| Summary: | The idea of an operational category over A generalizes the notions of tripleable and equational category over A, and also the dual notions of cotripleable and coequational category. An operational category, U:D (--->) A is given by a presentation ((theta),H) (UNFORMATTED TABLE FOLLOWS) === D C('T) === C('(theta)) === A C('B) === H*(TABLE ENDS) === where (theta) is a functor bijective on objects and D is a specified pullback. R:Op(A) (--->) Cat/A is defined as the category of operational categories (and functors) with given presentations. Another category, Op(,o)(A) over Cat/A of operational categories with standard presentations is also defined. There is a fixed theory (theta)(,o), employed in every standard presentation. Op(,o)(A) is a retract of Op(A) over Cat/A: (UNFORMATTED TABLE FOLLOWS) === i === Op(,o)(A) Op(A) === s === R(,o) R === Cat/A(TABLE ENDS) === i.e. every operational category (and functor) has a standard presentation (but not s(REVTURNST)i!). Also R(,o) has a left adjoint L(,o) and Op(,o)(A) is complete. Finally, there is a category of algebras, S(,*)-Alg over Cat/A such that Op(,o)(A) (DBLTURN) S(,*)-Alg over Cat/A. Thus, the operational categories can be determined by their internal structure, without reference to any presentation. Some properties of operational categories and some special cases are also examined.
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