higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
Generalised smooth spaces are, roughly speaking, generalisations of smooth manifolds. Their raison d’etre is the following
> Manifolds are fantastic spaces. It’s a pity that there aren’t more of them.
Many spaces that occur in mathematics aren’t manifolds but one would like to be able to treat them as if they were manifolds; in particular, they should have some smooth structure that goes beyond mere topology. By considering examples of these spaces and by considering what specifically one would like to do with or to them, it is possible to generalise the idea of a smooth manifold to encompass the new examples whilst preserving enough structure to retain the old tools. There have been several such generalisations in recent mathematical history. A (partial) list is below.
Often the examples of spaces that one would like to consider as manifolds are formed by applying a categorical construction to ordinary manifolds; such as limits, quotients, or function spaces. This leads one to ask for the categorical properties of each of the resulting categories of generalised smooth spaces.
Another obvious question to ask is what tools and techniques can be extrapolated from smooth manifolds to generalised smooth spaces. In addition, whilst some techniques have obvious generalisations there may be some hidden twists that are not apparent on just smooth manifolds.
According to the general nonsense of space and quantity, generalized smooth spaces may be determined by sheaves on smooth test spaces, in which case we call them smooth spaces here, or by co-(pre)sheaves on test spaces, in which case we call them structured generalized spaces here.
Chen spaces (called differentiable spaces in Chen’s works).
both
The relationships between (some) of the categories can be sumarised by the following diagram.
I subtracted $20$ from the $x$-coordinates on the names in the diagram so that they would stay in the boxes on my screen, but I'm not sure if this is right; the original looks fine to me as a free-standing diagram, so I don't know why it looks wrong here.
Anyway, if anybody finds that this version is worse than the previous one, then change it back to the previous one and chalk it up to an error in my browser. —Toby
Thanks, Toby. I was just heading over to see if I could fix it myself but you beat me to it. There seem to be a few subtleties over how Instiki imports SVG and I’m learning them by trial and error (and by bugging Jacques!). The picture in the Sandbox now looks right and, thanks to you, so does this one. Text boxes seems to be the trickiest to get right when doing TikZ-to-SVG conversion. —Andrew
It it helps any, I think that the problem was that the alphabetic text (but not the dates) began where it ought to have been centred. —Toby
Eventually the following will be a commented list – promised.
John Baez and Alexander Hoffnung, Convenient Categories of Smooth Spaces (arXiv, blog)
Patrick Iglesias-Zemmour, Diffeology (pdf)
Matthias Kreck, Stratifolds and differential algebraic topology (pdf)
William Lawvere, Taking categories seriously (pdf)
David Spivak, Quasi-smooth derived manifolds (pdf)
Andrew Stacey, Comparative Smootheology (arXiv)
Martin Laubinger, Differential Geometry in Cartesian Closed Categories of Smooth Spaces (pdf)
Alexander Hoffnung, Smooth spaces: convenient categories for differential geometry, (pdf slides)
Alexander Hoffnung, From Smooth Spaces to Smooth Categories, (pdf slides)
There are also Hofer’s polyfolds.
Concerning smooth ∞-stacks there is useful material in
Dual to generalized smooth spaces are generalized smooth algebras of functions on them, according to the general lore of space and quantity.
We had extensive discussion of generalized smooth spaces at the $n$-Café:
David Roberts: For those generalised smooth spaces which give rise to a topological space (e.g. a diffeological space), is the topology known to be locally contractible, or locally nice at all?
Andrew: That’s actually a question I’d quite like to study here. All of the definitions of “generalised smooth space” (that have underlying sets) induce a topology on that underlying set. Some have it built in (Chen’s early definitions, for example, and Smith spaces and differentiable modules) but even if it is not there you can induce it from the plots or functions. They are not, in general, going to be locally contractible but there are some pathologies that are ruled out.
David R: Clearly the philosophy behind smooth spaces means we have to keep what we get, and not fuss about how ugly the spaces might be. What interests me is what the fundamental group(oid) is going to look like. Will it be a profinite group? A pro-group? A smooth group? I suppose one could start with the smooth space of loops, and form the smooth quotient space under the relation of homotopy - but what does it look like?