Homotopy theory of algebras of substitudes and their localisation

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F22%3A00556688" target="_blank" >RIV/67985840:_____/22:00556688 - isvavai.cz</a>

Výsledek na webu

<a href="https://doi.org/10.1090/tran/8600" target="_blank" >https://doi.org/10.1090/tran/8600</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1090/tran/8600" target="_blank" >10.1090/tran/8600</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Homotopy theory of algebras of substitudes and their localisation

Popis výsledku v původním jazyce

We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler and operads coloured by a category) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant.nWe investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for -operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their -localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan stabilisation hypothesis for higher categories.

Název v anglickém jazyce

Homotopy theory of algebras of substitudes and their localisation

Popis výsledku anglicky

We study the category of algebras of substitudes (also known to be equivalent to the regular patterns of Getzler and operads coloured by a category) equipped with a (semi)model structure lifted from the model structure on the underlying presheaves. We are especially interested in the case when the model structure on presheaves is a Cisinski style localisation with respect to a proper Grothendieck fundamental localiser. For example, for the minimal fundamental localiser, the local objects in such a localisation are locally constant presheaves, and local algebras of substitudes are exactly algebras whose underlying presheaves are locally constant.nWe investigate when this localisation has nice properties. We single out a class of such substitudes which we call left localisable and show that the substitudes for -operads, symmetric, and braided operads are in this class. As an application we develop a homotopy theory of higher braided operads and prove a stabilisation theorem for their -localisations. This theorem implies, in particular, a generalisation of the Baez-Dolan stabilisation hypothesis for higher categories.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10101 - Pure mathematics

Projekt

—

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2022

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

American Mathematical Society. Transactions

ISSN

0002-9947

e-ISSN

1088-6850

Svazek periodika

375

Číslo periodika v rámci svazku

5

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

72

Strana od-do

3569-3640

Kód UT WoS článku

000807482000016

EID výsledku v databázi Scopus

2-s2.0-85127956772