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 Quoting George Janelidze <janelg@telkomsa.net>:
Dear Bill,
Indeed, there were no monoids in Vaughan's original message of February 28, but since you have mentioned them in your message of March 1, and since you were talking there about "...lacuna of explicitness ... in many papers on Galois theory...", I simply wanted to say that:
I do not see any relevance of these kinds of presentations in Galois theory (apart from the fact the internal pre-whatever-s in a category X form an X-valued presheaf category).
On the other hand Galois theory is not the end of the World, and I think the beauty and importance of those your ideas is clear to everyone who saw them.
Putting myself in risk of making my message boring, I would like to make one more remark concerning your last message and Galois theory:
You say: "...applying a non-exact functor F to a group..." - true and fine, but I have actually mentioned F(R) for R being not a group, but another extreme case of a groupoid, namely an equivalence relation. What seems to be most amazing is, that, because F preserves not-all-but-some pullbacks, there are beautiful examples where R is an equivalence relation and F(R) is a group; in simple words, F creates a group out of nothing! The classical example, as you know, is: if R is the kernel pair of a universal covering map E ---> B of a "good" connected topological space B, and F is the functor sending ("good") topological spaces to the sets of their connected components, then F(R) is the fundamental group of B. The same thing is true in other Galois theories of course.
George