Hi - Bill Lawvere mentioned that KT Chen had a cartesian closed category of smooth spaces. I've found this very useful in my work on geometry. I kept wanting more properties of this category, so finally my student Alex Hoffnung and I wrote a paper about it: Convenient Categories of Smooth Spaces http://arxiv.org/abs/0807.1704 Abstract: A "Chen space" is a set X equipped with a collection of "plots" - maps from convex sets to X - satisfying three simple axioms. While an individual Chen space can be much worse than a smooth manifold, the category of all Chen spaces is much better behaved than the category of smooth manifolds. For example, any subspace or quotient space of a Chen space is a Chen space, and the space of smooth maps between Chen spaces is again a Chen space. Souriau's "diffeological spaces" share these convenient properties. Here we give a unified treatment of both formalisms. Following ideas of Dubuc, we show that Chen spaces, diffeological spaces, and even simplicial complexes are examples of "concrete sheaves on a concrete site". As a result, the categories of such spaces are locally cartesian closed, with all limits, all colimits, and a weak subobject classifier. For the benefit of differential geometers, our treatment explains most of the category theory we use. In particular, at some point we break down and admit we're dealing with a "quasitopos". Best, jb
Bill Lawvere wrote:
By urging the study of the good geometrical ideas and constructions of Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier, Steenrod, I am of course not advocating the preferential resurrection of the particular categories they tentatively devised to contain the constructions.
I chose Chen's framework when Urs Schreiber and I were doing some work in mathematical physics and we needed a "convenient category" of smooth spaces. I decided to choose one that was easy to explain to people brainwashed by the "default paradigm", in which spaces are sets equipped with extra structure. Later I realized I needed to write a paper establishing some properties of Chen's framework. By doing that I guess I'm guilty of reinforcing the default paradigm, and for that I apologize. If I understand correctly, one can actually separate the objections to continuing to develop Chen's theory of "differentiable spaces" into two layers. Let me remind everyone of Chen's 1977 definition. He didn't state it this way, but it's equivalent: There's a category S whose objects are convex subsets C of R^n (n = 0,1,2,...) and whose maps are smooth maps between these. This category admits a Grothendieck pretopology where a cover is an open cover in the usual sense. A differentiable spaces is then a sheaf X on S. We think of X as a smooth space, and X(C) as the set of smooth maps from C to X. But the way Chen sets it up, differentiable spaces are not all the sheaves on S: just the "concrete" ones. These are defined using the terminal object 1 in S. Any convex set C has an underlying set of points hom(1,C). Any sheaf X on S has an underlying set of points X(1). Thanks to these, any element of X(C) has an underlying function from hom(1,C) to X(1). We say X is "concrete" if for all C, the map sending elements of X(C) to their underlying functions is 1-1. The supposed advantage of concrete sheaves is that the underlying set functor X |-> X(1) is faithful on these. So, we can think of them as sets with extra structure. But this advantage is largely illusory. The concreteness condition is not very important in practice, and the concrete sheaves form not a topos, but only a quasitopos. That's one layer of objections. Of course, *these* objections can be answered by working with the topos of *all* sheaves on S. This topos contains some useful non-concrete objects: for example, an object F such that F^X is the 1-forms on X. But now comes a second layer of objections. This topos of sheaves still lacks other key features of synthetic differential geometry. Most importantly, it lacks the "infinitesimal arrow" object D such that X^D is the tangent bundle of X. The problem is that all the objects of S are ordinary "non-infinitesimal" spaces. There should only be one smooth map from any such space to D. So as a sheaf on S, D would be indistinguishable from the 1-point space. So I guess the real problem is that the site S is concrete: that is, the functor assigning to any convex set C its set of points hom(1,C) is faithful. I could be jumping to conclusions, but it seems to me that that sheaves on a concrete site can never serve as a framework for differential geometry with infinitesimals. Best, jb
[Note from moderator: with apologies to the poster, this is being resent since some of you will have received a version with some corrupted characters.] There are no > objections to continuing to develop Chen's theory of "differentiable > spaces" Indeed on 8/17, 8/26, and 8/27 I urged the continuation of the development of Chen's theory (for example the smooth space of piecewise smooth paths), making use of recent experience of the range of possible categories. It is possible that >sheaves on a concrete site can never serve as a framework > for differential geometry with infinitesimals. But a proof would require a definition of what is meant by infinitesimals, as well as the constraint on the framework that the maps 1->R and R->R are the standard ones. Otherwise nonstandard analysis might fit. The nilpotents or germs capture the Heraclitian nature of motion in a way that abstract sets do not directly. The misplaced concreteness, according to which >spaces are [single] sets equipped with extra structure is only a "second aspect of the default paradigm. The first aspect, successfully overcome by the named pioneers, is the one generalizing the default category of topological spaces (or locales). Here "default" refers to the habitual response to the frequently occurring need to specify a background category of cohesion in which to interpret our algebra. The generalization from Sierpinski-valued functions (open sets) to real-valued, has also been proposed, but that sort of attempt never succeeded in yielding a simple theory of map spaces. In contrast to this "function-algebra X/R as primary" paradigm, the semi-dual "figure-geometry S/X as primary" has led to good map spaces (including internal function algebras) for many authors (Sebastiao e Silva, Fox, Hurewicz 60 years ago and several more recent). I believe that attempting to force nearly-perfect duality has in general not led to good results, but of course one studies the extent to which a monad (presumed identity on models S) approximates the identity on general spaces. For example, Froelicher’s duality condition applies not only to the line R but to the function space R^R, a non-trivial fact about the smooth case, derived by LSZ from a study of distributions of compact support (so citing it is not just name-dropping). Bill On Tue 09/02/08 6:00 PM , John Baez baez@math.ucr.edu sent:
Bill Lawvere wrote:
By urging the study of the good geometrical ideas and constructions of>Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier,>Steenrod, I am of course not advocating the preferential resurrection of>the particular categories they tentatively devised to contain the>constructions.
I chose Chen's framework when Urs Schreiber and I were doing some work in mathematical physics and we needed a "convenient category" of smoothspaces. I decided to choose one that was easy to explain to people brainwashed by the "default paradigm", in which spaces are sets equippedwith extra structure. Later I realized I needed to write a paper establishing some properties of Chen's framework. By doing that I guessI'm guilty of reinforcing the default paradigm, and for that I apologize. If I understand correctly, one can actually separate the objections to continuing to develop Chen's theory of "differentiable spaces"into two layers.
Let me remind everyone of Chen's 1977 definition. He didn't state it this way, but it's equivalent:
There's a category S whose objects are convex subsets C of R^n (n = 0,1,2,...) and whose maps are smooth maps between these. This category admits a Grothendieck pretopology where a cover is an open cover in the usual sense.
A differentiable spaces is then a sheaf X on S. We think of X as a smooth space, and X(C) as the set of smooth maps from C to X.
But the way Chen sets it up, differentiable spaces are not all the sheaves on S: just the "concrete" ones.
These are defined using the terminal object 1 in S. Any convex set C has an underlying set of points hom(1,C). Any sheaf X on S has an underlying set of points X(1). Thanks to these, any element of X(C) has an underlying function from hom(1,C) to X(1). We say X is "concrete"if for all C, the map sending elements of X(C) to their underlying functions is 1-1.
The supposed advantage of concrete sheaves is that the underlying set functor X |-> X(1) is faithful on these. So, we can think of them as sets with extra structure.
But this advantage is largely illusory. The concreteness condition is not very important in practice, and the concrete sheaves form not a topos, but only a quasitopos.
That's one layer of objections. Of course, *these* objections can be answered by working with the topos of *all* sheaves on S. This topos contains some useful non-concrete objects: for example, an object F such that F^X is the 1-forms on X.
But now comes a second layer of objections. This topos of sheaves still lacks other key features of synthetic differential geometry. Most importantly, it lacks the "infinitesimal arrow" object D suchthat X^D is the tangent bundle of X.
The problem is that all the objects of S are ordinary "non-infinitesimal"spaces. There should only be one smooth map from any such space to D. So as a sheaf on S, D would be indistinguishable from the 1-point space. So I guess the real problem is that the site S is concrete: that is, the functor assigning to any convex set C its set of points hom(1,C) is faithful. I could be jumping to conclusions, but it seems to me that that sheaves on a concrete site can never serve as a framework for differential geometry with infinitesimals.
Best, jb
There are no > objections to continuing to develop Chen's theory of "differentiable >spaces” Indeed on 8/17, 8/26, and 8/27 I urged the continuation of the development of Chen’s theory (for example the smooth space of piecewise smooth paths), making use of recent experience of the range of possible categories. It is possible that >sheaves on a concrete site can never serve as a framework > for differential geometry with infinitesimals. But a proof would require a definition of what is meant by infinitesimals, as well as the constraint on the framework that the maps 1->R and R->R are the standard ones. Otherwise nonstandard analysis might fit. The nilpotents or germs capture the Heraclitian nature of motion in a way that abstract sets do not directly. The misplaced concreteness, according to which >spaces are [single] sets equipped with extra structure is only a “second aspect” of the default paradigm. The first aspect, successfully overcome by the named pioneers, is the one generalizing the default category of topological spaces (or locales). Here “default” refers to the habitual response to the frequently occurring need to specify a background category of cohesion in which to interpret our algebra. The generalization from Sierpinski-valued functions (open sets) to real-valued, has also been proposed, but that sort of attempt never succeeded in yielding a simple theory of map spaces. In contrast to this “function-algebra X/R as primary” paradigm, the semi-dual “figure-geometry S/X as primary” has led to good map spaces (including internal function algebras) for many authors (Sebastiao e Silva, Fox, Hurewicz 60 years ago and several more recent). I believe that attempting to force nearly-perfect duality has in general not led to good results, but of course one studies the extent to which a monad (presumed identity on models S) approximates the identity on general spaces. For example, Froelicher’s duality condition applies not only to the line R but to the function space R^R, a non-trivial fact about the smooth case, derived by LSZ from a study of distributions of compact support (so citing it is not just name-dropping). Bill On Tue 09/02/08 6:00 PM , John Baez baez@math.ucr.edu sent:
Bill Lawvere wrote:
By urging the study of the good geometrical ideas and constructions of>Chen and Froelicher, as well as of Bott, Brown, Hurewicz, Mostow, Spanier,>Steenrod, I am of course not advocating the preferential resurrection of>the particular categories they tentatively devised to contain the>constructions.
I chose Chen's framework when Urs Schreiber and I were doing some work in mathematical physics and we needed a "convenient category" of smoothspaces. I decided to choose one that was easy to explain to people brainwashed by the "default paradigm", in which spaces are sets equippedwith extra structure. Later I realized I needed to write a paper establishing some properties of Chen's framework. By doing that I guessI'm guilty of reinforcing the default paradigm, and for that I apologize. If I understand correctly, one can actually separate the objections to continuing to develop Chen's theory of "differentiable spaces"into two layers.
Let me remind everyone of Chen's 1977 definition. He didn't state it this way, but it's equivalent:
There's a category S whose objects are convex subsets C of R^n (n = 0,1,2,...) and whose maps are smooth maps between these. This category admits a Grothendieck pretopology where a cover is an open cover in the usual sense.
A differentiable spaces is then a sheaf X on S. We think of X as a smooth space, and X(C) as the set of smooth maps from C to X.
But the way Chen sets it up, differentiable spaces are not all the sheaves on S: just the "concrete" ones.
These are defined using the terminal object 1 in S. Any convex set C has an underlying set of points hom(1,C). Any sheaf X on S has an underlying set of points X(1). Thanks to these, any element of X(C) has an underlying function from hom(1,C) to X(1). We say X is "concrete"if for all C, the map sending elements of X(C) to their underlying functions is 1-1.
The supposed advantage of concrete sheaves is that the underlying set functor X |-> X(1) is faithful on these. So, we can think of them as sets with extra structure.
But this advantage is largely illusory. The concreteness condition is not very important in practice, and the concrete sheaves form not a topos, but only a quasitopos.
That's one layer of objections. Of course, *these* objections can be answered by working with the topos of *all* sheaves on S. This topos contains some useful non-concrete objects: for example, an object F such that F^X is the 1-forms on X.
But now comes a second layer of objections. This topos of sheaves still lacks other key features of synthetic differential geometry. Most importantly, it lacks the "infinitesimal arrow" object D suchthat X^D is the tangent bundle of X.
The problem is that all the objects of S are ordinary "non-infinitesimal"spaces. There should only be one smooth map from any such space to D. So as a sheaf on S, D would be indistinguishable from the 1-point space. So I guess the real problem is that the site S is concrete: that is, the functor assigning to any convex set C its set of points hom(1,C) is faithful. I could be jumping to conclusions, but it seems to me that that sheaves on a concrete site can never serve as a framework for differential geometry with infinitesimals.
Best, jb
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John Baez -
wlawvere＠buffalo.edu