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.

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

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