Filtered cocategories

TL;DR

This paper extends the theory of graded and conilpotent cocategories to an $bL$-filtered setting, allowing broader classes of filtrations and modules, and describes morphisms and coderivations in this context.

Contribution

It introduces a more general framework for filtered cocategories without restrictions on modules, broadening the scope of existing theories and including new classes of filtrations.

Findings

Descriptions of cofunctors and coderivations in completed filtered cocategories.

Construction of a partial internal hom as a tensor cocategory of coderivation quivers.

Extension of theory to include directed group filtrations.

Abstract

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 $\Lambda$-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.