Synthetic differential geometry
A smooth algebra or -ring is an algebra over the reals for which not only the product operation lifts to the algebra product , but for which every smooth map (morphism in Diff) lifts to a map in a compatible way.
In short this means that is
The smoothness of such -rings is witnessed by the fact that this Lawvere theory is even a Fermat theory.
The opposite category of the category of -rings is the category of smooth loci. This and its subcategories play a major role as sites for categories of sheaves that serve as models for synthetic differential geometry.
For a smooth manifold, the assignment
of the set of smooth -valued functions on is clearly covariant and hence yields a co-presheaf on CartSp Diff: a functor
Since the hom-functor sends limits to limits in its second argument this is clearly product preserving.
If as usual we write for the set of just -valued smooth functions, then the usual pointwise product of functions
can be regarded as the image of our co-presheaf under the muliplication map on the algebra of real numbers:
CartSp is the full subcategory of Diff on manifolds of the form .
Definition (smooth tensor product)
The coproduct in we call the smooth tensor product
More generally, for and two morphisms in , we call the pushout
the smooth tensor product over of and .
Finitely generated -rings
For a smooth manifold, the smooth algebra is the functor
(finitely generated and finitely presented -rings)
For a -ring, and an ideal in the underlying ordinary ring, there is a canonical -ring structure on the ordinaryy quotient ring .
A -ring is called finitely generated if it is of the form for and an ideal in .
It is finitely presented if also is finitely generated, as an ideal, with .
This is equivalent to being a pushout of the form
(germ-determined finitely generated / fair )
be the natural projection onto the smooth algebra of germs of functions at .
A -ring is called fair or finitely generated and germ-determined if it is finitely generated and the ideal has the property that is an element of if (and hence precisely if) for all the germ is in the germ of the ideal.
For any smooth topos , there is an internal notion of generalized smooth algebra:
Definition (internal generalized smooth algebra)
For a topos equipped with an internal ring object (possibly but not necessarily a smooth topos), let be the full subcategory of on objects of the form for . Then a -algebra is a product-preserving functor .
All constructions on smooth algebras generalize to -algebras. In particular for any object we have the function -algebra
The following remark asserts that when is itself a sufficiently nice category of sheaves on formal duals of -algebras, then the internal notion of smooth function algebras on formal duals of external smooth algebras reproduces these external smooth algebras.
Let be a finitely generated -ring, its incarnation as an object in and its incarnation in , with the Yoneda embedding and using the assumption that the Grothendieck topology used to form is subcanonical.
Also suppose that the line object is represented by
Then we have for all that
This is a straightforward manipulation:
the first step expresses the nature of the line object in the models under consideration
the second step expresses that the embedding is a full and faithful functor
the third step expresses that the Yoneda embedding is a full and faithful functor
the fourth step is the definition of as the opposite category of
the fifth step expresses that is the free generalized smooth algebra on generators (MSIA, chaper I, prop 1.1)
The category CartSp carries a natural Grothendieck topology.
A smooth algebra
is a local algebra if sends covering families to epimorphism families: for each covering the morphism
is an epimorphism.
Local smooth algebras are precisely the “local Archimedian” algebras (…).
This is (BungeDubuc, prop. 2.1).
Functions on smooth manifolds
For a smooth manifold, the smooth algebra is the functor
Weil algebras: functions on small infinitesimal spaces
A Weil algebra in this context is a finite-dimensional commutative -algebra with a maximal ideal such that and for some .
There is a unique -ring structure on a Weil algebra . It makes a finitely presented -ring.
Functions on germs of manifolds
For , the algebra of germs of smooth -valued functions at carries an evident -ring structure .
With the ideal of functions that vanish on a neighbourhood of we have
yielding a finitely generated but not (for ) finitely presented -ring.
Limits and colimits
All limits and all directed colimits in are computed objectwise in as limits in Set.
As discussed at limits and colimits by example, all limits and colimits in are computed objectwise, so the remaining question is if they preserve the property of functors to preserved products. The claim follows from the observation that limits and directed colimits do commute with products.
See also MSIA, p. 22.
The underlying ordinary algebra
There is a forgetful functor
from generalized smooth algebras to ordinary algebras which is given by evaluation on
and equipping the set with the algebra structure induced on it:
the product and sum on is the image of the corresponding operations on the algebra
Moreover there is canonically a morphism of rings
This makes an -algebra.
The forgetful functor fits into an adjunction
This statement may be understood as a special case of the following more general statement:
(In the case under discussion, is the free algebra monad on , is the free smooth algebra monad, and is induced from the obvious inclusion which interprets an -ary algebra operation (in the theory ) as a smooth operation in the theory . See finitary monad for discussion on the connection between finitary monads and Lawvere theories .)
The desired left adjoint takes an algebra to the reflexive coequalizer exhibited as a diagram
in the category of -algebras, where is the monad multiplication. The coequalizer is denoted to emphasize the analogy with pushing forward -modules along a ring homomorphism to get -modules; the proof below is an arrow-theoretic transcription of the usual proof of the adjunction between pushing forward and pulling back in the context of rings and modules.
A finitary monad preserves reflexive coequalizers, so that there is a canonical isomorphism
It follows that -algebra (-module) maps , i.e., maps that render commutative the diagram
are in bijection with maps that render commutative
(that is to say, -algebra maps ) which additionally coequalize the parallel pair in the diagram
Since is a monad morphism, we have commutativity of parallel squares in
so that coequalizes the bottom pair. However, because is a -algebra map, its pullback defines an -algebra map . This -algebra map factors through the coequalizer of the bottom pair of maps in , i.e., factors uniquely through an -algebra map . This establishes the adjunction .
Finitely presented -rings
Proposition (MSIA, prop. 1.1)
is the free smooth algebra on generators, in that for every and every smooth algebra there is an adjunction isomorphism
- Every finitely presented -ring is fair/germ determined.
We have a chain of inclusions
finitely presented -rings
finitely generated -rings
Points of smooth loci
An -point of a -ring is a point of the corresponding smooth locus, i.e. a morphism .
Points of a -ring are in bijection with points of the underlying -algebra , i.e. with ordinary -algebra morphisms .
In particular every Weil algebra has a unique point : every Weil algebra is the algebra of functions on an infinitesimal thickening of an ordinary point.
By the properties of for a smooth manifold discussed below, the -points of are precisely the ordinary points of the manifold .
Smooth function algebras on smooth manifolds
Proposition (MSIA, prop. 2.5, 2.6 )
Let and be transversal maps of smooth manifolds. Then the functor sends the pullback
to the pushout
In particular this implies (for )that the the smooth tensor product of functions on and is the algebra of functions on the product :
Turning this inclusion into an equivalence is usually called a completion of the algebraic tensor product. Therefore we see:
The smooth tensor product is automatically the completed tensor product.
In summary this yields the following characterization of smooth function algebras on manifolds.
Theorem (MSIA, theorem 2.8)
Deformation theory of smooth algebras
For any category whose objects we think of as “functions algebras on test spaces”, such as , there is a general intrinsic notion of tangent complex and deformation theory of such objects.
As describe there, the key structure of interest from which all the other structure here is induced is the tangent category
This is the codomain fibration of “fiberwise stabilized”, meaning that in each fiber one takes it to consist of , the abelian group objects in the overcategory.
We now first recall what this means for ordinary rings and how it induces the ordinary notion of derivations and modules for ordinary rings by setting CRing, and then look at what it implies for -rings by setting .
By an old argument by Quillen, for CRing we have that is the bifibration of modules over rings, there is a natural equivalence
This is induced by the functor that sends an -module to the corresponding object in the square-0-extension . (See module).
From this structure alone a lot of further structure follows:
a derivation is precisely a section of the corresponding morphism in , in the category namely a ring homomorphism
The forgetful functor has a left adjoint
that sends each ring to its module of Kähler differentials.
The fact that it is left adjoint is the universal property of the Kähler differentials as the objects co-representing derivations
So every derivation uniquely corresponds to a module morphism , namely the one that sends .
This abstract story remains precisely the same for -rings (and in fact for everything else!) but what it means concretely changes.
The crucial observation is (as one can show) that an abelian group object in is a square-0 extension for an (ordinary) module of the underlying -algebra . This square-0-extension happens to be uniquely equipped with the -Ring-structure given by
This uniquely induced smooth structure on objects in then in turn affects the nature of the notion of derivation and of Kähler differentials, as those are defined by general abstract reasoning from the former.
First of all it follows that a derivation – by general abstract definition a morphism of -rings – is a morphism that satisfies for all that
For ordinary rings only the compatibility with the single product operation is required. Here, however, compatibility with infinitely more operations is demanded.
Accordingly, then, the Kähler differentials as defined with respect to such derivations are different from the purely ring-theoretic ones: they produce the right notion of smooth 1-forms here, whereas the ring-theoretic one does not.
The generalization of the notion of smooth algebra for (∞,1)-category theory is
The generalization to supergeometry is
A standard textbook reference is chapter 1 of
The concept of -rings in particular and that of synthetic differential geometry in general was introduced in
Bill Lawvere, Categorical dynamics
in Anders Kock (eds.) Topos theoretic methods in geometry, volume 30 of Various Publ. Ser., pages 1-28, Aarhus Univ. (1997)
but examples of the concept are older. A discussion from the point of view of functional analysis is in
- G. Kainz, A. Kriegl, Peter Michor, -algebras from the functional analytic view point Journal of pure and applied algebra 46 (1987) (pdf)
A characterization of those -rings that are algebras of smooth functions on some smooth manifold is given in
Lawvere’s ideas were later developed by Eduardo Dubuc, Anders Kock, Ieke Moerdijk, Gonzalo Reyes, and Gavin Wraith.
Studies of the properties of -rings include
The notion of the spectrum of a -ring and that of -schemes is discussed in
and more generally in
Ieke Moerdijk, Gonzalo Reyes, Rings of smooth functions and their localization II , in Mathematical logic and theoretical computer science page 275 (Google books)
Marta Bunge, Eduardo Dubuc, Archimedian local -rings and models of synthetic differential geometry Cahiers de Topologie et Géométrie Différentielle Catégoriques, 27 no. 3 (1986), p. 3-22 (numdam).
More recent developments along these lines are in
The higher geometry generalization to a theory of derived smooth manifolds – spaces with structure sheaf taking values in simplicial C∞-rings – was initiated in
based on the general machinery of structured (∞,1)-toposes in
where this is briefly mentioned in the very last paragraph.
See also the references at Fermat theory, of which -rings are a special case. And the references at smooth locus, the formal dual of a -ring. And the references at super smooth topos, which involves generalizations of -rings to supergeometry.