FOM: Deep theorem in category theory

[Till Mossakowski]

Steve Simpson (Sun May  2 22:08:40 1999) asked for a deep theorem
in category theory.

I want to mention the following correspondence theorems

equationally defined classes of algebras - monads
implicationally defined classes of algebras - algebraic categories
implicationally defined classes of partial algebras - essentially algebraic categories

Moreover, implicationally defined subcategories can be related
to epi-reflective subcategories and the existence of adjoints
in a very general setting, applicable to algebras, partial algebras,
compact Hausdorff spaces and other categories.

Till Mossakowski