Toby Bartels writes:
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?
I'm confused: in my understanding, a sketch basically amounts to a way of giving generators and relations for a category with products, Different sketches give the same category with products, not vice versa. Your example gives two sketches, but one category with products. In this sense, a sketch is more like an "idea" than you seem to be giving it credit for. By the way, in response to Lawvere's comments: My use of the term "Platonic idea of X" for the free category/category with products/monoidal category/2-category/whatever on an X was not meant as an endorsement of "Platonism" in the philosophy of mathematics - especially since "Platonism" means many things to many people. It was also not meant to suggest that Plato had this idea. It was basically meant to get people thinking about abstract generals versus concrete particulars. Best, John Baez
Certainly I did not mean to suggest that either John or Andree were supporting platonism as a philosophy of mathematics. In fact I had momentarily even forgotten that John had used the term. In my 1972 Perugia Notes I had made an attempt to characterize the relation between these sorts of mathematical considerations and philosophy by saying that while platonism is wrong on the relation between Thinking and Being, something analogous is correct WITHIN the realm of Thinking. The relevant dialectic there is between abstract general and concrete general. Not concrete particular ("concrete" here does not mean "real").There is another crucial dialectic making particulars (neither abstract nor concrete) give rise to an abstract general; since experiments do not mechanically give rise to theory, it is harder to give a purely mathematical outline of how that dialectic works, though it certainly does work. A mathematical model of it can be based on the hypothesis that a given set of particulars is somehow itself a category (or graph), i.e., that the appropriate ways of comparing the particulars are given but that their essence is not. Then their "natural structure" (analogous to cohomology operations) is an abstract general and the corresponding concrete general receives a Fourier-Gelfand-Dirac functor from the original particulars. That functor is usually not full because the real particulars are infinitely deep and the natural structure is computed with respect to some limited doctrine; the doctrine can be varied, or "screwed up or down" as James Clerk Maxwell put it, in order to see various phenomena. From: baez@math.ucr.edu
To: categories@mta.ca (categories) Subject: categories: Sketches and Platonic Ideas Date: Mon, 3 Dec 2001 19:42:40 -0800 (PST)
Toby Bartels writes:
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.
...
who has the right idea?
I'm confused: in my understanding, a sketch basically amounts to
...
By the way, in response to Lawvere's comments:
My use of the term "Platonic idea of X" for the free
...
versus concrete particulars.
Best, John Baez
There are a number of definitions of sketch around, some of which require it to be a category with finite products. In one of Ehresmann's (and Bastiani's, I believe) there is mentioned the possibility of its being what they called a quasicategory (or some such substructure term) in which composition is a partly defined multi-ary operation (in other words, fgh could be defined without fg or gh being defined). Charles and I realized that this was equivalent to what we called a graph with diagrams, which seemed a more useable notion. So what we called a sketch was a graph with diagrams as well as certain cones and cocones that were singled out to be taken to limits and colimits, resp. Peter Johnstone criticized us for doing the equivalent of replacing groups by generators and relations, which is correct, but it was a conscious decision and there were reasons for it. I had never heard the term "idea" in this connection or we might have used it. But anyway, "sketch" is used in different ways and I guess Charles and I contributed to this, but didn't create it. On Mon, 3 Dec 2001 baez@math.ucr.edu wrote:
Toby Bartels writes:
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?
I'm confused: in my understanding, a sketch basically amounts to a way of giving generators and relations for a category with products, Different sketches give the same category with products, not vice versa. Your example gives two sketches, but one category with products. In this sense, a sketch is more like an "idea" than you seem to be giving it credit for.
By the way, in response to Lawvere's comments:
My use of the term "Platonic idea of X" for the free category/category with products/monoidal category/2-category/whatever on an X was not meant as an endorsement of "Platonism" in the philosophy of mathematics - especially since "Platonism" means many things to many people. It was also not meant to suggest that Plato had this idea. It was basically meant to get people thinking about abstract generals versus concrete particulars.
Best, John Baez
participants (3)
-
baez＠math.ucr.edu -
F. William Lawvere -
Michael Barr