(for martin)

so let x be a symmetric monoidal cocomplete k-linear category and m : y -> z a morphism in x. then to universally require m = 0 (without disturbing the symmetric monoidal cocomplete k-linear structure) is equivalent to universally requiring 1 : z -> z to be its cokernel, which in turn is equivalent to universally requiring the quotient map z -> cok(m) to be invertible. and the universal symmetric monoidal cocontinuous k-linear functor in question can be explicitly constructed as the left adjoint reflector onto the full subcategory of x consisting of just those objects that "believe" that all of the tensor translates of the diagram y -> z -> z -> 0 are exact; where the arrows in the diagram are respectively m, 1, and 0; and where an object w "believes" that a diagram d is exact iff homming into w turns d into an exact diagram.

i think that i can prove the above claim, though as usual there are details that i need to check carefully. the general principle is that given a cocomplete category x, and given some class of cocones in x, the universal cocontinuous functor that takes the cocones in that class to colimit cocones is the left adjoint reflector onto the full subcategory consisting of those objects that "believe" that all of the cocones in that class are colimit cocones.

(i _think_ that that principle is true in general, without extra "set-theoretic" assumptions; but i'm not certain of it.)

so for example take x to be the graded modules of polynomials in n+1 degree 1 variables, and take m to be the projection from the unit graded module to its minimal quotient... this may give a useful way of thinking about the quasicoherent sheaves over projective space ...

i hope that i'm not making huge mistakes here... i'm pretty awake now, but still capable of making pretty big mistakes ...