On 21/09/08 3:34 AM, "Zinovy Diskin" <zdiskin@swen.uwaterloo.ca> wrote:
Okay, sketches are presentations of theories but Steve's claim was that they are not mathematical objects. Michael's and mine bewilderment is about why does the former imply the latter? (at least, why "of course" :)
Zinovy
What I actually said was this: "Sketches are not mathematical objects in their own right, in the same sense that groups or spaces are. They are presentations (for theories), and have status similar to other sorts of presentations (for groups, rings, etc.) Of course that is in no way meant to suggest that they are not important and worthy of study." So I did not say that "they are not mathematical objects", and I used the words "of course" only in clarifying that I was not suggesting that they were unimportant. What I was saying was that they have a different flavour to such mathematical objects as groups or spaces. I was saying this in response to the observation that sketches did not seem to fit into the Bourbaki notion of structure, and so in particular, that the notion of isomorphism of sketch was not as crucial as that of isomorphism of group. Michael Barr asked what the content of the statement might be. I certainly wasn't trying to make a precise mathematical statement, although Michael himself indicated one that could be made. I guess that my second sentence (that sketches are presentations) is the content. So the content, if you like, is "whatever status you give to group presentations, you should give the same to sketches". For my part, I think that presentations are extremely important technical tools, which need to be studied and understood; but which nonetheless are just that: technical tools for dealing with the real objects of study (the things they present). Steve Lack.