A quantale can be defined as a monoid in the monoidal category $\mathbf{Sup}$ of complete join semilattices, equipped with tensor product.

What are examples of non-diagonal comonoids in $\mathbf{Sup}$?

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    $\begingroup$ For instance, any localic group has an underlying Hopf algebra. $\endgroup$
    – Zhen Lin
    Dec 26, 2014 at 11:54
  • $\begingroup$ @ZhenLin : I do not quite understand. I see that a localic group is a monoid in $(\mathbf{Loc},\times)$ and this is the same thing as a comonoid in $(\mathbf{Frm},\oplus)$. How do we get a comonoid in $(\mathbf{Sup},\otimes)$ from that? Is the forgetful functor $\mathbf{Frm}\to\mathbf{Sup}$ lax comonoidal? $\endgroup$ Dec 26, 2014 at 13:04
  • 1
    $\begingroup$ A frame is a monoid in $\mathbf{Sup}$, and the coproduct in $\mathbf{Frm}$ is the tensor product in $\mathbf{Sup}$. $\endgroup$
    – Zhen Lin
    Dec 26, 2014 at 13:06
  • $\begingroup$ @ZhenLin: I get it. Just like abelian groups and commutative rings. Thanks. $\endgroup$ Dec 26, 2014 at 13:22


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