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@article{1642196,
author = {Bourke, John Denis and Garner, Richard Henry George},
article_location = {SAN DIEGO},
article_number = {JUL 31 2019},
doi = {http://dx.doi.org/10.1016/j.aim.2019.05.016},
keywords = {Monad; Lawvere theory; Nerve},
language = {eng},
issn = {0001-8708},
journal = {Advances in Mathematics},
title = {Monads and theories},
url = {https://www.sciencedirect.com/science/article/pii/S0001870819302580},
volume = {351},
year = {2019}
}
TY - JOUR
ID - 1642196
AU - Bourke, John Denis - Garner, Richard Henry George
PY - 2019
TI - Monads and theories
JF - Advances in Mathematics
VL - 351
IS - JUL 31 2019
SP - 1024-1071
EP - 1024-1071
PB - ACADEMIC PRESS INC ELSEVIER SCIENCE
SN - 00018708
KW - Monad
KW - Lawvere theory
KW - Nerve
UR - https://www.sciencedirect.com/science/article/pii/S0001870819302580
L2 - https://www.sciencedirect.com/science/article/pii/S0001870819302580
N2 - 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.
ER -
BOURKE, John Denis a Richard Henry George GARNER. Monads and theories. Online. \textit{Advances in Mathematics}. SAN DIEGO: ACADEMIC PRESS INC ELSEVIER SCIENCE, 2019, roč.~351, JUL 31 2019, s.~1024-1071. ISSN~0001-8708. Dostupné z: https://dx.doi.org/10.1016/j.aim.2019.05.016. [citováno 2024-04-24]
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