Galois Connections

About 1830 E. Galois discovered and investigated a connection, for a given field extension K -* L, between the collection of all subfields of L containing K and the collection of all automorphisms of L leaving K pointwise fixed. The formal properties of this connection remain valid in more abstract settings. In 1940 G. Birkhoff [1] associated wi th any relation a connection, which he called a polarity. Generalizing this concept, O. Ore [8] introduced in 1944 Galois connexions between par t ia l ly ordered sets. These, as we l l as the polarities, have a contravariant form. Its covariant version was introduced in 1953 by J. Schmidt [11] under the name Galots connections of mixed type. Categorists observed that these connections are nothing else but adj0int s i tuations between par t ia l ly ordered sets, considered in the s tandard w a y as categories (see S. Mac Lane [7]). Unfor tuna te ly most properties of Galois connections, in fact a l l of the interesting ones, are no longer val id in the realm of adjoint situations. Hence, for Galois connections, adjoint functors form an inappropriate level of generality. The aim of this note is to provide suitable levels of generality.