Re: Undirected graph citation

["George Janelidze" <janelg@telkomsa.net>]

Dear Bill,

> By a presentation in mathematics I mean generators and relations for an
> algebraic structure of a certain kind. Occasionally we are fortunate to
> have also another more direct description of the same algebra, which it is
> useful to make explicit; a well known example of the usefulness of making
> explicit such a conceptual (as opposed to syntactical) description is the
> pair of definitions for the algebra of operators that defines the notion
> of simplicial set.

By a presentation in mathematics I mean exactly the same thing, I agree with
every word you said in this paragraph, and I know that what we call today
Lawvere theories was your beautiful discovery along this lines to thoughts.

> In your 2001 book with Borceux, Definition 7.2.1 involves five generators
> and five relations. Sometimes this is augmented by symmetry.
>
> What is actually being presented ?  a certain full finite subcategory of
> the category of finite sets.

I am sorry, what is "presented" in Definition 7.2.1 can of course be
considered as a subcategory of the category of finite sets, but certainly
not full (because, say, there are no arrows from C_1 to C_2). I suppose you
have noticed this, and so you are suggesting to modify the Definition
7.2.1 - since in "all" examples in fact there are more arrows. Well, one
could do so, but there is also a good reason not to do so: those five
generators and five relations is exactly the minimum needed to define
internal actions (I have actually first used this definition in my CT90
paper "Precategories and Galois theory" and many other people used similar
definitions for other purposes, probably long before).

Having said this, I again agree with every word of the rest of your message.
Can you accept the fact I agree with it and at the same time I do like
Definition 7.2.1? Note that if we go one step down in dimension, there will
be reflexive graphs whose "theory" is a full subcategory of sets and just
graphs whose "theory" is not. Are you telling me that this is a good reason
to forget the notion of graph, and use only reflexive graphs?

George

----- Original Message -----
From: <wlawvere@buffalo.edu>
To: <categories@mta.ca>
Sent: Monday, March 06, 2006 10:08 PM
Subject: categories: Re: Undirected graph citation


>
> Dear George
>
> By a presentation in mathematics I mean generators and relations for an
> algebraic structure of a certain kind. Occasionally we are fortunate to
> have also another more direct description of the same algebra, which it is
> useful to make explicit; a well known example of the usefulness of making
> explicit such a conceptual (as opposed to syntactical) description is the
> pair of definitions for the algebra of operators that defines the notion
> of simplicial set.
>
> In your 2001 book with Borceux, Definition 7.2.1 involves five generators
> and five relations. Sometimes this is augmented by symmetry.
>
> What is actually being presented ?  a certain full finite subcategory of
> the category of finite sets. Why should diagrams of this shape occur so
> often and be transported by functors even when they do not satisfy any
> exactness ? That is especially evident in the case of the Amitsur complex
> on page 264: the family of powers of a given object is a functor of the
> exponents, which are sets from that little category.
>
> That groupoids form a subcategory of the topos permits to take images, in
> the topos, of maps between groupoids; surprisingly, that can be useful.
>
> I prefer to consider one more finite set, so that "associativity" is a
> structure even when it is not an exact property (and analogously in the
> case of categories vs truncated simplicial sets - the question is how
> truncated). Then to be a groupoid is just a pullback-preservation
> condition.
>
> Bill
>