Anindya wrote:

> I'm wondering why we insist that \\(\mathcal{V}\\) be symmetric in the definition of a \\(\mathcal{V}\\)-enriched category. The symmetry doesn't seem to play any explicit role in the definition.

Excellent question! There is no reason to insist that \\(\mathcal{V}\\) be symmetric.. Some more advanced theorems require it, but the concept of \\(\mathcal{V}\\)-enriched category makes perfect sense whenever \\(\mathcal{V}\\) is a monoidal poset.

In fact, it works just fine whenever \\(\mathcal{V}\\) is a monoidal preorder! The only reason I said "symmetric monoidal poset" is that Fong and Spivak restrict their attention to this case, so I'm assuming they will use these hypotheses at some point, and I don't want to have to worry about exactly when they need them.

> I'm wondering why we insist that \\(\mathcal{V}\\) be symmetric in the definition of a \\(\mathcal{V}\\)-enriched category. The symmetry doesn't seem to play any explicit role in the definition.

Excellent question! There is no reason to insist that \\(\mathcal{V}\\) be symmetric.. Some more advanced theorems require it, but the concept of \\(\mathcal{V}\\)-enriched category makes perfect sense whenever \\(\mathcal{V}\\) is a monoidal poset.

In fact, it works just fine whenever \\(\mathcal{V}\\) is a monoidal preorder! The only reason I said "symmetric monoidal poset" is that Fong and Spivak restrict their attention to this case, so I'm assuming they will use these hypotheses at some point, and I don't want to have to worry about exactly when they need them.