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Stream: deprecated: recommendations

Topic: smooth sets

Evan Patterson (Feb 06 2022 at 22:00):

On the nLab I read about smooth sets, which are sheaves on the site of cartesian spaces. The page geometry of physics — smooth sets has a longer exposition, but hardly any references are given.

I am intrigued by this viewpoint on differential geometry. Are there published papers, books, etc where I can read about it?

Fawzi Hreiki (Feb 06 2022 at 22:17):

The smooth sets POV on manifolds is really only a minor modification of the locally ringed space POV

Fawzi Hreiki (Feb 06 2022 at 22:18):

Which is developed in Wedhorn’s book ‘Manifolds, Sheaves, and Cohomology’

Fawzi Hreiki (Feb 06 2022 at 22:18):

I’m not sure about smooth sets, but there is the more refined idea of C-infinity algebras and schemes (which includes smooth sets as a special case)

Fawzi Hreiki (Feb 06 2022 at 22:19):

There is also the slightly more restrictive idea of diffeology

Evan Patterson (Feb 06 2022 at 22:37):

Fawzi Hreiki said:

The smooth sets POV on manifolds is really only a minor modification of the locally ringed space POV

Which is developed in Wedhorn’s book ‘Manifolds, Sheaves, and Cohomology’

Thanks, that's promising. I've been reading Wedhorn's book lately and I like it pretty well. It is my first introduction to locally ringed spaces. (I know the basics of differential geometry but nothing about algebraic geometry.)

In a ringed space, you have a sheaf on a "small" site, the poset of open sets, which tells you the smooth maps out of the space. In a smooth set, you have a sheaf on a "large" site, the category of cartesian spaces or even the category of all smooth manifolds, which tells you the smooth maps into your space. On the face of it, both differences seem fairly significant to me. Can you elaborate on why you say smooth sets are a minor modification of locally ringed spaces?

Todd Trimble (Feb 06 2022 at 22:48):

Have you looked at the book by Moerdijk and Reyes, Models for Smooth Infinitesimal Analysis?

Fawzi Hreiki (Feb 06 2022 at 22:51):

There are smooth sets that are not locally $\mathbb{R}$-ringed spaces and vice versa (akin to how there are Zariski sheaves on $\text{CRing}^{op}$ that are not locally ringed spaces). However when you restrict to spaces with an atlas, both sides are the same.

Fawzi Hreiki (Feb 06 2022 at 22:53):

Sheaves on Cartesian spaces are the same as sheaves on manifolds, so that aspect doesn’t make a difference.

Zhen Lin Low (Feb 06 2022 at 22:57):

§3.3 of my thesis is essentially about defining manifolds as smooth sets. §3.5 deals with schemes as Zariski or fppf sheaves, in the same framework.

Evan Patterson (Feb 06 2022 at 22:59):

Todd Trimble said:

Have you looked at the book by Moerdijk and Reyes, Models for Smooth Infinitesimal Analysis?

Thanks, I hadn't seen that one. Ch II on "$C^\infty$-rings as variable spaces" seems very relevant.

Evan Patterson (Feb 06 2022 at 23:20):

Zhen Lin Low said:

§3.3 of my thesis is essentially about defining manifolds as smooth sets. §3.5 deals with schemes as Zariski or fppf sheaves, in the same framework.

Thanks, your thesis looks very nice. I see that the introduction discusses the two different sheaf-theoretic approaches to defining manifold-like objects: locally ringed spaces and the "functor of points" approach, a phrase I did not know.

For my own future reference, the nLab page on functorial geometry also talks about this distinction. Smooth sets are the manifestation in differential geometry of the functor of points idea.

Fawzi Hreiki (Feb 06 2022 at 23:36):

Another good introduction to C^infinity algebraic geometry is this survey paper

Fawzi Hreiki (Feb 06 2022 at 23:37):

Unlike smooth sets, sheaves on (the duals of) C^infinity algebras actually form a model for synthetic differential geometry.

John Baez (Feb 07 2022 at 00:37):

Evan Patterson said:

On the nLab I read about smooth sets, which are sheaves on the site of cartesian spaces. The page geometry of physics — smooth sets has a longer exposition, but hardly any references are given.

I am intrigued by this viewpoint on differential geometry. Are there published papers, books, etc where I can read about it?

Urs Schreiber uses this approach all the time, so his work is where you'd see a lot about it.

I wrote a paper with Alex Hoffnung about a particular subcategory of the smooth sets: the 'diffeological spaces'. These are the 'concrete' sheaves on the site of cartesian spaces, meaning that they're given by an actual set $X$, together with for each $\mathbb{R}^n$ a set of actual functions from $\mathbb{R}^n$ to $X$.

There is a lot to read about diffeological spaces, such as Iglesias-Zemmour's book.

It's up to you whether you want the extra power gained by working with sheaves that are not concrete, or the extra ease of visualization gained by focusing on the concrete case. The arbitrary sheaves form a topos, as usual, while the concrete ones form a 'quasitopos' (see my paper for what that means).

Evan Patterson (Feb 07 2022 at 01:26):

Your nice and very readable paper with Hoffnung is partly what got me into in this! I particularly liked the complementary result characterizing simplicial complexes as concrete in the (pre)sheaf topos of symmetric simplicial sets. As you know, I've been doing computational stuff with discrete differential geometry, but the impression I've formed is that the discrete exterior calculus and related formalisms are in some ways ad hoc and unsatisfactory. Reading your paper made me think that sheaf theory might offer a unifying abstraction for smooth and discrete differential geometry, which, if luck holds, would help clarify the discrete theory.

The reason I'm interested in smooth sets, not just diffeological spaces, is that I really care not just about the geometrical spaces but about the quantities that live on them, especially differential forms. As Urs' exposition shows, the topos of smooth sets is a setting that encompasses both spaces and forms, in an elegant way.

John Baez (Feb 07 2022 at 02:40):

Okay, then I think Urs' writings are exactly what you need to read.