What is $\mathbf{B}\Omega A$, where $A$ is a pointed object of an $(\infty,1)$ category with point $*\to A$, $\Omega A$ is the loop space of $A$, and $\mathbf{B}X$ is the delooping of $X$?

The closest I have come to finding anything about this is in this $n$lab entry, titled looping, it is mentioned that the based loop space object of $A$, i.e., $\mathbf{B}\Omega_{\mathrm{pt}}A\simeq A$.

Edit: How does one prove the following statement: there is a map $\mathbf B\Omega A \to A$ which is a weak equivalence onto the connected component of the base point, which is given in an answer by John Klein below?

  • 5
    $\begingroup$ Regarding your edit: look up adjoint functors, in particular the book of May Simplicial Objects in Algebraic Topology (djvu format). To echo the words of May: learn about simplicial sets and spaces first; then you can actually appreciate what is happening in the $\infty$-categorical setting. $\endgroup$ – David Roberts May 12 '14 at 23:14
  • $\begingroup$ @DavidRoberts Thanks for the book. I'll try to read it. $\endgroup$ – user62675 May 12 '14 at 23:23

A simple example should indicate the general phenomenon: Let $A$ be a discrete based set. The $\Omega A$ is a point, so $B \Omega A$ is a point.

The general phenomenon is this: $B\Omega A$ is always connected, whereas $A$ needn't be. The statement which is true is that there's a map $B\Omega A \to A$ which is a weak equivalence onto the connected component of the base point.

  • $\begingroup$ Thanks for the quick answer. How could one prove the last statement? I'll add that into the question. $\endgroup$ – user62675 May 12 '14 at 22:53
  • 2
    $\begingroup$ @SanathDevalapurkar, look at the homotopy groups: $\Omega$ shifts them down by one (forgetting about $\pi_0$), and $B$ shifts them back up by one. $\endgroup$ – Mike Shulman May 15 '14 at 17:54
  • $\begingroup$ @mikeshulman Thanks for providing me with that geometrical interpretation. I understand it now! $\endgroup$ – user62675 May 15 '14 at 17:59

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.