish a basis for studying homotopy types of almost and generalized complex manifolds.

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arXiv2024

A prop structure on partitions

Motivated by its link with functor homology, we study the prop freely generated by the operadic suspension of the operad Com. We exhibit a particular family of generators, for which the composition and the symmetric group actions admit simple descriptions. We highlight associated subcategories of its Karoubi envelope which allows us to compute extensions groups between simple functors from free groups. We construct a particular prop structure on partitions whose composition corresponds to the Yoneda product of extensions between exterior power functors.

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arXiv2023

Homotopy theory of spectral sequences

Let $R$ be a commutative ring with unit. We consider the homotopy theory of the category of spectral sequences of $R$-modules with the class of weak equivalences given by those morphisms inducing a quasi-isomorphism at a certain fixed page. We show that this admits a structure close to that of a category of fibrant objects in the sense of Brown and in particular the structure of a partial Brown category with fibrant objects. We use this to compare with related structures on the categories of multicomplexes and filtered complexes.

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arXiv2022

Rigidification of cubical quasi-categories

We construct a cubical analogue of the rigidification functor from quasi-categories to simplicial categories present in the work of Joyal and Lurie. We define a functor from the category of cubical sets of Doherty-Kapulkin-Lindsey-Sattler to the category of (small) simplicial categories. We show that this rigidification functor establishes a Quillen equivalence between the Joyal model structure on cubical sets (as it is called by the four authors) and Bergner's model structure on simplicial categories. We follow the approach to rigidification of Dugger and Spivak, adapting their framework of necklaces to the cubical setting.

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arXiv2021

Model category structures on multicomplexes

We present a family of model structures on the category of multicomplexes. There is a cofibrantly generated model structure in which the weak equivalences are the morphisms inducing an isomorphism at a fixed stage of an associated spectral sequence. Corresponding model structures are given for truncated versions of multicomplexes, interpolating between bicomplexes and multicomplexes. For a fixed stage of the spectral sequence, the model structures on all these categories are shown to be Quillen equivalent.

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arXiv2019

On the spectral sequence associated to a multicomplex

A multicomplex, also known as a twisted chain complex, has an associated spectral sequence via a filtration of its total complex. We give explicit formulas for all the differentials in this spectral sequence.

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arXiv2018

Model category structures and spectral sequences

Let k be a commutative ring with unit. We endow the categories of filtered complexes and of bicomplexes of k-modules, with cofibrantly generated model structures, where the class of weak equivalences is given by those morphisms inducing a quasi-isomorphism at a certain fixed stage of the associated spectral sequence. For filtered complexes, we relate the different model structures obtained, when we vary the stage of the spectral sequence, using the functors shift and décalage.

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arXiv2017

Derived A-infinity algebras and their homotopies

The notion of a derived A-infinity algebra, considered by Sagave, is a generalization of the classical notion of A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy, for non-negative integers r, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as multicomplexes) first since it is of independent interest and this simpler case clearly exemplifies the structure we study. We also give two new interpretations of derived A-infinity algebras as A-infinity algebras in twisted complexes and as A-infinity algebras in split filtered cochain complexes.

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arXiv2016

On the deformation complex of homotopy affine actions

An affine action of an associative algebra $A$ on a vector space $V$ is an algebra morphism $A \to V \rtimes {\rm End}(V)$, where $V$ is a vector space and $V \rtimes {\rm End}(V)$ is the algebra of affine transformations of $V$. The one dimensional version of the Swiss-Cheese operad, denoted ${\mathrm{\bf{sc}}}_1$, is the operad that governs affine actions of associative algebras. This operad is Koszul and admits a minimal model denoted by $({\mathrm{\bf{sc}}}_1)_\infty$. Algebras over this minimal model are called Homotopy Affine Actions, they consist of an $A_\infty$-morphism $A \to V \rtimes {\rm End}(V)$, where $A$ is an $A_\infty$-algebra. In this paper we prove a relative version of Deligne's conjecture. In other words, we show that the deformation complex of a homotopy affine action has the structure of an algebra over an ${\rm SC}_2$ operad. That structure is naturally compatible with the ${\rm E}_2$ structure on the deformation complex of the $A_\infty$-algebra.

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arXiv2015

Non-formality of the Swiss-Cheese operad

In this note, we prove that the Swiss-cheese operad is not formal. We also give a criteria in terms of Massey operadic product for the non-formality of a topological operad.

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arXiv2014

Representations of derived A-infinity algebras

The notion of a derived A-infinity algebra arose in the work of Sagave as a natural generalisation of the classical A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We develop some of the basic operadic theory of derived A-infinity algebras, building on work of Livernet-Roitzheim-Whitehouse. In particular, we study the coalgebras over the Koszul dual cooperad of the operad dAs, and provide a simple description of these. We study representations of derived A-infinity algebras and explain how these are a two-sided version of Sagave's modules over derived A-infinity algebras. We also give a new explicit example of a derived A-infinity algebra.

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arXiv2014

Generalised bialgebras and entwined monads and comonads

Jean-Louis Loday has defined generalised bialgebras and proved structure theorems in this setting which can be seen as general forms of the Poincaré-Birkhoff-Witt and the Cartier-Milnor-Moore theorems. It was observed by the present authors that parts of the theory of generalised bialgebras are special cases of results on entwined monads and comonads and the corresponding mixed bimodules. In this article the Rigidity Theorem of Loday is extended to this more general categorical framework.

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arXiv2013

$A_\infty$-actions and Recognition of Relative Loop Spaces

We show that relative loop spaces are recognized by $A_\infty$-actions. A certain version of the 2-sided bar construction is used to prove such recognition theorem. The operad ${\sf Act}_\infty$ of $A_\infty$-actions is presented in terms of the Boardman-Vogt resolution of the operad ${\sf Act}$. We exhibit an operad homotopy equivalence between such resolution and the $1$-dimensional Swiss-cheese operad $\mathcal{SC}_1$.

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arXiv2012

Derived A-infinity algebras in an operadic context

Derived A-infinity algebras were developed recently by Sagave. Their advantage over classical A-infinity algebras is that no projectivity assumptions are needed to study minimal models of differential graded algebras. We explain how derived A-infinity algebras can be viewed as algebras over an operad. More specifically, we describe how this operad arises as a resolution of the operad dAs encoding bidgas. This generalises the established result describing the operad A-infinity as a resolution of the operad As encoding associative algebras. We further show Sagave's definition of morphisms agrees with the infinity-morphisms of dA-infinity algebras arising from operadic machinery. We also study the operadic homology of derived A-infinity algebras.

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arXiv2012

On the Spectral Sequence of the Swiss-cheese Operad

We prove that the homology of the Swiss-cheese operad is a Koszul operad. As a consequence, we obtain that the spectral sequence associated to the stratification of the compactification of points on the upper half plane collapses at the second stage, proving a conjecture by A. Voronov. However, we prove that the operad obtained at the second stage differs from the homology of the Swiss-cheese operad.

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arXiv2012

OCHA and Leibniz Pairs, towards a Koszul duality

In this paper we study the homology of 2 versions of the swiss-cheese operad. We prove that the zeroth homology of these two versions are Koszul operads and relate this to strong homotopy Lebiniz pairs and OCHA, defined by Kajiura and Stasheff.

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