Filtered cocategories

We recall the notions of a graded cocategory, conilpotent cocategory, morphisms of such (cofunctors), coderivations and define their analogs in $\mathbb{L}$-filtered setting. The difference with the existing approaches: we do not impose any restriction on $\mathrm{\Lambda }$-modules of morphisms (unlike Fukaya and collaborators), we consider a wider class of filtrations than De~Deken and Lowen (including directed groups $\mathbb{L}$). Results for completed filtered conilpotent cocategories include: cofunctors and coderivations with value in completed tensor cocategory are described, a partial internal hom is constructed as the tensor cocategory of certain coderivation quiver, when the second argument is a completed tensor cocategory.