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,-):RegCat→Cat 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.
Read more about this topic: Regular Category
Famous quotes containing the words regular, logic and/or categories:
“It was inspiriting to hear the regular dip of the paddles, as if they were our fins or flippers, and to realize that we were at length fairly embarked. We who had felt strangely as stage-passengers and tavern-lodgers were suddenly naturalized there and presented with the freedom of the lakes and woods.”
—Henry David Thoreau (18171862)
“It is the logic of our times,
No subject for immortal verse
That we who lived by honest dreams
Defend the bad against the worse.”
—Cecil Day Lewis (19041972)
“Kitsch ... is one of the major categories of the modern object. Knick-knacks, rustic odds-and-ends, souvenirs, lampshades, and African masks: the kitsch-object is collectively this whole plethora of trashy, sham or faked objects, this whole museum of junk which proliferates everywhere.... Kitsch is the equivalent to the cliché in discourse.”
—Jean Baudrillard (b. 1929)