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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 $X$?
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The closest I have come to finding anything about this is in [this $n$lab entry, titled *looping*](http://ncatlab.org/nlab/show/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?



What is the delooping of a looping?
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.


