TITLE: What do $\Gamma$-sets classify?
QUESTION [16 upvotes]: 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?

REPLY [2 votes]: Of course the above answer of @AndreasBlass exhausts it all, but it may be still amusing to compute directly the statement in the previous comment by @CharlesRezk.
One additional piece of information is this: for an object $X$ in a topos $T$, giving a geometric morphism from a topos $E$ to $T/X$ is equivalent to giving a geometric morphism $f:E\to T$ together with a point $1\to f^*(X)$. 
Applying this to $T=Psh(fSet^{op})$ and $X=$ the generic object, which is the embedding $X:(fSet^{op})^{op}\to Set$ gives that the category of pointed objects of $E$ is equivalent to the category of gm's from $E$ to $Psh(fSet^{op})/X$. Now the latter is equivalent to presheaves over the Grothendieck construction of $X$ which in turn is equivalent to the category of finite pointed sets (objects are pairs $(n,i)$ where $n$ is a finite set and $i$ is an element of $X(n)$, i.~e. of $n$, etc....), as a particular case of the equivalence
$$
Psh(C^{op})/\hom_C(c,-)\simeq Psh((c/C)^{op})
$$