nLab Gamma-set

Idea

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

The topos $\Set^{\Gamma^{op}}$ of $\Gamma$-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 $\Gamma$-set to be a pointed functor $X\colon \Gamma^{op}\to Set_{\ast}$, the category of pointed sets. The category of $\Gamma$-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.

Related notions

• Gamma-space
• Segal's category
• symmetric set

References

• Graeme Segal, Categories and cohomology theories, Topology 13 (1974).
• Alain Connes, Caterina Consani, Absolute algebra and Segal’s Gamma sets, [arxiv:1502.05585]