In his paper "[Categories and cohomology theories](http://nlab.mathforge.org/nlab/files/SegalCategoriesAndCohomologyTheories.pdf)" Graeme Segal gives examples how to construct a Gamma category and therefore also a Gamma space from a strict monoidal category like finite chain complexes of complex fin.-dim. vector bundles with the alternating sum over the dimensions equal to 1 and chain homotopy equivalences as morphisms. This yields a nice model for $BU\_{\otimes}$ and proves that this is an infinite loop space. My question is



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> Can you still construct a Gamma category if the monoidal structure is not strict, for example if there is a non-trivial associator? If such a Gamma category $C$ exists, what would $C(3)$ and the induced morphisms look like?
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Gamma spaces and monoidal categories
If your monoidal category is not strict you can first form a multicategory out of it. This process involves some choices (how to bracket higher tensor products) but they are not essential (see e.g. "Tom Leinster - Higher Categories, Higher Operads", chapter 3.3 for discussion). In nature symmetric monoidal categories very often come from multicategoires, like the category of vector bundles etc. So in some sense the underlying multicategory is the even more natural objects.


As a next step you can extract a gamma category out of a multicategory, see "Permutative categories, multicategories, and algebraic K-theory" by Elmendorf and Mandell. This is a direct generalization of the Gamma category of Segal associated to a permutative category. In particular if your category from the start was permutative it gives back the old construction of Segal.


