The category $\Gamma^{\mathrm{op}}$ is defined to be a skeleton of the category of finite pointed sets (see also this question). Then $\Gamma$-spaces, meaning space-valued presheaves $\Gamma^{\mathrm{op}}\to \mathrm{Spaces}$, can be used to present spaces with commutative and associative multiplication up to all higher homotopies. This is similar to how the category $\Delta^{\mathrm{op}}$ can be defined as a skeleton of the category of finite total orders with distinct endpoints, and presheaves $\Delta^{\mathrm{op}}\to \mathrm{Spaces}$ (simplicial spaces) can be used to present spaces with associative multiplication up to all higher homotopies.

It is known that the topos of simplicial sets is the classifying topos for total orders with distinct endpoints. Does the topos of set-valued presheaves on $\Gamma$ have a similar interpretation?