Two remarks on spaces of maps between operads of little cubes

TL;DR

This paper investigates the properties of derived mapping spaces between little cubes operads, revealing their equivalence to non-unitary versions and characterizing all endomorphisms as automorphisms, with discussions on variants.

Contribution

It establishes key equivalences and automorphism properties of derived mapping spaces between little cubes operads, extending to variants with localizations and tangential structures.

Findings

Mapping spaces between $E_d$ operads are equivalent to those between non-unitary versions.

All endomorphisms of $E_d$ are automorphisms.

Results extend to variants with localizations and tangential structures.

Abstract

We record two facts on spaces of derived maps between the operads $E_d$ of little $d$-cubes. Firstly, these mapping spaces are equivalent to the mapping spaces between the non-unitary versions of $E_d$. Secondly, all endomorphisms of $E_d$ are automorphisms. We also discuss variants for localisations of $E_d$ and for versions with tangential structures.