The ghost length and duality on the chain and cochain type levels

TL;DR

This paper explores dualities between chain and cochain levels in algebraic topology, establishing equalities, conditions for finiteness, and bounds related to Lie groups and Koszul duality in derived categories.

Contribution

It introduces duality relations between chain and cochain levels, providing new equalities, finiteness conditions, and bounds in the context of derived and coderived categories.

Findings

Established equalities between cochain and chain level maps.

Derived conditions for finiteness of levels based on duality.

Provided a lower bound for the cochain level of the diagonal map on classifying spaces.

Abstract

We establish equalities between cochain and chain type levels of maps by making use of exact functors which connect appropriate derived and coderived categories. Relevant conditions for levels of maps to be finite are extracted from the equalities which we call duality on the levels. Moreover, we give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by considering the ghostness of a shriek map which appears in derived string topology. A variant of Koszul duality for a differential graded algebra is also discussed.