Searcharxiv⌕ Search

arXiv subjects

Kensuke Arakawa

Publications and source records attributed to Kensuke Arakawa.

17 recordsLinked to original sources
arXiv2026

Hammock localization via Segal animae

We give a short, conceptual account of Dwyer--Kan's hammock localization in the setting of $\infty$-categories. Starting from Mazel-Gee's formula for localization via the relative Rezk nerve, we show that its Segalification can be described explicitly as a Segal anima of zig-zags. We also compute its mapping animae and recover and generalize Dwyer--Kan's hammock formula. As an application, we show that, when a relative $\infty$-category supports fractions, the mapping animae of localization admit a simple description. This gives a unifying treatment for several formulas of this type in the existing literature.

math.CT↗
arXiv2026

Relative dendroidal Rezk nerve and applications

We extend the dendroidal Rezk nerve to the setting of relative $\infty$-operads. Our main theorem relates it to localization of $\infty$-operads, generalizing a theorem of Mazel-Gee. By exploiting the relation, we obtain a surprisingly effective tool to prove localization results in operadic contexts. As applications, we obtain a number of new results on operadic localizations, including a generalization of Willwacher's recent result on cyclic operads and operadic modules, and a description of locally constant factorization algebras on spheres in terms of discrete geometry.

math.AT↗
arXiv2026

On the equivalence of two approaches to multiplicative homotopy theories

We study the relation of two frameworks for multiplicative homotopy theories: Presentably symmetric monoidal $\infty$-categories and combinatorial symmetric monoidal model categories. Our main theorem establishes an equivalence of their homotopy theories. As consequences, we solve Pavlov's conjecture and obtain a solution to a special case of Hovey's 10th problem. We also prove several variations of the main theorem, such as an analog for non-symmetric monoidal semi-model categories.

math.CT↗
arXiv2026

On the equivalence of Brantner's and Chu--Haugseng's approaches to enriched $\infty$-operads

We prove that two models of (monochromatic) enriched $\infty$-operads, due to Brantner and Chu--Haugseng, are equivalent. We show this as a consequence of the equivalence of two models of monoidal $\infty$-categories of symmetric sequences and the composition product, due to Brantner and Haugseng. As a consequence, constructions and results formulated in either framework, such as notions of algebra and Koszul duality, are also shown to be equivalent.

math.AT↗
arXiv2026

A Context for Manifold Calculus

We develop Weiss's manifold calculus in the setting of $\infty$-categories, where we allow the target $\infty$-category to be any $\infty$-category with small limits. We will establish the connection between polynomial functors, Kan extensions, and Weiss sheaves, and will classify homogeneous functors. We will also generalize Weiss and Boavida de Brito's theorem to functors taking values in arbitrary $\infty$-categories with small limits.

math.AT↗
arXiv2026

Monoidal Relative Categories Model Monoidal $\infty$-Categories

We prove that the homotopy theory of monoidal relative categories is equivalent to that of monoidal $\infty$-categories, and likewise in the symmetric monoidal setting. As an application, we give a concise and complete proof of the fact that every presentably monoidal or presentably symmetric monoidal $\infty$-category is presented by a monoidal or symmetric monoidal model category, which, in the monoidal case, was sketched by Lurie, and in the symmetric monoidal case, was proved by Nikolaus--Sagave.

math.CT↗
arXiv2026

Homotopy Limits and Homotopy Colimits of Chain Complexes

We give a formula for homotopy limits and homotopy colimits of diagrams of chain complexes using the cobar and bar constructions, also known as the Bousfield--Kan formula. Along the way, we show that the Bousfield--Kan formula computes homotopy colimits in any framed model category.

math.AT↗
arXiv2025

Relative operads model $\infty$-operads

Given a (colored) operad and a set of unary operations, we can form an associated $\infty$-operad via localization. We show that localization determines an equivalence of homotopy theories of relative operads and $\infty$-operads. As an application, we give an affirmative answer to an open question by Harpaz, proving that Lurie's operadic nerve functor determines an equivalence of homotopy theories of simplicial operads and Lurie's $\infty$-operads.

math.AT↗
arXiv2025

Cubical models of $\infty$-presheaves and the Bousfield-Kan formula

We construct the covariant and the cocartesian model structures on the slice categories of cubical sets and marked cubical sets, respectively. As an application, we derive a version of the Bousfield-Kan formula for arbitrary cofibrantly generated monoidal model categories satisfying Muro's axiom.

math.AT↗
arXiv2025

Classification diagrams of simplicial categories

We show that the classification diagram of a relative $\infty$-category arising from a relative simplicial category is equivalent to the levelwise nerve. Applications include the comparison of the diagonal of the levelwise nerve and the homotopy coherent nerve, and a result on the levelwise localizations of simplicial categories.

math.AT↗
arXiv2025

Derived mapping spaces of $\infty$-categories

We prove the Derived Mapping Space Lemma, which generalizes the central theorem of Cisinski's work on calculus of fractions for $\infty$-categories, and allows us to provide a unified framework for analyzing mapping spaces in localizations of ($\infty$-)categories. As an application, we give a sufficient condition for when a cubical or simplicial category is the localization of its underlying category at homotopy equivalences.

math.AT↗
arXiv2025

Monoidal Envelopes of Families of $\infty$-Operads and $\infty$-Operadic Kan Extensions

We provide details of the proof of Lurie's theorem on operadic Kan extensions. Along the way, we generalize the construction of monoidal envelopes of $\infty$-operads to families of $\infty$-operads and use it to construct the fiberwise direct sum functor, both of which we characterize by certain universal properties. Aside from their uses in the proof of Lurie's theorem, these results and constructions have their independent interest.

math.CT↗
arXiv2025

Universal Properties of Variations of the Little Cubes Operads

Given a map $B\to B\mathrm{Top}(n)$ of spaces, one can define a version $\mathbb{E}_{B}$ of the little cubes operad, whose construction is due to Lurie. We show that $\mathbb{E}_{B}$ enjoys the universal property that, for every $\infty$-operad $\mathcal{O}$, an operad map $\mathbb{E}_{B}\to\mathcal{O}$ is equivalent to a $\mathrm{Top}(n)$-equivariant map $B\times_{B\mathrm{Top}(n)}E\mathrm{Top}(n)\to\operatorname{Map}(\mathbb{E}_{n},\mathcal{O})$. This gives us an explicit diagram exhibiting $\mathbb{E}_{B}$ as a colimit of $\mathbb{E}_{n}$ parametrized by $B$. It also shows that locally constant factorization algebras satisfy descent, reproving a recent theorem of Matsuoka.

math.AT↗
arXiv2024

The Grothendieck Construction for $\infty$-Categories Fibered over Categorical Patterns

We show how to treat families of $\infty$-categories fibered in categorical patterns (e.g., $\infty$-operads and monoidal $\infty$-categories) in terms of fibrations by relativizing the Grothendieck construction. As applications, we construct an analog of the universal cocartesian fibration and explain how to compute limits and colimits of $\infty$-categories fibered in categorical patterns.

math.CT↗
arXiv2024

Classification Diagrams of Marked Simplicial Sets

We prove that the classification diagram functor from the category of marked simplicial sets to the category of bisimplicial sets carries cartesian equivalences to Rezk equivalences. As a corollary, we obtain Mazel-Gee's theorem on localizations of relative $\infty$-categories.

math.AT↗