Co-operational bivariant theory

TL;DR

This paper introduces a dual concept called co-operational bivariant theory, extending the framework of operational bivariant theory to contravariant functors, and relates it to generalized cohomology operations.

Contribution

It constructs a co-operational bivariant theory for contravariant functors, providing a dual perspective to Fulton and MacPherson's operational bivariant theory.

Findings

Defines co-operational bivariant groups for contravariant functors.

Shows that for cohomology theories, these groups correspond to cohomology operations.

Establishes a dual framework enriching the theory of bivariant theories.

Abstract

For a covariant functor W. Fulton and R. MacPherson defined \emph{an operational bivariant theory} associated to this covariant functor. In this paper we will show that given a contravariant functor one can similarly construct a ``dual" version of an operational bivariant theory, which we call a \emph{co-operational} bivariant theory. If a given contravariant functor is the usual cohomology theory, then our co-operational bivariant group for the identity map consists of what are usually called ``cohomology operations". In this sense, our co-operational bivariant theory consists of \emph{``generalized"} cohomology operations.