\chapter{Galois connections}
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dual group and Pontryagin duality:
Hom_Grp(G,R) = Hom_Grp(G,R^*) = Hom_Rng(ZG,R)
characters are group of units of dual ring;
dual ring is group ring of characters.
Unfortunately, this is generally too small.
(EG of simple group, or S_n)
Hence need something fancier -- get to Tanaka duality!
(Look at whole category!)
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\section{Definition and general concepts}
- closure
Galois connections yield closures,
and many interesting closures come from
\section{Examples}
2 notable classes:
- group actions
- dualities
Galois theory:
- subgroups of Galois group
- extension fields
Covering spaces:
\subsection{Dualities}
Annihilators / perps:
Given $S \subset M$, $\Ann S < M^*$.
Spec:
ideals and zero sets
close
\subsection{Others}
Lattices: can define lattice in terms of a closure operator
EG in topological spaces (closure or dually, interior)
\section{Categorical generalization}
- Galois connections and closures
generalize to
- Adjoint functors and monads
- Galois connections and closures
are the poset example
Topoi and representable functors?