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We continue our study of tensor products in the operator system category. Wedefine operator system quotients and exactness in this setting and refine thenotion of nuclearity by studying operator systems that preserve various pairsof tensor products. One of our main goals is to relate these refinements ofnuclearity to the Kirchberg conjecture. In particular, we prove that theKirchberg conjecture is equivalent to the statement that every operator systemthat is (min,er)-nuclear is also (el,c)-nuclear. We show that operator systemquotients are not always equal to the corresponding operator space quotientsand then study exactness of various operator system tensor products for theoperator system quotient. We prove that an operator system is exact for the mintensor product if and only if it is (min,el)-nuclear. We give manycharacterizations of operator systems that are (min,er)-nulcear,(el,c)-nuclear, (min,el)-nulcear and (el,max)-nuclear. These characterizationsinvolve operator system analogues of various properties from the theory ofC*-algebras and operator spaces, including the WEP and LLP.

Quotients, exactness, and nuclearity in the operator system category