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Dwyer–Kan homotopy theory for cyclic operads

Part of: Homotopy theory Applied homological algebra and category theory Categories with structure

Published online by Cambridge University Press:  14 January 2021

Gabriel C. Drummond-Cole  and
Philip Hackney
Gabriel C. Drummond-Cole
Affiliation:
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang37673, Republic of Korea ([email protected])
Philip Hackney
Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA, USA ([email protected])
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Abstract

We introduce a general definition for coloured cyclic operads over a symmetric monoidal ground category, which has several appealing features. The forgetful functor from coloured cyclic operads to coloured operads has both adjoints, each of which is relatively simple. Explicit formulae for these adjoints allow us to lift the Cisinski–Moerdijk model structure on the category of coloured operads enriched in simplicial sets to the category of coloured cyclic operads enriched in simplicial sets.

Keywords

cyclic operadQuillen model categorycoloured operadadjoint functorDwyer–Kan equivalence

MSC classification

Primary: 55P48: Loop space machines, operads
Secondary: 55U35: Abstract and axiomatic homotopy theory 18D50: Operads 18D20: Enriched categories (over closed or monoidal categories)
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 1 , February 2021 , pp. 29 - 58
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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