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If we apply the day convolution to , we get an asymmetric monoidal product from what appears to be a symmetric monoidal product (ordinal sum) on the augmented simplex category. The strange thing is, there's nothing that appears to be asymmetric in the formula for the day convolution, so why do we end up with an asymmetric monoidal product (the join)?
Is the ordinal sum symmetric? I have a feeling that there is something strange there. may be wrong ... often am!
Yeah, I'm not sure either. Maybe someone will be able to shed some light on it.
Try all the axioms of a sym. tensor product. That cannot do any harm! It may be symm. but it is worth checking.
The point is surely similar to the order reversal on ordinals being an involution but is does not give an involution on the functor category.
(Look at p. 13 of Phil Ehlers thesis.)
I think the point is that the `obvious' isomorphism is NOT in the category $\Delta$
For a monoidal product to be symmetric, does it need to actually induce isomorphisms between the two objects and , or does there just need to exist one?
The two 'bifunctors' have to be naturally isomorphic at least. It is not clear what`induce isomorphisms' might mean.
Oh, I see! They're braided (but not symmetric).
I think it is worse than that. There is no natural isomorphism between the two sides.
What's an example of a symmetric monoidal product then? I can't think of one offhand. Maybe the tensor product? If not, the symmetric product? (of modules, of course).
Those are examples but why be that complicated. Cartesian product will do.
The equivalent definition stated at join of quasi-categories
exhibits the directed structure of the join more manifestly: it's all in the orientation of .
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