# $\Gamma$-sets vs $\Gamma$-spaces

I know that every $$\Gamma$$-space is stably equivalent to a discrete $$\Gamma$$-space, i.e. a $$\Gamma$$-set.

For example, Pirashvili proves, as theorem 1.2 of Dold-Kan Type Theorem for $$\Gamma$$-Groups, that every $$\Gamma$$-space is stably equivalent to a discrete $$\Gamma$$-group (i.e. a group object in the category of $$\Gamma$$-sets).

What I would like to know if this can be done in a "nice way," similar to how a quasi-category can be obtained from a Segal space. (For example, Pirashvili's argument is quite indirect and the its functoriality or universal properties are unclear). More precisely:

Consider the Bousfield-Friedlander model structure on $$\Gamma$$-spaces (here "space", will mean pointed simplicial sets) whose equivalences are the stable equivalences.

If $$X$$ is a simplicial set, denote by $$X_0$$ its set of vertices $$X([0])$$. If $$X$$ is a $$\Gamma$$-space, let $$X_0$$ be the application of this construction levelwise in $$\Gamma$$''. i.e.:

$$X_0 : \Gamma^{op} \overset{X}{\rightarrow} sSet_* \overset{(\_)_0}{\rightarrow} Set_*$$

Question: Given $$X$$ a $$\Gamma$$-space fibrant in the Bousfield Friedlander model structure, is the natural morphism:

$$X_0 \rightarrow X$$

a (stable) weak equivalence ?

I believe this the right question to ask, but if there is something very similar with a positive answer (maybe "fibrant in the Bousfield-Friedlander model structure" is not quite the right condition?), please let me know!

Note that this exactly how the equivalence between complete Segal spaces (seen as simplicial-spaces satisfying a fibrancy condition) and quasi-categories works.