Topological linear compactness for Grothendieck categories. Theorem of Tychonoff. Applications to coalgebras

We show the Tychonoff's theorem for a Grothendieck category with a set of small projective generators. Strictly quasi-finite objects for semiartinian Grothendieck categories are characterized. We apply these results to the study of the Morita duality of dual algebra of a coalgebra.