The "symmetric" property of Day convolution.

Question

This question has to be divided into the following parts:

1. The definition of Day convolution in nlab

   To define Day convolution, it assumes that $V$ be a closed symmetric monoidal category with all small limits and colimits, and $C$ be a monoidal category.

   see https://ncatlab.org/nlab/show/Day+convolution#definition

   Notice that nlab doesn't says that $C$ must be symmetric.

2. Day convolution form a monoidal category in nlab

   see https://ncatlab.org/nlab/show/Day+convolution#DayConvolutionYieldsMonoidalCategoryStructure

   That means, if we there is a tensor unit $y(I)$, then the category $([C,V], ⊗_{Day}, y(I))$ form a monoidal category automatically.

   Notice that nlab doesn't say that $C$ must be symmetric.

3. The definition of Day convolution in wikipedia

   To define Day convolution, it assumes that $C$ be a symmetric monoidal category. (Of course, $V$ must be monoidal category, because enriched)

   see https://en.wikipedia.org/wiki/Day_convolution

   Notice that wikipedia doesn't say that $V$ must be symmetric.

4. Day convolution form a monoidal category in wikipedia

   It says that

   If the category $V$ is a symmetric monoidal closed category, we can show this defines an associative monoidal product.

   see https://en.wikipedia.org/wiki/Day_convolution

   Since a monoidal category must satisfy associative law, that means if we expect that the category $([C,V], ⊗_{Day}, y(I))$ form a monoidal category, then $V$ must be symmetric, i.e. $C$ and $V$ are both symmetric monoidal category.

   It also provides a proof for this associative law, in which, it seems that the two symmetric /commutative laws be used.

My questions are:

1. Why the definition of Day convolution in nlab and wikipedia are different?

   I mean that, to define Day convolution, why nlab require $V$ to be a symmetric monoidal category, but wikipedia doesn't require symmetric on $V$ and vice versa...

2. Why the condition of "Day convolution form a monoidal category" in nlab and wikipedia are different?

   I mean that, to form a monoidal category under Day convolution, why wikipedia require both $C$ and $V$ are symmetric, but nlab doesn't require this condition?

3. Why Day convolution need some sort of "symmetric" property?

   I didn't see any symmetry intuition from this Day convolution formula:

   $F*G = \int^{x,y \in C} C(x \otimes y, -) \otimes Fx \otimes Gy$

PS: I apologize if the question is silly, I'm a category theory beginner, but this definition make me confusion...

Very thanks.

Answer 1

The description on the nLab is correct: $\mathscr C$ does not need to be symmetric, but $\mathscr V$ does. If $\mathscr C$ is symmetric, then the Day convolution tensor product on $[\mathscr C, \mathscr V]$ will also be symmetric. Wikipedia actually does require $\mathscr V$ to be symmetric, but delays stating this to establish why symmetry is important: it's necessary for the induced tensor product to be associative (and hence be monoidal). This matches Day's original setting.

As of the time of writing, Wikipedia does state that $\mathscr C$ should be symmetric, but this is unnecessary. Anyone can edit Wikipedia, so this could easily be addressed.