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Stream: learning: questions

Topic: Day convolution and Kan extensions

Naso (Nov 10 2021 at 13:26):

the Day convolution can be described as a left Kan extension of the 'external tensor product' along the tensor product (https://ncatlab.org/nlab/show/Day+convolution#InTermsOfCoends) . What happens if you take the right Kan extension of the same diagram?

Nathanael Arkor (Nov 10 2021 at 13:30):

You would get Day convolution for copresheaves (corresponding to free completion of small categories) rather than presheaves (corresponding to free cocompletion of small categories).

Naso (Nov 10 2021 at 13:33):

say we are in simplicial sets, the Day convolution corresponds to the join operation. Would the right Kan extension do anything interesting to simplicial sets?

Nathanael Arkor (Nov 10 2021 at 13:38):

The reason Day convolution can be used to define operations on simplicial sets is because simplicial sets form a presheaf category. (Note that the join is Day convolution of ordinal sums: you need to specify which operation you're convolving.) If you wanted to use right Kan extensions, you would need to be interested in a copresheaf category.

Zhen Lin Low (Nov 10 2021 at 13:44):

@Nathanael Arkor I wouldn't say that. The free completion of a small category $\mathcal{C}$ is $[\mathcal{C}, \textbf{Set}]^\textrm{op}$, which is definitely not to be confused with the free cocompletion of $\mathcal{C}^\textrm{op}$.

Naso (Nov 10 2021 at 13:47):

For simplicial sets, in that linked page, should $\mathcal{C} = \Delta^{op}$ or $\mathcal{C} = \Delta$

Naso (Nov 10 2021 at 13:50):

or is this the right diagram image.png

Nathanael Arkor (Nov 10 2021 at 13:50):

@Zhen Lin Low: thanks, what I meant to say was the objects look the same, but the difference in morphisms is crucial.

Nathanael Arkor (Nov 10 2021 at 13:54):

Nasos Evangelou-Oost said:

For simplicial sets, in that linked page, should $\mathcal{C} = \Delta^{op}$ or $\mathcal{C} = \Delta$

It should be $\Delta^{op}$. The notation on that page is a little unconventional, I would say.

Nathanael Arkor (Nov 10 2021 at 13:54):

Because you want an operation on presheaves, i.e. functors $\Delta^{op} \to \mathbf{Set}$.

Naso (Nov 10 2021 at 13:55):

Ok, but what stops me from taking the right Kan extension there, won't i still be a simplicial set?

Naso (Nov 10 2021 at 13:56):

$+$ is still a monoidal operation on $\Delta^{op}$?

Nathanael Arkor (Nov 10 2021 at 13:59):

The right Kan extension will give a monoidal structure on $[\Delta^{op}, \mathbf{Set}]^{op}$, rather than on $[\Delta^{op}, \mathbf{Set}]$.

Naso (Nov 10 2021 at 14:00):

Oh ok, so it would still define an operation on objects but it would turn the morphisms around the wrong way?

Nathanael Arkor (Nov 10 2021 at 14:01):

Yes: that's the crucial difference between presheaves and copresheaves.

Naso (Nov 10 2021 at 14:01):

I see now, thanks