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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.

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