10
$\begingroup$

Derived mapping spaces between little $d$-disks operads $E_d$ play an important role in embedding calculus. For example, Dwyer-Hess expresses the homotopy of framed long knots as loop spaces such mapping spaces, a result which was generalized by Boavida de Brito-Weiss.

When stating these results, the $E_d$-operad is usually unitary: its space of 0-ary operations is contractible (in fact a point). There is also a non-unitary version $E_d^{nu}$, where replace the $0$-ary operations by an empty set. There is a natural forgetful map from the derived mapping space of unitary operads to that of non-unitary operads

$$Map^h_{\mathsf{Op}}(E_m,E_n) \to Map^h_{\mathsf{Op}}(E_m^{nu},E_n^{nu}).$$

Is this a weak equivalence?

This is known if we assume $m=n$ and we replace the operads by their rationalizations, by Section 7 of Fresse-Willwacher.

$\endgroup$
1
  • 2
    $\begingroup$ It was pointed out to me that arxiv.org/pdf/1111.2723.pdf proves that in the case $m=1$ it is an injection on $\pi_0$ and a weak equivalence onto the path components it hits. $\endgroup$
    – skupers
    Nov 26, 2020 at 2:28

1 Answer 1

4
$\begingroup$

A positive answer is the main theorem of my paper with Krannich and Horel, Two remarks on spaces of maps between operads of little cubes. The proof uses a result of Haugseng and Kock to reduce it to a theorem of Lurie.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.