A spectral sequence for spaces of maps between operads

Göppl, Florian; Weiss, Michael

doi:10.2140/agt.2024.24.1655

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A spectral sequence for spaces of maps between operads

Florian Göppl and Michael Weiss

Algebraic & Geometric Topology 24 (2024) 1655–1690

DOI: 10.2140/agt.2024.24.1655

Abstract

Under mild conditions on topologically enriched operads $P$ and $Q$ , the derived mapping space $RHom (P, Q)$ is the limit (sequential homotopy inverse limit) of a tower whose $n^{th}$ layer admits a description in terms of certain (small) diagrams $J_{n} (P)$ and $J_{n} (Q)$ . More precisely $J_{n} (P)$ is a $3$ –term diagram of spaces with action of $Σ_{n}$ of the form ${bound}_{n} (P) \to P (n) \to {cobound}_{n} (P)$ , where $P (n)$ is the space of $n$ –ary operations in $P$ . The statement takes some inspiration from manifold calculus, but the proof relies on the homotopical theory of dendroidal spaces and the concept of dendroidal nerve of an operad.

Keywords

operad, dendroidal space, spectral sequence

Mathematical Subject Classification

Primary: 18M75, 55P48

Secondary: 18N60

References

Publication

Received: 15 June 2022

Revised: 23 October 2022

Accepted: 18 November 2022

Published: 28 June 2024

Authors

Florian Göppl
Mathematics Institute University of Münster Frankfurt Germany

Michael Weiss
Mathematics Institute University of Münster Frankfurt Germany

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Volume 24, issue 3 (2024)