Dikran Dikranjan

Functorial subgroups of topological groups

The connected component c(G) of a topological group G has the following obvious property: 
(\dag) f(c(G)) \subseteq c(H) for every continuous group homomorphism f: G \to H.
A functorial subgroup is a subgroup \mathtt{r}(G) assigned to every topological group G with the property (\dag) with c(-) replaced by \mathtt{r}(-).
Other prominent examples of functorial subgroups are the quasi-component q(G) and the arc-component a(G) of G.  The term functorial subgroup is explained by the categorical interpretation of the property (\dag) which simply says that the assignment G \mapsto \mathtt{r}(G) is a functor from the category TG of topological groups to itself. 
The union \mathrm{comp}(G) of all compact subgroups of a topological group G satisfies (\dag), but \mathrm{comp}(G) need not be a subgroup of G when G is not abelian (this is why one may speak of a functorial subset, in similar cases). 
The aim of this mini-course is to introduce various  functorial subgroups and  functorial subsets of the topological groups and discuss: 
(a) general properties of  functorial subsets (with particular emphasis on the subcategory TAG of topological abelian groups); 
(b) relevant examples of functorial subgroups (mainly in TAG) and their interrelations; 
(c) applications of functorial subgroups in the structure theory of topological groups (e.g., the Resolution Theorem, functorial topologies, the Open Mapping Theorem, etc.) 
(d) classification of a class of functorial subgroups in TAG with some natural  properties. 

The choice of the topic and many of the results are inspired by my recent joint work with Wayne Lewis, Peter Loth and Adolf Mader [1,2].

Bibliography
[1] D. Dikranjan, W. Lewis, P. Loth and A. Mader, A distinguished subgroup of compact abelian groups, Axioms 2022, 11, 200.
[2] D. Dikranjan, W. Lewis, P. Loth and A. Mader, Fat Delta 2: functorial subgroups of topological abelian groups, Topology Proc. 2022, to appear.