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Félix Loubaton

Publications and source records attributed to Félix Loubaton.

9 recordsLinked to original sources
arXiv2026

Effectivity of Generalized Double $\infty$-Categories

We construct an adjunction between $m$-categories internal to $(\infty,n)$-categories, called $(n,m)$-double $\infty$-categories, and filtrations $A_0\to \dots\to A_m$ where for all $i<m$, $A_i$ is a $(n+i)$-category. We show that this adjunction induces an equivalence between $(n,m)$-double $\infty$-categories admitting enough companions and filtrations such that each morphism $A_i\to A_{i+1}$ is essentially surjective on cells of dimension lower than or equal to $i$. This result can be seen as a $(\infty,n)$-categorical generalization of the equivalence between internal groupoids and effective epimorphisms in the category of $\infty$-groupoids proven by Rezk and Lurie. In the case $n=0$, this recovers the characterization of flagged $m$-categories given by Ayala-Francis, and in the case $n=1$, it allows us to prove some conjectures concerning the square functor and its variants, stated by Gaitsgory-Rozenblyum in the appendix of their book on Derived Algebraic Geometry.

math.CT↗
arXiv2026

Day convolution for algebraic patterns

We characterize the exponentiable objects for a wide range of structures prevalent in $\infty$-categorical algebra, extending the construction of Day convolution to more general structures than $\infty$-operads. More precisely, we give a criterion that is both necessary and sufficient for many of these structures encountered in practice, such as (equivariant) $\infty$-operads and virtual double $\infty$-categories. We work within the framework of algebraic patterns of Chu-Haugseng that describe these structures in terms of weak Segal fibrations. As part of the proof, we give a new description of weak Segal fibrations in terms of generalized Segal spaces on certain "tree" categories. We also define the "underlying graph" of a weak Segal fibration, extending the notion of the underlying $\infty$-category for $\infty$-operads, and explicitly describe the underlying graph of exponential objects in weak Segal fibrations.

math.CT↗
arXiv2025

On the squares functor and the Gaitsgory-Rozenblyum conjectures

In the seminal work of Gaitsgory and Rozenblyum on derived algebraic geometry, eight conjectures regarding the theory of $(\infty,2)$-categories are stated. This paper aims to clarify the status of these claims, and to provide a proof for the last remaining open one. Along the way, we demonstrate the universal property of the so-called squares functor, a construction that plays an important role in the $(\infty,2)$-categorical foundations of Gaitsgory-Rozenblyum.

math.CT↗
arXiv2024

Categorical Theory of $(\infty,ω)$-Categories

This text is dedicated to the development of the theory of $(\infty,ω)$-categories. We present generalizations of standard results from category theory, such as the lax Grothendieck construction, the Yoneda lemma, lax (co)limits and lax Kan extensions, among others.

math.CT↗
arXiv2024

A model for the coherent walking $ω$-equivalence

We prove that a certain $ω$-category, which was constructed in previous work by the third and fourth author, is a model for the fully coherent walking $ω$-equivalence. Further, appropriate truncations of it give models for the fully coherent walking $n$-equivalence for each $n\geq1$.

math.CT↗
arXiv2023

Theory and models of $(\infty,ω)$-categories

This thesis is divided into two parts. In the first part, we study models of $(\infty,ω)$-categories. The main result is to establish a Quillen equivalence between Rezk's complete Segal $Θ$-spaces and Verity's complicial sets. In the second part, we study the $(\infty,1)$-category corresponding to these two model structures, denoted by $(\infty,ω)$-cat. Its connection with Rezk's complete Segal $Θ$-spaces allows us to use the globular language, while its connection with complicial sets gives us access to a fundamental operation, the Gray tensor product. The objective will be to implement standard categorical constructions in the context of $(\infty,ω)$-categories. A special emphasis will be placed on the Grothendieck construction.

math.CT↗
arXiv2022

Conditions de Kan sur les nerfs des $ω$-catégories

We show that the Street nerve of a strict $ω$-category $C$ is a Kan complex (respectively a quasi-category) if and only if the $n$-cells of $C$ for $n\geq 1$ (respectively $n> 1$) are weakly invertible. Moreover, we equip $\mathcal{N}(C)$ with a structure of saturated complicial set where the $n$-simplices correspond to morphisms from the $n^{th}$ oriental to $C$ sending the unique non-trivial $n$-cell of the domain to a weakly invertible cell of $C$.

math.CT↗
arXiv2022

Dualities in the complicial model of $\infty$-categories

In this note, we study the connection between Gray tensor product and suspension. We derive a characterization of weak equivalences as fully faithful and essentially surjective functors. We construct the $co$ duality, a weak involution that reverses the direction of even dimensional cells. We conclude by studying Grothendieck fibrations of complicial sets.

math.CT↗