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I think there was some terminological confusion, where the nLab defined Segal’s category $\Gamma$ to be a skeleton of finite pointed sets; I think it should be the category opposite to that. I’ve made edits at this article and at Gamma-space.
I have a memory (possibly false) of seeing both $\Gamma$ and its opposite called Segal’s category, so the main thing is to be reasonably consistent through any pages that use the term.
Thanks Todd, for looking into this. In my experience (say from the beginnings of some seminar talks), as Tim says, people tend to disagree on this convention. But I suppose there is of course one that is really the original convention, and to the extend that we manage to, it would be good to try to stick to that, as you do. Thanks.
The problem, of course, is that the original definition of $\Gamma$ makes it analogous not to $\Delta$ but to $\Delta^{op}$. That makes it very tempting to redefine “$\Gamma$” (perhaps even accidentally) so that it agrees with all other shape categories like $\Delta$, where the “geometric shapes” in question are presheaves on the category. One solution is to just discard the notation $\Gamma$ and use something else like $FinSet_\ast$ or $\mathcal{F}$…
This may all be true, but unfortunately the articles were not even consistent with themselves! E.g. under the definition which was originally there, it was claimed that presheaves on $\Gamma$ classify pointed objects. But this is true only if we change to Segal’s original definition, where the forgetful functor $\Gamma^{op} = FinSet_\ast \to Set$ is the generic model.
Oh certainly; I didn’t mean to imply otherwise. Thanks for fixing it!
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