Dan - no choice of quantale lets us do that. A quantale is a very nice monoidal _preorder_, but we need a very nice monoidal _category_: the category \\(\mathbf{Set}\\).

\\(\mathbf{Set}\\) is not a quantale, but Equation 2.97 generalizes, and gives this:

\\[ (M * N)(x, z) := \bigsqcup_{y \in Y} M(x, y) \times N(y, z). \\]

where \\(\bigsqcup\\) means disjoint union of sets, and \\(\times\\) means Cartesian product of sets. \\(\bigsqcup\\), also known as 'coproduct', is a generalization of 'join'. \\(\times\\) is a special case of a monoidal structure \\(\otimes\\) in a monoidal category.

Remember, Fong and Spivak do the funny thing of first discussing categories enriched over preorders (for example quantales), then categories enriched over \\(\mathbf{Set}\\) (which we're discussing now - they're plain old categories), and finally categories enriched over arbitrary monoidal categories (which include both the previous examples).

Only at the final step will we be in the position to write down a matrix multiplication formula that specializes to handle both quantale-enriched categories and plain old categories!

And you will need to remind me to do this, if you want to see it, because Fong and Spivak say very little if anything about this.

However, you can already clearly see the analogy between the quantale-enriched formula

\\[ (M * N)(x, z) := \bigvee_{y \in Y} M(x, y) \otimes N(y, z) \\]

and the \\(\mathbf{Set}\\)-enriched formula

\\[ (M * N)(x, z) := \bigsqcup_{y \in Y} M(x, y) \times N(y, z). \\]

\\(\mathbf{Set}\\) is not a quantale, but Equation 2.97 generalizes, and gives this:

\\[ (M * N)(x, z) := \bigsqcup_{y \in Y} M(x, y) \times N(y, z). \\]

where \\(\bigsqcup\\) means disjoint union of sets, and \\(\times\\) means Cartesian product of sets. \\(\bigsqcup\\), also known as 'coproduct', is a generalization of 'join'. \\(\times\\) is a special case of a monoidal structure \\(\otimes\\) in a monoidal category.

Remember, Fong and Spivak do the funny thing of first discussing categories enriched over preorders (for example quantales), then categories enriched over \\(\mathbf{Set}\\) (which we're discussing now - they're plain old categories), and finally categories enriched over arbitrary monoidal categories (which include both the previous examples).

Only at the final step will we be in the position to write down a matrix multiplication formula that specializes to handle both quantale-enriched categories and plain old categories!

And you will need to remind me to do this, if you want to see it, because Fong and Spivak say very little if anything about this.

However, you can already clearly see the analogy between the quantale-enriched formula

\\[ (M * N)(x, z) := \bigvee_{y \in Y} M(x, y) \otimes N(y, z) \\]

and the \\(\mathbf{Set}\\)-enriched formula

\\[ (M * N)(x, z) := \bigsqcup_{y \in Y} M(x, y) \times N(y, z). \\]