The notion of pointlike sets provides a way to “measure the essential differences” between a given semigroup and those belonging to some pseudovariety of interest. 

We introduce the foundations for a general theory of pointlike sets. Roughly speaking, this theory consists of a “web” of commuting Galois connections between various lattices of functors and the lattice of pseudovarieties. Many universal motifs in the study of pointlikes—including the notion of “pointlike sets” itself—are exhibited as “naturally occurring features” of these Galois connections. 

In this talk, I will describe how the fundamental concepts and constructions of the theory are “extracted” from ubiquitous notions in the study of pointlikes—and, in particular, how these motifs alone are sufficient to recover the standard theory of pointlikes.

This is joint work with Karsten Henckell.