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Louis Martini

Publications and source records attributed to Louis Martini.

9 recordsLinked to original sources
arXiv2026

Free fibrations, lax colimits and Kan extensions for $(\infty,2)$-categories

In the first part of this paper we study fibrations of $(\infty,2)$-categories. We give a simple characterization of such fibrations in terms of a certain square being a pullback, and apply this to show that in some cases $(\infty,2)$-categories of functors and partially (op)lax transformations preserve fibrations. We also describe free fibrations of $(\infty,2)$-categories, including in the case where we only ask for (co)cartesian lifts of specified 1- and 2-morphisms in the base, and describe the right adjoint to pullback from fibrations to such partial fibrations along an arbitrary functor. In the second part of the paper we apply these results to study colimits and Kan extensions of $(\infty,2)$-categories. Most notably, we give a fibrational description of both partially (op)lax and weighted (co)limits of $(\infty,2)$-categories and construct partially lax Kan extensions. Among other results, we also include a model-independent version of cofinality for $(\infty,2)$-categories and briefly consider presentable $(\infty,2)$-categories, characterizing them as accessible localizations of presheaves of $\infty$-categories.

math.CT↗
arXiv2025

The condensed homotopy type of a scheme

We study a condensed version of the \'etale homotopy type of a scheme, which refines both the usual \'etale homotopy type of Friedlander-Artin-Mazur and the pro\'etale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a description of the condensed homotopy type using the Galois category of a scheme introduced by Barwick-Glasman-Haine. In the second part, we focus on the fundamental group of the condensed homotopy type in more detail. We show that, unexpectedly, the fundamental group of the condensed homotopy type of the affine line $\mathbf{A}^1_{\mathbf{C}}$ over the complex numbers is nontrivial. Nonetheless, its Noohi completion recovers the pro\'etale fundamental group of Bhatt-Scholze. Moreover, we show that a mild correction, passing to the quasiseparated quotient, fixes most of this group's quirks. Surprisingly, this quotient is often a topological group.

math.AG↗
arXiv2024

$(\infty,2)$-Topoi and descent

We set the foundations of a theory of Grothendieck $(\infty,2)$-topoi based on the notion of fibrational descent, which axiomatizes both the existence of a classifying object for fibrations internal to an $(\infty,2)$-category as well as the exponentiability of these fibrations. As our main result, we prove a 2-dimensional version of Giraud's theorem which characterizes $(\infty,2)$-topoi as those $(\infty, 2)$-categories that appear as localizations of $\mathfrak{C}\!\operatorname{at}$-valued presheaves in which the localization functor preserves certain partially lax finite limits which we call oriented pullbacks. We develop the basics of a theory of partially lax Kan extensions internal to an $(\infty,2)$-topos, and we show that every $(\infty,2)$-topos admits an internal version of the Yoneda embedding. Our general formalism recovers the theory of categories internal to a $(\infty,1)$-topos (as develop by the second author and Sebastian Wolf) as a full sub-$(\infty,2)$-category of the $(\infty,2)$-category of $(\infty,2)$-topoi. As a technical ingredient, we prove general results on the theory of presentable $(\infty,2)$-categories, including lax cocompletions and 2-dimensional versions of the adjoint functor theorem, which might be of independent interest.

math.CT↗
arXiv2024

Colimits and cocompletions in internal higher category theory

We develop a number of basic concepts in the theory of categories internal to an $\infty$-topos. We discuss adjunctions, limits and colimits as well as Kan extensions for internal categories, and we use these results to prove the universal property of internal presheaf categories. We furthermore construct the free cocompletion of an internal category by colimits that are indexed by an arbitrary class of diagram shapes.

math.CT↗
arXiv2023

Proper morphisms of $\infty$-topoi

We characterise proper morphisms of $\infty$-topoi in terms of a relativised notion of compactness: we show that a geometric morphism of $\infty$-topoi is proper if and only if it commutes with colimits indexed by filtered internal $\infty$-categories in the target. In particular, our result implies that for any $\infty$-topos, the global sections functor is proper if and only if it preserves filtered colimits. As an application, we show that every proper and separated map of topological spaces gives rise to a proper morphism between the associated sheaf $\infty$-topoi, generalising a result of Lurie. Along the way, we develop some aspects of the theory of localic higher topoi internal to an $\infty$-topos, which might be of independent interest.

math.CT↗
arXiv2023

Internal higher topos theory

We develop the theory of topoi internal to an arbitrary $\infty$-topos $\mathcal B$. We provide several characterisations of these, including an internal analogue of Lurie's characterisation of $\infty$-topoi, but also a description in terms of the underlying sheaves of $\infty$-categories, and we prove a number of structural results about these objects. Furthermore, we show that the $\infty$-category of topoi internal to $\mathcal B$ is equivalent to the $\infty$-category of $\infty$-topoi over $\mathcal B$, and use this result to derive a formula for the pullback of $\infty$-topoi. Lastly, we use our theory to relate smooth geometric morphisms of $\infty$-topoi to internal locally contractible topoi.

math.CT↗
arXiv2022

Presentability and topoi in internal higher category theory

The goal of this article is to develop the theory of presentable categories and topoi internal to an arbitrary $\infty$-topos $\mathcal{B}$. Our main results are internal analogues of Lurie's and Lurie-Simpson's characterisations of presentable $\infty$-categories and $\infty$-topoi. In the process, we introduce a theory of internal filteredness and accessible internal categories and establish a number of structural results about presentable $\mathcal{B}$-categories such as adjoint functor theorems and the existence of an internal analogue of the Lurie tensor product. We also compare these internal notions with external variants. We show that $\mathcal{B}$-modules embed fully faithfully into presentable $\mathcal{B}$-categories and prove that there is an equivalence between topoi internal to $\mathcal{B}$ and $\infty$-topoi over $\mathcal{B}$. We also include a number of applications of our results, such as a general version of Diaconescu's theorem for $\infty$-topoi and a characterisation of locally contractible geometric morphisms in terms of smoothness.

math.CT↗
arXiv2022

Yoneda's lemma for internal higher categories

We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal categories.

math.CT↗