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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?

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    $\begingroup$ 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
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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.

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  • $\begingroup$ 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
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    $\begingroup$ @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

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