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Ali, you have misunderstood Steven's answer. Take seriously that sets are discrete spaces and geometric realization of a simplicial or bisimplicial set is a special case of geometric realization of a simplicial or bisimplicial space. That is much more natural and prevents mistakes such as saying that the realization of a bisimplicial set is a simplicial set. It isn't.
Everything else I've published is on my web page. There are already pirated versions on line (the first one I saw classified the book as science fiction!), but please do not download one. Reputable publishers will go out of business if they do not have enough time to at least recoup costs of production and distribution before their books go on line. Therefore I have not put it on line yet.
Do you really mean ``commute''? In any case, probably the closest thing would be to understand moduli spaces of infinite loop structures (there is relevant spectrum level work of Goerss and Hopkins). By way of example, there is a beautiful but very special case where there is an answer: Adams and Priddy proved that BSU and BSO have unique infinite loop structures (whereas BU and BO do not: $\oplus$ and $\otimes$ give inequivalent infinite loop structures)
It was a long time ago and I don't know that I have records, but Frank Quinn followed essentially this path in trying to dualize my recognition principle for n-fold loop spaces. I found a mistake right away and his preprint never saw the light of day. Probably at least 30 years ago, but the moral is that this is not something that you can expect to work: it's easy to dualize formally, but the dual claim will likely fail if there is any non-formal, calculational, content to the proof.
