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Stream: learning: questions

Topic: A question on notation

Keith Elliott Peterson (Jan 28 2024 at 05:41):

For displayed categories, we use the esh/integral symbol, $\int F$, and for the corresponding lax normal functor, the derivative symbol is used, $\mathbf{d}F$.

Is this just notation, or is there something more here?

Mike Shulman (Jan 28 2024 at 06:58):

I think that's an integral, not an esh. (The esh is used for topological shape which I think is something different.)

I've always figured the idea of $\int$ for Grothendieck constructions is that you are "summing up" all the fibers in a sense similar to a coproduct $\sum$ but where the morphisms in the base "connect the fibers together", just as a numerical integral is similar to a discrete sum but with the points "connected together" by topology.

I've never seen the symbol $\mathrm{d}$ used for the opposite direction, though.

Keith Elliott Peterson (Jan 28 2024 at 08:39):

Mike Shulman said:

I've never seen the symbol $\mathrm{d}$ used for the opposite direction, though.

You can notice it here on the nlab: https://ncatlab.org/nlab/show/displayed+category#properties_of_functors_through_properties_of_the_reindexing

Matteo Capucci (he/him) (Jan 28 2024 at 11:10):

That notation is taken from the paper of Benabou's cited there

Matteo Capucci (he/him) (Jan 28 2024 at 11:11):

Afaiu is just suggestive: $\int$ evokes summation, and therefore its opposite can be suggestively denoted as $d$.