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LSpice
LSpice
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I believe you meant to write $Q(BG_+)$ in the first paragraph of your post, where $Q = \Omega^\infty\Sigma^\infty$. The this result is really a folk theorem and is sometimes called the "Barratt-Priddy-Quillen-Segal""Barratt–Priddy–Quillen–Segal" theorem. This is not a folk theorem but rather it is a theorem with a folk authorship in that none of these people wrote down the theorem in this context as far as I know.

One way to deduce it is to use the the Group Completion Theorem (see e.g., McDuff, D.; Segal, G. Homology fibrations and the "group-completion'' theoremHomology fibrations and the “group-completion” theorem. Invent. Math. 31 (1975/76), no. 3, 279–284).

The classifying space the category of finite free $G$-sets and their isomorphisms defines a topological monoid $M$. The group completion theorem tells us in this case that $\Omega B M$ coincides with $Q(BG_+)$.

I believe you meant to write $Q(BG_+)$ in the first paragraph of your post, where $Q = \Omega^\infty\Sigma^\infty$. The this result is really a folk theorem and is sometimes called the "Barratt-Priddy-Quillen-Segal" theorem. This is not a folk theorem but rather it is a theorem with a folk authorship in that none of these people wrote down the theorem in this context as far as I know.

One way to deduce it is to use the the Group Completion Theorem (see e.g., McDuff, D.; Segal, G. Homology fibrations and the "group-completion'' theorem. Invent. Math. 31 (1975/76), no. 3, 279–284).

The classifying space the category of finite free $G$-sets and their isomorphisms defines a topological monoid $M$. The group completion theorem tells us in this case that $\Omega B M$ coincides with $Q(BG_+)$.

I believe you meant to write $Q(BG_+)$ in the first paragraph of your post, where $Q = \Omega^\infty\Sigma^\infty$. The this result is really a folk theorem and is sometimes called the "Barratt–Priddy–Quillen–Segal" theorem. This is not a folk theorem but rather it is a theorem with a folk authorship in that none of these people wrote down the theorem in this context as far as I know.

One way to deduce it is to use the the Group Completion Theorem (see e.g., McDuff, D.; Segal, G. Homology fibrations and the “group-completion” theorem. Invent. Math. 31 (1975/76), no. 3, 279–284).

The classifying space the category of finite free $G$-sets and their isomorphisms defines a topological monoid $M$. The group completion theorem tells us in this case that $\Omega B M$ coincides with $Q(BG_+)$.

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John Klein
John Klein
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I believe you meant to write $Q(BG_+)$ in the first paragraph of your post, where $Q = \Omega^\infty\Sigma^\infty$. The this result is really a folk theorem and is sometimes called the "Barratt-Priddy-Quillen-Segal" theorem. This is not a folk theorem but rather it is a theorem with a folk authorship in that none of these people wrote down the theorem in this context as far as I know.

One way to deduce it is to use the the Group Completion Theorem (see e.g., McDuff, D.; Segal, G. Homology fibrations and the "group-completion'' theorem. Invent. Math. 31 (1975/76), no. 3, 279–284).

The classifying space the category of finite free $G$-sets and their isomorphisms defines a topological monoid $M$. The group completion theorem tells us in this case that $\Omega B M$ coincides with $Q(BG_+)$.