nLab Gamma-set

Redirected from "$\Gamma$-sets".

Idea

Let Γ op\Gamma^{op} (see Segal's category) be the skeleton of the category of finite pointed sets. We write n̲\underline{n} for the finite pointed set with nn non-basepoint elements. Then a Γ\Gamma-set is a functor X:Γ opSetX\colon \Gamma^{op}\to Set.

The topos Set Γ op\Set^{\Gamma^{op}} of Γ\Gamma-sets is the classifying topos for pointed objects (MO question). For more on this see also at classifying topos for the theory of objects.

By contrast, in (Connes & Consani 15, 2.1) the authors define a Γ\Gamma-set to be a pointed functor X:Γ opSet *X\colon \Gamma^{op}\to Set_{\ast}, the category of pointed sets. The category of Γ\Gamma-sets in this sense is no longer a topos, but it is a symmetric monoidal closed category. This construction is used in their approach to the field with one element.

References

Last revised on April 16, 2024 at 13:23:49. See the history of this page for a list of all contributions to it.