Regular Category - Regular Logic and Regular Categories

Regular Logic and Regular Categories

Regular logic is the fragment of first-order logic that can express statements of the form


,


where and are regular formulae i.e. formulae built up from atomic formulae, the truth constant, binary meets and existential quantification. Such formulae can be interpreted in a regular category, and the interpretation is a model of a sequent


,


if the interpretation of factors through the interpretation of . This gives for each theory (set of sequences) and for each regular category C a category Mod(T,C) of models of T in C. This construction gives a functor Mod(T,-):RegCatCat from the category RegCat of small regular categories and regular functors to small categories. It is an important result that for each theory T and for each category C, there is a category R(T) and an equivalence


,


which is natural in C. Up to equivalence any small regular category C arises this way as the classifying category, of a regular theory.

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