Sketches

Sketches

[Charles Wells]

This is in reply to Toby Bartels, quoted below.  I don't believe that those 
of us who have written about "ideas" in Ehresmann's sense ever conceived 
that each theory (sketch) was based on one right idea.  There is no 
"correct" idea for a given sketch.

I want to add, for those new to the subject, that the word "sketch" has 
been used with at least three meanings.  Ehresmann and his students use it 
for a structure which is a weakening of the concept of category (the 
composite may not be defined for all composable pairs) plus specified cones 
and/or cocones.  Many others have used the word sketch to refer to a 
category with specified cones and/or cocones.  Michael Barr and I in our 
two books used "sketch" to mean a graph with specified cones and/or cocones 
plus some commutativity conditions on paths; that is in the same spirit as 
Ehresmann's "idea".

--Charles Wells

>Andree Ehresmann wrote in part:
>
> >He thought
> >first of calling a sketch an idea, but then reserved the word "idea" for
> >the smallest part which helps reconstruct the sketch; for instance for a
> >category, the arrows which 'represent' the domain and codomain maps and the
> >composition law.
>
>There could be multiple ideas that generate the same sketch;
>how do we decide which is the correct idea among equivalent ones?
>OTOH, if we take equivalence classes of ideas, then we're taking sketches.
>For example, one could define the idea of multiplication in a monoid
>as a binary operation and a nullary operation
>or alternatively as an operation on finite tuples.
>The former is more common, but I prefer the latter;
>who has the right idea?
>
>
>-- Toby
>    toby@math.ucr.edu




Charles Wells,
Emeritus Professor of Mathematics, Case Western Reserve University
Affiliate Scholar, Oberlin College
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Sketches

[John Baez]

Zinovy Diskin wrote:

>Okay, sketches are presentations of theories but Steve's claim was
>that they are not mathematical objects.

He didn't say that.  He wisely said something much more cautious.

Best,
jb