duality theory

duality theory

[Natalie]

I'm interested in  duality theory for an arbitrary category,
especially for "classical" algebraic categories( e.g., SEMI, the
category of semigroups and their homomorphisms).

I want to obtain such result: to build general "dualization
algorithm" (for varieties), and "to hang" fundamental
operations and identities at each step of this "algorithm", where they
arise.

But I haven't possibility to get books
(I'm think these books would help me)
such as
Borceux, "Categorical Algebra";
Clark/Davey, "Natural Dualities for the Working Algebraist";
Johnstone, "Stone Spaces";
Manes, "Algebraic theories"
and more others.

IS THERE DUALITY THEORY FOR THE CATEGORIES described above?


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My ideas in this direction are restricted only by the next:
1. Using factorization systems(in particular, via congruences lattice)
   for the category of algebras(but HOW in general situation, without
   special methods?)
2. Using inclusion of the category TH^op
   (considering as theory in the sense of (Barr/Wells)'s "Toposes, triples and theories")
   in the category MOD(TH) of models for this theory.
3. Via iso of categories (SET^(W))^op = CABA_(W^op)
  (for given endofunctor( or, narrow concept, functor part of triple) W on SET).
4.(main!!) Via generalization of the standart duality example
  (ComRing1)^op ~=~ AffSchemes
  What is the role of Birkhoff's subdirect representation theorem for
  algebras in the construction of the topological space SPEC, how we
  can construct (in general situation) the sheaf of algebras on this
  space?
  And the main: what the grounds of this construction( if it is
  possible)?
  How to prove directly the duality between algebraic and geometric
  theories ( if it is available)?
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The next questions/exersices  parallels this "algorithm":
A.
  The best test for this general theory --- to apply it for the  well-known
  duality  (ComRing1)^op ~=~ AffSchemes, mentioned above.
B.
  If (4.) is available, how we can in general terms to obtain the equivalence
  between the category CABA_(W^op) in (3.) and the correspondent
  category given by construction in (4.)?
C.(deeper)
  How the duality theory connect algebra, logics and topology?
D.
  What is this "algorithm in terms 2-categories?"
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 Natalie            natalie_reznik@myrealbox.com