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Let $M$ be a topological monoid with product $\mu$. Then $H_*(M)$ is a Hopf algebra with product $\mu_*$ and coproduct $\Delta_*$. The group-completion theorem by McDuff-Segal, 1976 gives that as a Pontrjagin ring, the localization $$H_*(M)[\pi_0M^{-1}]$$ is isomorphic to $$ H_*(\Omega BM)$$ through the map on homology induced by the canonical map $M\to \Omega BM$. It is a ring isomorphism mapping $\mu_*$ to the product of $H_*(\Omega BM)$.

Question:

(1). What is the coproduct structure of the localization $$H_*(M)[\pi_0M^{-1}]$$ induced by $\Delta_*$?

(2). Is the above ring isomorphism $$ H_*(M)[\pi_0M^{-1}]\to H_*(\Omega BM)$$ also an isomorphism of coalgebras preserving the coproduct $\Delta_*$?

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  • $\begingroup$ what is the antipode in $H_{\ast}(M)$ ? Is it not just a bialgebra ? $\endgroup$
    – Max
    Aug 22 '15 at 10:39
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  1. List item $H_*(M)[\pi _0(M)^{-1}]$ inherits its coalgebra structure from $H_*(M)$. i.e., there is a unique coalgebra structure so that the localization becomes a map of coalgebras. Concretely we "extend" the diagonal by declaring elements of $\pi _0(M)^{-1}$ to be group-like.
  2. List item Since the map $M\rightarrow \Omega BM$ is a map of spaces, it induces a map of coalgebras. So you get an isomorphism of coalgebras.
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