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Renaud Gauthier

Publications and source records attributed to Renaud Gauthier.

At least 19 recordsLinked to original sources
arXiv2026

Consciousness in a Higher Categorical Context

We provide two representations of the Segal category $\mathcal{X}$ modeling natural phenomena, the first one being based on the concept of micro-reversibility, producing a long sequence $\Sigma$ of categories as a resolution of $\mathcal{X}$, the second one providing graded categories cofibered in groupoids over the categories of $\Sigma$, using the concept of consciousness as impetus. We show those two representations are dual to each other.

math.CT↗
arXiv2025

Homotopy Wave Function in Algebraic Geometry

We propose the homotopy shape of the Segal topos of derived stacks over simplicial k-algebras as the higher homotopical generalization of the concept of wave function in Quantum Mechanics

math.AG↗
arXiv2025

Construction and Properties of the Ground State of Natural Phenomena

We construct an $\infty$-category $\mathcal{G}$ as a model for the Ground State of physical phenomena and we provide properties of its manifestations $\chi = \text{Fun}(\mathcal{G}, \text{Cat}_{\infty})$ in $\text{Cat}_{\infty}$ as well as of its $\infty$-category of spectra $\text{Sp}(\chi)$.

math.CT↗
arXiv2023

Introduction to the Category of Derived Motivic Spectra

We formalize an abstraction of Grothendieck's philosophy of motives and construct a category of derived motivic spectra in the Segal category $\mathbb{R} \underline{\text{Hom}} ((\text{dSt}_k)^{\text{op}}_{/F}, \text{Top})$ ($\text{dSt}_k$ the Segal category of derived stacks on s$k$-Alg, Top = $L \text{Set}_{\Delta}$ the Segal category of simplicial sets), thereby providing a starting point for a construction of a stable motivic homotopy category in the Segal setting.

math.AG↗
arXiv2022

A Spectral "schematization" of Homotopy Types

Toen has interpreted the schematization problem as originally imagined by Grothendieck in "Pursuing Stacks" in such a way that solution(s) to this problem could be given. As he pointed out, there are many solutions available, and he gave two constructions solving this problem. What we do in the present work is reconsider what Grothendieck initially had in mind and develop a formalism that provides a concept of "schematization" and corresponding homotopy groups of "schematized" homotopy types. In our view this can be realized if we generalize topological spaces to symmetric spectra, and the mapping spaces Map(S,A) in the category of symmetric sequences play the role of homotopy groups, S the sphere spectrum, A a symmetric spectrum.

math.AG↗
arXiv2022

The Physical Mathematics of Segal Topoi and Strings

We introduce a notion of dynamics in the setting of Segal topos, by considering the Segal category of stacks $\mathcal{X} = \text{dAff}_{\mathcal{C}}^{\, \sim, \tau}$ on a Segal category $\text{dAff}_{\mathcal{C}}=$ L(Comm($\mathcal{C})^{op})$ as our system, and by regarding objects of $\mathbb{R}\underline{\text{Hom}}(\mathcal{X}, \mathcal{X})$ as its states. We develop the notion of quantum state in this setting and construct local and global flows of such states. In this formalism, strings are given by equivalences between elements of commutative monoids of $\mathcal{C}$, a base symmetric monoidal model category. The connection with standard string theory is made, and with M-theory in particular.

math.CT↗
arXiv2021

Goodwillie Calculus and Geometric Stacks

We show Goodwillie's calculus of functors and $n$-geometric $D^{-}$-stacks share similar features by starting to focus on the convergence of Taylor towers for homotopy functors and the fact that $\mathbb{R} F(A) \cong \text{holim} \mathbb{R} F(A_{\leq n})$ for geometric stacks, where $\{ A_{\leq n} \}$ provides a Postnikov tower of some given $A \in s \text{k-Alg}$. From there we show parallel results, such as similar homotopy fibers of connecting maps in towers, as well as polynomial approximations, pointwise approximations and reconstruction theorems for towers.

math.AT↗
arXiv2021

Internal Bousfield Localizations

We develop the notion of left and right Bousfield localizations in proper, cellular symmetric monoidal model categories with cofibrant unit, using homotopy function complexes defined by internal Hom objects instead of Hom sets.

math.CT↗
arXiv2021

A Dual Representation in Spectral Algebraic Geometry

Given a spectral Deligne-Mumford stack $X$, we define a perception of $X$ to be a collection of a certain class of morphisms $Y \rightarrow X$. For the class of affine morphisms in SpDM, we show that from QCoh($X$) on can extract the affine perception $\text{Aff}_X$ of $X$ on the one hand, and a subcategory of an $\infty$-category of representations $\text{Rep}_{\mathfrak{g}_*}$ of a dg Lie algebra $\mathfrak{g}_*$ associated with $X$ on the other. For the class of local morphisms $\text{Spét } R \rightarrow X$, the local perception of $X$ is given by the functor $\mathbf{X} = \text{Hom}(\text{Spét}(-), X)$ it represents. If $\mathbf{X}$ is a geometric stack, Tannaka duality allows us to recover $\mathbf{X}$ from $\text{QCoh}(\mathbf{X})$, from which we can also get, after base change, a subcategory of $\text{Rep}_{\mathfrak{g}_*}$. We generalize those results by considering functors $\mathbf{X}: \text{CAlg}^{\text{cn}} \rightarrow \mathcal{S}$ that are representable in accordance with the spectral Artin representability theorem of Lurie.

math.AG↗
arXiv2021

Supersymmetric Derived Stacks

Stacks have become a prevalent tool in studying problems with connections to String Theory, hence we see a need to develop a theory of supersymmetric stacks proper. We first define derived stacks on $\mathbb{Z}_2$-bi-graded k-modules (objects of sk-sMod$_*$) following the exposition of Toen and Vezzosi on ungraded modules in HAG I & II. We then define $\text{Top}_{* \centerdot}$-valued maps on those supermodules ($\text{Top}_{* \centerdot}$ $\mathbb{Z}_2$-bi-graded), and show how they behave under supersymmetry transformations in the base. For $Ψ: M \rightarrow X$ one such map, $M \in $ sk-sMod$_*$, $X \in \text{Top}_{* \centerdot}$, we argue that defining a prestack $F$ of simplicial sets over simplicial graded k-superalgebras object-wise by $F(M) = \{Ψ(σ, θ) | σ, θ\in M \}$ with the induced topology, one can call $F$ a supersymmetric stack if it is a derived stack.

math.AG↗
arXiv2020

Segal Topoi and Natural Phenomena: Universality of Physical Laws

J. Lurie proved in Higher Topos Theory that for $K$ a simplicial set, $\mathcal{C}$ a simplicial category, $f: \mathfrak{C}[K] \rightarrow \mathcal{C}^{\text{op}}$ an equivalence of simplicial categories, we have a Quillen equivalence $(\text{Set}^+_Δ)_{/K} \rightleftarrows (\text{Set}^+_Δ)^{\mathcal{C}}$. We prove a partial converse to this theorem at the level of Segal categories, namely that if $L(\text{Set}^+_Δ)_{/K}$ is isomorphic to $L(\text{Set}^+_Δ)^{\mathcal{C}}$ in Ho(SePC), then $L \mathfrak{C}[K]^{\text{op}}$ and $L\mathcal{C}$ are equivalent as Segal pre-categories relative to Segal categories of pre-stacks. We interpret this as indicating that the Segal category of pre-stacks $L(\text{Set}^+_Δ)^{\mathcal{C}} \cong \mathbb{R} \underline{\text{Hom}} (L \mathcal{C}, L \text{Set}^+_Δ)$ on $L \mathcal{C}$ is equivalently given by a choice of simplicial set $K$, relative to which phenomena in $\text{Top}^+ = L \text{Set}^+_Δ$ are considered, a sort of relativity principle. If we further take the Bousfield localizations of $L(\text{Set}^+_Δ)^{\mathfrak{C}[K]^{\text{op}}} \cong L( \text{Set}^+_Δ)_{/K}$ and $L(\text{Set}^+_Δ)^{\mathcal{C}}$ with respect to hypercovers, then regarding $L_{\text{Bous}}(L(\text{Set}^+_Δ)^{\mathcal{C}})$ as the Segal topos of natural phenomena on $L\mathcal{C}$, we also obtain an isomorphism $L_{\text{Bous}}(L(\text{Set}^+_Δ)^{\mathfrak{C}[K]^{\text{op}}}) \cong L_{\text{Bous}} (L(\text{Set}^+_Δ)^{\mathcal{C}})$ of Segal topoi of stacks. This provides two representations of the same natural phenomena, concurrently with the equivalence $L \mathfrak{C}[K]^{\text{op}} \simeq L\mathcal{C}$ relative to prestacks, which we interpret as a weak universality of natural laws.

math.CT↗
arXiv2020

Higher Galois for Segal Topos and Natural Phenomena

Toen and Vezzosi showed that $RHom^{geom}(T,lX)$ is a Segal groupoid, for $T$ a Segal topos, $lX = Loc(X)$ the Segal category of locally constant stacks on a CW complex $X$. Taking the realization of such a groupoid defines a pro-object $H_T = |RHom^{geom}(T, -)|$ that is defined to be the homotopy shape of the topos $T$. What we do instead is fix a Segal topos $X$, we let $T$ vary, and use the fact that $RHom^*_{Lex}(X,T) = RHom^{geom}(T,X)$ is a fundamental $\infty$-groupoid. We then prove that $X$ is a localization of the Segal category of local systems on $RHom^{geom}(T,X)$, in the spirit of Hoyois' work in his "Higher Galois Theory" paper, where it is proved, morally, that local systems on $H_T$ are equivalent to $T$ itself. We provide one application of this formalism, regarding the Segal topos $X=dSt(k)$ of derived stacks, for $k$ a commutative ring, as corresponding to manifestations of natural laws, themselves modeled by simplicial algebras, objects of $sk-CAlg$.

math.AG↗
arXiv2020

Motifs in Derived Algebraic Geometry

We formalize the concept of sheaves of sets on a model site by considering variables thereof, or motifs, and we construct functorially defined derived algebraic stacks from them, thereby eliminating the necessity to choose derived extensions.

math.AG↗
arXiv2019

Modular Model Categories

To any model category $\mathcal{M}$, we associate a modular model category, a functor of points $\mathcal{M}[-]:$ Cat $\rightarrow$ Cat, that associates to any small category $\mathcal{C}$ a functor category $\mathcal{M}[\mathcal{C}] = \text{Fun}_{fes}(\mathcal{C}, \mathcal{M})$ of full and essentially surjective functors from $\mathcal{C}$ to $\mathcal{M}$, providing parametrizations of a same model category $\mathcal{M}$ by different small categories. We are in particular interested in using schemes as parameters. We consider $\mathbb{Z}$Sm$/k$ the category of linear combinations of smooth separated schemes of finite type over Spec($k$), $k$ a field, referred to as $\mathbb{Z}$-schemes, and let $\mathcal{C} = Sh(\mathbb{Z} \text{Sm}/k, \text{Nis})$. We contrast this with using the $\mathbb{A}^1$-homotopy category of $\mathbb{Z}$-schemes as a parametrizing category.

math.AG↗
arXiv2018

$\infty$-topoi and Natural Phenomena: Generation

We show that the Segal topos of derived stacks over simplicial commutative $k$-algebras, which can be used to model natural phenomena, has a subobject classifier, something we regard as being a source from which dynamics is generated. This is done by considering the $\infty$-category associated to such a Segal topos, which turns out to be an $\infty$-topos. At this point we have the formalism of Higher topoi at our disposal to deal with Higher Category Theory concepts in a transparent manner.

math.AG↗
arXiv2015

Giraud's Theorem and Categories of Representations

We present an alternate proof of Giraud's Theorem based on the fact that given the conditions on a category E for being a topos, its objects are sheaves by construction. Generalizing sets to R-modules for R a commutative ring, we prove that a category with small hom-sets and finite limits is equivalent to a category of sheaves of R-modules on a site if and only if it satisfies Giraud's axioms and in addition is enriched in a certain symmetric monoidal category parametrized by an R-module.

math.AG↗