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 $\mathbf{Sup}$?
$\begingroup$
$\endgroup$
4
-
2$\begingroup$ For instance, any localic group has an underlying Hopf algebra. $\endgroup$– Zhen LinDec 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? $\endgroup$– Gejza JenčaDec 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 LinDec 26, 2014 at 13:06
-
$\begingroup$ @ZhenLin: I get it. Just like abelian groups and commutative rings. Thanks. $\endgroup$– Gejza JenčaDec 26, 2014 at 13:22
Add a comment
|