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 On Fri, Sep 19, 2008 at 6:27 PM, <Mark.Weber@pps.jussieu.fr> wrote:
Dear Michael,
Semantically, as Lawvere observed long ago, a monad gives rise not just to a category of algebras but also to a forgetful functor into the category on which the monad acts. For any category C the functor
"semantics" : Mnd(C)^op --> CAT/C
whose object map sends a monad on C to its associated forgetful functor is full and faithful. Thus a pair of monads on C giving rise to isomorphic forgetful functors must necessarily be isomorphic. So your observations about different monads giving rise to the same algebras, while correct, do not tell the whole story on the semantic side.
The situation is of course different for sketches: they too give rise to forgetful functors (into Set), but this does not suffice to determine a given sketch up to isomorphism in the same way, and this justifies Steve Lack's perspective of "sketches as presentations of theories".
Mark Weber
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.
Yes, all this "discussion" is mainly misunderstanding, and I apologize if I've contributed to it. It seems it was triggered by this piece: On Wed, Sep 17, 2008 at 12:36 AM, <Andre.Rodin@ens.fr> wrote:
Dear Steve,
Sketches are not mathematical objects in their own right, in the same sense that groups or spaces are.
Of course, they are not.
... So, the issue is closed. Still there is some point to mention, and I again apologize if I'm peering into it too much. Our entire mis-discussion is, perhaps, a result of two different attitudes. CT favors and prefers to work in a presentation-free setting while engineering applications are all about presentations; and this mismatch may contribute to the disdain of CT from the practitioners' side. (Of course, this is not meant to anyhow diminish the elegance, value and usefulness even for practical problems such concepts as triple or classifying category :). Zinovy On Sun, Sep 21, 2008 at 7:55 PM, Steve Lack <s.lack@uws.edu.au> wrote:
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.
The ontological status of "presentations of structures" is a key issue in the history of mathematics in the 20th century. I wrote a number of historical material about it. The following (in French) are available on the Internet : http://people.math.jussieu.fr/~burroni/mapage/Burroni.pdf http://www.univ-nancy2.fr/poincare/colloques/symp02/abstracts/ageron.pdf Pierre Ageron
participants (3)
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pierre.ageron＠math.unicaen.fr -
Steve Lack -
Zinovy Diskin