The partition poset complex and the Goodwillie derivatives of the identity in spaces

TL;DR

This paper constructs a homotopy-coherent operad structure on the derivatives of the identity functor in spaces, offering a new perspective and connecting it to algebraic structures in spectra.

Contribution

It introduces a canonical operad structure on derivatives of the identity functor and relates primitives of commutative coalgebras to this operad, expanding understanding of homotopy-coherent structures.

Findings

Operad structure on derivatives of the identity functor in spaces

New description of the operad structure via cosimplicial pairings

Primitives of commutative coalgebras form an algebra over this operad

Abstract

We produce a canonical highly homotopy-coherent operad structure on the derivatives of the identity functor in spaces via a pairing of cosimplicial objects, providing a new description of an operad structure on such objects first described by Ching. In addition, we show the derived primitives of a commutative coalgebra in spectra form an algebra over this operad.