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Javier J. Gutiérrez

Publications and source records attributed to Javier J. Gutiérrez.

At least 19 recordsLinked to original sources
arXiv2026

A new model structure on dendroidal spaces for the theory of $\infty$-operads

We introduce a new model structure on the category of dendroidal spaces, designed to provide a further model for the homotopy theory of $\infty$-operads. This model is directly analogous to a recent construction on the category of simplicial spaces by Moser and Nuiten, and can be seen as its dendroidal counterpart. In our new model structure, the fibrant objects are the dendroidal Segal spaces, while the cofibrations form a subclass of those in the Segal space model structure. The weak equivalences between fibrant objects are precisely the Dwyer--Kan equivalences, and the fibrations between them are the isofibrations. We prove that this model structure is Quillen equivalent to the Cisinski--Moerdijk model structures on dendroidal sets and dendroidal spaces, thereby establishing a compatible extension of the theory of $\infty$-categories to the operadic setting.

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arXiv2025

Cloning systems and action operads

Action operads and cloning systems are, respectively, the main ingredients in two approaches for axiomatically constructing Thompson-like groups due to Thumann and Witzel-Zaremsky. In this paper, we prove that action operads are equivalent to cloning systems that admit a certain extra structure, and which we call bilateral cloning systems. In addition, we describe their relation with crossed interval groups and product categories.

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arXiv2025

Sketchable infinity categories

A sketch is a category equipped with specified collections of cones and cocones. Its models are functors to the category of sets that send the distinguished cones and cocones to limit cones and colimit cocones, respectively. Sketches provide a categorical formalization of theories, interpreting logical operations in terms of limits and colimits. Gabriel and Ulmer showed that categories of models of sketches involving only cones (called limit sketches) are precisely the locally presentable categories, while Lair extended this correspondence to sketches including both cones and cocones, thereby characterizing accessible categories. In this article, we discuss a homotopy-coherent generalization of sketches in the context of $\infty$-categories and prove that presentable $\infty$-categories are the $\infty$-categories of models of limit sketches, whereas accessible $\infty$-categories arise as the $\infty$-categories of models of arbitrary sketches. As illustrations, we make the corresponding sketches explicit for a wide range of $\infty$-categories, including complete Segal spaces, $\infty$-operads, $A_\infty$-algebras, $E_\infty$-algebras, spectra, and higher sheaves.

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arXiv2024

Homotopy reflectivity is equivalent to the weak Vopěnka principle

Homotopical localizations with respect to (possibly proper) classes of maps are known to exist assuming the validity of a large-cardinal axiom from set theory called Vopěnka's principle. In this article, we prove that each of the following statements is equivalent to an axiom of lower consistency strength than Vopěnka's principle, known as weak Vopěnka's principle: (a) Localization with respect to any class of maps exists in the homotopy category of simplicial sets; (b) Localization with respect to any class of maps exists in the homotopy category of spectra; (c) Localization with respect to any class of morphisms exists in any presentable $\infty$-category; (d) Every full subcategory closed under products and fibres in a triangulated category with locally presentable models is reflective. Our results are established using Wilson's 2020 solution to a long-standing open problem concerning the relative consistency of weak Vopěnka's principle within the large-cardinal hierarchy.

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arXiv2019

Morita homotopy theory for $(\infty,1)$-categories and $\infty$-operads

We prove the existence of Morita model structures on the categories of small simplicial categories, simplicial sets, simplicial operads and dendroidal sets, modelling the Morita homotopy theory of $(\infty,1)$-categories and $\infty$-operads. We give a characterization of the weak equivalences in terms of simplicial presheaves, simplicial algebras and slice categories. In the case of the Morita model structure for simplicial categories and simplicial operads, we also show that each of these model structures can be obtained as an explicit left Bousfield localization of the Bergner model structure on simplicial categories and the Cisinski--Moerdijk model structure on simplicial operads, respectively.

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arXiv2018

Encoding Equivariant Commutativity via Operads

In this paper, we prove a conjecture of Blumberg and Hill regarding the existence of $N_\infty$-operads associated to given sequences $\mathcal{F} = (\mathcal{F}_n)_{n \in \mathbb{N}}$ of families of subgroups of $G\times Σ_n$. For every such sequence, we construct a model structure on the category of $G$-operads, and we use these model structures to define $E_\infty^{\mathcal{F}}$-operads, generalizing the notion of an $N_\infty$-operad, and to prove the Blumberg-Hill conjecture. We then explore questions of admissibility, rectification, and preservation under left Bousfield localization for these $E_\infty^{\mathcal{F}}$-operads, obtaining some new results as well for $N_\infty$-operads.

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arXiv2017

Bousfield localisations along Quillen bifunctors

Consider a Quillen adjunction of two variables between combinatorial model categories from $\mathcal{C}\times\mathcal{D}$ to $\mathcal{E}$, and a set $\mathcal{S}$ of morphisms in $\mathcal{C}$. We prove that there is a localised model structure $L_{\mathcal{S}}\mathcal{E}$ on $\mathcal{E}$, where the local objects are the $\mathcal{S}$-local objects in $\mathcal{E}$ described via the right adjoint. These localised model structures generalise Bousfield localisations of simplicial model categories, Barnes and Roitzheim's familiar model structures, and Barwick's enriched Bousfield localisations. In particular, we can use these model structures to define Postnikov sections in more general left proper combinatorial model categories.

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arXiv2016

On autoequivalences of the (\infty, 1)-category of \infty-operads

We study the (\infty, 1)-category of autoequivalences of \infty-operads. Using techniques introduced by Toën, Lurie, and Barwick and Schommer-Pries, we prove that this (\infty, 1)-category is a contractible \infty-groupoid. Our calculation is based on the model of complete dendroidal Segal spaces introduced by Cisinski and Moerdijk. Similarly, we prove that the (\infty, 1)-category of autoequivalences of non-symmetric \infty-operads is the discrete monoidal category associated to Z/2Z. We also include a computation of the (\infty, 1)-category of autoequivalences of (\infty, n)-categories based on Rezk's Θ_n-spaces.

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arXiv2016

Towers and fibered products of model structures

Given a left Quillen presheaf of localized model structures, we study the homotopy limit model structure on the associated category of sections. We focus specifically on towers and fibered products (pullbacks) of model categories. As applications we consider Postnikov towers of model categories, chromatic towers of spectra and Bousfield arithmetic squares of spectra. For stable model categories, we show that the homotopy fiber of a stable left Bousfield localization is a stable right Bousfield localization.

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arXiv2015

Homotopy transfer and rational models for mapping spaces

By using homotopy transfer techniques in the context of rational homotopy theory, we show that if $C$ is a coalgebra model of a space $X$, then the $A_\infty$-coalgebra structure in $H_*(X;\mathbb{Q})\cong H_*(C)$ induced by the higher Massey coproducts provides the construction of the Quillen minimal model of $X$. We also describe an explicit $L_\infty$-structure on the complex of linear maps ${\rm Hom}(H_*(X; \mathbb{Q}), π_*(ΩY)\otimes\mathbb{Q})$, where $X$ is a finite nilpotent CW-complex and $Y$ is a nilpotent CW-complex of finite type, modeling the rational homotopy type of the mapping space ${\rm map}(X, Y)$. As an application we give conditions on the source and target in order to detect rational $H$-space structures on the components.

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arXiv2014

On solid and rigid monoids in monoidal categories

We introduce the notion of solid monoid and rigid monoid in monoidal categories and study the formal properties of these objects in this framework. We show that there is a one to one correspondence between solid monoids, smashing localizations and mapping colocalizations, and prove that rigid monoids appear as localizations of the unit of the monoidal structure. As an application, we study solid and rigid ring spectra in the stable homotopy category and characterize connective solid ring spectra as Moore spectra of subrings of the rationals.

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arXiv2013

A generalization of Ohkawa's theorem

A theorem due to Ohkawa states that the collection of Bousfield equivalence classes of spectra is a set. We extend this result to arbitrary combinatorial model categories.

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arXiv2012

Are all localizing subcategories of stable homotopy categories coreflective?

We prove that, in a triangulated category with combinatorial models, every localizing subcategory is coreflective and every colocalizing subcategory is reflective if a certain large-cardinal axiom (Vopenka's principle) is assumed true. It follows that, under the same assumptions, orthogonality sets up a bijective correspondence between localizing subcategories and colocalizing subcategories. The existence of such a bijection was left as an open problem by Hovey, Palmieri and Strickland in their axiomatic study of stable homotopy categories and also by Neeman in the context of well-generated triangulated categories.

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arXiv2012

Motivic slices and colored operads

Colored operads were introduced in the 1970's for the purpose of studying homotopy invariant algebraic structures on topological spaces. In this paper we introduce colored operads in motivic stable homotopy theory. Our main motivation is to uncover hitherto unknown highly structured properties of the slice filtration. The latter decomposes every motivic spectrum into its slices, which are motives, and one may ask to what extend the slice filtration preserves highly structured objects such as algebras and modules. We use colored operads to give a precise solution to this problem. Our approach makes use of axiomatic setups which specialize to classical and motivic stable homotopy theory. Accessible t-structures are central to the development of the general theory. Concise introductions to colored operads and Bousfield (co)localizations are given in separate appendices.

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arXiv2011

Cellularization of structures in stable homotopy categories

We describe the formal properties of cellularization functors in triangulated categories and study the preservation of ring and module structures under these functors in stable homotopy categories in the sense of Hovey, Palmieri and Strickland, such as the homotopy category of spectra or the derived category of a commutative ring. We prove that cellularization functors preserve modules over connective rings but they do not preserve rings in general (even if the ring is connective or the cellularization functor is triangulated). As an application of these results, we describe the cellularizations of Eilenberg-Mac Lane spectra and compute all acyclizations in the sense of Bousfield of the integral Eilenberg-Mac Lane spectrum.

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arXiv2011

Transfer of algebras over operads along derived Quillen adjunctions

Let V be a cofibrantly generated monoidal model category and let M be a monoidal V-model category. Given a cofibrant C-coloured operad P in V, we give sufficient conditions for the fibrant replacement and cofibrant replacement functors in M^C to preserve P-algebra structures. In particular, we show how P-algebra structures can be transferred along derived Quillen adjunctions of monoidal V-model categories, and we apply this result to the Quillen adjunctions defined by enriched Bousfield localizations and colocalizations on M. As an application, we prove that in the category of symmetric spectra the n-connective cover functor preserves A_{\infty} and E_{\infty} module spectra over connective ring spectra, for every integer n.

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