Could anybody point to me a good reference for defining the $\Gamma$-category associated to a permutative category (better also with some illustrative examples)?
Dan Freed in his notes does provide a definition but many details are left out.
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There are several different, provably equivalent, definitions. Construction 10 in http://www.math.uchicago.edu/~may/PAPERS/23.pdf is one example. It is used to prove the uniqueness of a machine taking permutative categories to spectra. Therefore, any two reasonable constructions of $\Gamma$-categories from permutative categories give rise to equivalent spectra. Section 3 of Segal's original paper ``Categories and cohomology theories'' gives examples, but it is unclear whether you are asking for examples of permutative categories (plentiful since any small symmetric monoidal category is equivalent to a permutative category) or of examples of different constructions of the associated $\Gamma$-category. One different, more categorical construction, is given in Theorem 3.4 of http://www.math.uchicago.edu/~may/PAPERS/32.pdf. (I should apologize for referring to my old papers since there are many other sources; that is just quickest for me.)
answered Aug 14, 2016 at 22:10
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$\begingroup$ Thank you so much for the references Prof May! I am actually asking for examples of constructions of the associated $\Gamma$-category. I will read through your and Segal's paper to find them out. I will come back to ask for more if needed. $\endgroup$ Commented Aug 14, 2016 at 22:48
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$\begingroup$ I have a related question, math.stackexchange.com/questions/4910494/…. Could Prof May give an answer? $\endgroup$ Commented May 4 at 4:28