Question about the meaning of Δ𝔽 in "Functorial Data Migration"

[Fred Eisele]

Here is a simple diagram describing Δ𝔽.

𝕍/𝕎 : database schema

𝕀/𝕁 : instances of 𝕍/𝕎 in 𝕊𝕖𝕥

𝕍-𝕊𝕖𝕥/𝕎-𝕊𝕖𝕥 : Fun(𝕍/𝕎 𝕊𝕖𝕥) : the category of instances 𝕀/𝕁 on 𝕎

How is Δ𝔽 a pullback?

A pullback is defined as the limit of a pair of functions

f : X → Z and g : Y → Z having a common codomain Z.

Δ𝔽 is a functor so in order to be a limit (an object) we must be in the categories of functors.

𝔽 appears to be one of the pullback functions as the phrase "pullback along 𝔽" is used.

This leads me to thinking the pullback would be something like the following?

If that is correct then what are {A B C D}?

[Fred Eisele]

Errata:

𝕍-𝕊𝕖𝕥/𝕎-𝕊𝕖𝕥 : Fun(𝕍/𝕎 𝕊𝕖𝕥) : the category of instances 𝕀/𝕁 on 𝕍/𝕎

[Fred Eisele]

[David Spivak]

Hi Fred,

The functor Delta_F is called a pullback because, even though F goes from V to W, it transfers data backwards from W-instances to V-instances. So it's named more because of how it seems to act.

However, it turns out it is in fact a pullback in the sense you mention also. Given a functor J: W-->Set, one can perform an operation called the Grothendieck Construction on J or take the category of elements of J. It is written ∫J, and it always comes with a functor π:∫J --> W. This is in my book if you're interested. It kind of turns the database tables into a graph database where every node in the graph ∫J maps to an object (table) in the schema W, and every edge in ∫F maps to a morphism (column) in W. From this point of view, given a functor F:V-->W, we perform a pullback of categories

 

X---> ∫J 

|     |

V---->W

and X = ∫ Delta_F(J).

Best,

David

On Fri, Mar 16, 2018 at 12:41 PM, Fred Eisele <fredric...@gmail.com> wrote:

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[Fred Eisele]

Thanks, that is pretty cool.

However, it turns out it is in fact a pullback in the sense you mention also.

Given a functor J: W-->Set, one can perform an operation called the Grothendieck Construction on J or take the category of elements of J.

 

Section 6.2.2: in "Category Theory for the Sciences"