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We know that there the so called smooth algebras also known as $C^\infty$-rings. They can play an important role in modern treatment of differential geometry. Is there a coring analogue?

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    $\begingroup$ In some sense "naive smooth coalgebras" of nonzero dimension cannot exist. What I mean is that induced comultiplication on the (algebraic, not topological) dual vector space to a smooth ring never lands inside non-completed tensor square; it's easy to see that by multiplying together bump functions on small disjoint disks. With a bit more care one can say the same thing even for l.c.s. dual. $\endgroup$
    – Denis T
    Commented Apr 23 at 20:15

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