Maths - Sheaf Example - Equalities

This is an example for the page about sheaves which is here.


In model theory (see this page) a 'theory' is a set of first-order sentences, for instance:
  • α: for allx.(x+0=x)
  • β: for allx.(0+x=x)
  • γ: for allxfor ally.(x+y=y+x)

In this diagram we have a box for each set of these sentances:

The arrows mean implies. A set with more sentances can imply a set with less sentances because we are using Intuitionistic logic so we don't use the excluded middle rule. So if a sentance is not included in a set it doesn't mean its false it just means we are not saying anything about it.

I have left out arrows if they are given by composing the above arrows.

diagram external
There is more we can imply from these sentances because α and γ implies β so we can add the red arrows: diagram
In a similar way we can imply α from β and γ. We can't however imply γ from β and α. diagram
If there is an implication arrow going in both directions we can treat the sets as being the same. So here I have put them in the same box: diagram

We can now draw this as a presheaf.

Each entry in set represents a different value for the varable 'y'. We can also have every possible value for the varable 'x' but, to avoid complicating things I have not done this yet.

diagram equalities

Equalities as a Shape

Each equation can be a loop in some space.

What is a 'variable' in a space?

x=x+0 <=> 0=0+0,1=1+0,2=2+0,3=3+0...

Equalities as a Group or Groupoid

A 'group presentation' represents a group as a set of equations (see page here).

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see also:

Michael Robinson - Youtube from two-day short course on Applied Sheaf Theory:

  1. Lecture 1
  2. Lecture 2 - about simplicies & topology (sound on video is not good - quiet and delayed)
  3. Lecture 3
  4. Lecture 4
  5. Lecture 5
  6. Lecture 6
  7. Lecture 7
  8. Lecture 8
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Book Shop - Further reading .

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cover Introduction to Topological Manifolds (Graduate Texts in Mathematics S.)

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