Monads and theories

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216224%3A14310%2F19%3A00113518" target="_blank" >RIV/00216224:14310/19:00113518 - isvavai.cz</a>

Výsledek na webu

<a href="https://www.sciencedirect.com/science/article/pii/S0001870819302580" target="_blank" >https://www.sciencedirect.com/science/article/pii/S0001870819302580</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.aim.2019.05.016" target="_blank" >10.1016/j.aim.2019.05.016</a>

Jazyk výsledku

angličtina

Název v původním jazyce

Monads and theories

Popis výsledku v původním jazyce

Given a locally presentable enriched category epsilon together with a small dense full subcategory A of arities, we study the relationship between monads on and identity-on-objects functors out of A, which we call A-pretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised on the one side as the A-nervous monads-those for which the conclusions of Weber's nerve theorem hold-and on the other, as the A-theories which we introduce here. The resulting equivalence between A-nervous monads and A-theories is best possible in a precise sense, and extends almost all previously known monad-theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak omega-groupoids. Besides establishing our general correspondence and illustrating its reach, we study good properties of A-nervous monads and A-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Mellies and Weber.

Název v anglickém jazyce

Monads and theories

Popis výsledku anglicky

Given a locally presentable enriched category epsilon together with a small dense full subcategory A of arities, we study the relationship between monads on and identity-on-objects functors out of A, which we call A-pretheories. We show that the natural constructions relating these two kinds of structure form an adjoint pair. The fixpoints of the adjunction are characterised on the one side as the A-nervous monads-those for which the conclusions of Weber's nerve theorem hold-and on the other, as the A-theories which we introduce here. The resulting equivalence between A-nervous monads and A-theories is best possible in a precise sense, and extends almost all previously known monad-theory correspondences. It also establishes some completely new correspondences, including one which captures the globular theories defining Grothendieck weak omega-groupoids. Besides establishing our general correspondence and illustrating its reach, we study good properties of A-nervous monads and A-theories that allow us to recognise and construct them with ease. We also compare them with the monads with arities and theories with arities introduced and studied by Berger, Mellies and Weber.

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

2019

Kód důvěrnosti údajů

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

Název periodika

Advances in Mathematics

ISSN

0001-8708

e-ISSN

—

Svazek periodika

351

Číslo periodika v rámci svazku

JUL 31 2019

Stát vydavatele periodika

US - Spojené státy americké

Počet stran výsledku

48

Strana od-do

1024-1071

Kód UT WoS článku

000475548900028

EID výsledku v databázi Scopus

2-s2.0-85066289226