Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
I have added an answer to another mathoverflow post that is very much related. mathoverflow.net/a/462976/76299 To spare you the click: In the case of a symmetric monoidal groupoid, the S^-1 S construction is the unstraigthening of the diagonal action of S on S x S. This characterization can be completely generalized in the setting of infinity-categories
Not an answer, but a clarification: The reason the telescope construction is often used is because the only input one needs is the homology of the underlying space one started with and some knowledge about $\pi_0$ of the space. This is good enough to characterize some of the standard examples (Say for the groupoid of finite sets, one computes the homology of the infinite symmetric group). Your construction doesn't seem to have good computational properties at a first glance: To compute its homology one would need to apply the homotopy colimit spectral sequence and the Künneth theorem.
