# Gamma spaces and monoidal categories

In his paper "Categories and cohomology theories" 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

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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You can, as long as you can strictify associators... –  Fernando Muro Sep 29 '11 at 13:51
My hope was that one can somehow avoid strictifications of the category and see directly what happens, but maybe that is too naive. –  Ulrich Pennig Sep 29 '11 at 14:26
On the contrary I'd say it would be very complicated. Gamma spaces are too strict. Either you strictify associators or weaken the notion of Gamma space by means of a "homotopy coherent" version. –  Fernando Muro Sep 29 '11 at 17:31