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On autoequivalences of the \((\infty , 1)\)-category of \(\infty \)-operads

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Abstract

We study the \((\infty , 1)\)-category of autoequivalences of \(\infty \)-operads. Using techniques introduced by Toën, Lurie, and Barwick and Schommer-Pries, we prove that this \((\infty , 1)\)-category is a contractible \(\infty \)-groupoid. Our calculation is based on the model of complete dendroidal Segal spaces introduced by Cisinski and Moerdijk. Similarly, we prove that the \((\infty , 1)\)-category of autoequivalences of non-symmetric \(\infty \)-operads is the discrete monoidal category associated to \(\mathbf {Z}/2\mathbf {Z}\). We also include a computation of the \((\infty ,1)\)-category of autoequivalences of \((\infty , n)\)-categories based on Rezk’s \(\Theta _n\)-spaces.

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Authors and Affiliations

  1. Institute for Mathematics, Astrophysics and Particle Physics, Radboud Universiteit Nijmegen, Heyendaalseweg 135, 6525 AJ, Nijmegen, The Netherlands

    Dimitri Ara, Moritz Groth & Javier J. Gutiérrez

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  1. Dimitri Ara
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  2. Moritz Groth
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  3. Javier J. Gutiérrez
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Correspondence to Dimitri Ara.

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The authors were supported by the Dutch Science Foundation (NWO). The third-named author was also supported by the Spanish Ministry of Education and Science under MEC-FEDER Grant MTM2010-15831 and by the Generalitat de Catalunya as a member of the team 2009 SGR 119.

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Ara, D., Groth, M. & Gutiérrez, J.J. On autoequivalences of the \((\infty , 1)\)-category of \(\infty \)-operads. Math. Z. 281, 807–848 (2015). https://doi.org/10.1007/s00209-015-1509-5

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