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smooth topos

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Synthetic differential geometry

differential geometry

synthetic differential geometry

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topos theory

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Idea

A smooth topos or smooth lined topos is the kind of topos studied in synthetic differential geometry, a category of generalized smooth spaces for which a notion of infinitesimal space exists.

It is defined to be a category of objects that behave like spaces, one of which — the line object RR — is equipped with the structure of a commutative algebra, such that for infinitesimal objects SR nS \subset R^n all morphisms SRS \to R are linear — i.e. such that the Kock-Lawvere axiom holds.

Definition

There is a standard definition and various straightforward variations.

standard definition

Definition

For (𝒯,R)(\mathcal{T},R) a lined topos there is the obvious notion of RR-algebra objects AA in TT.

For AA and BB any two RR-algebra objects, there is the subobject RAlg T(A,B)B AR Alg_T(A,B) \subset B^A of morphisms ABA \to B that are algebra homomorphisms.

Write

Spec(A):=RAlg 𝒯(A,R) Spec(A) := R Alg_{\mathcal{T}}(A,R)

for the algebra spectrum of AA in 𝒯\mathcal{T}.

An RR-Weil algebra WW is an RR-algebra of the form W=RJW = R \oplus J, where JJ is an RR-finite-dimensional nilpotent ideal.

Definition

(smooth topos)

A lined topos (𝒯,R)(\mathcal{T},R) is a smooth topos if

Examples

well-adapted models

A smooth topos (𝒯,R)(\mathcal{T},R) is called a well adapated model if there is a full and faithful functor

Diff𝒯 Diff \hookrightarrow \mathcal{T}

from the category Diff of smooth manifolds into it, that takes the real line \mathbb{R} to the line object RR.

In these well adapted models ordinary differential geometry is therefore faithfully embedded.

For a list of examples of well adapted models see

Notice that by far not all models are of this form, as the following examples show. On the contrary, the axioms of synthetic differential geometry may be regarded as providing a unified framework in particular for differential geometry of manifolds and algebraic geometry of algebraic spaces, schemes and other objects.

models for supergeometry

It is straightforward to slightly enhance the axioms of smooth toposes such as to incorporate the step from differential geometry to supergeometry, one just requires that algebra structure on the line object RR is further refined to thatr of a superalgebra. The result is called a super smooth topos. See there for a list of models of these.

models for algebraic geometry

A simple model of a smooth topos that may be regarded as a context inside which much of algebraic geometry takes place is the following:

Let kk be a field and let (kAlg finp) op (k-Alg^{finp})^{op} be the opposite category of the category of finitely presented kk-algebras. Then the presheaf category 𝒯=PSh((kAlg finp) op)\mathcal{T} = PSh((k-Alg^{finp})^{op}) equipped with the line object R=k[T]R = k[T] (the algebra of polynomials over kk in one variable TT)

(𝒯=PSh((kAlg finp) op),R=k[T]) ( \mathcal{T} = PSh((k-Alg^{finp})^{op}), R = k[T] )

is a smooth topos. This is described in section 9.3 of

Notice that despite the name of that book, this model is not a well adapted model in that ordinary smooth manifolds do not embed full and faithfully into this topos.

Instead, interpreting the internal notion of manifold described in that book – called formal manifolds in the model 𝒯=PSh((kAlg finp) op)\mathcal{T} = PSh((k-Alg^{finp})^{op}) produces something like formal schemes over kk.

Indeed, much of algebraic geometry over kk may be thought of as being concerned with this model for a smooth topos. A main difference is that in algebraic geometry attention is usually focused on particularly well behaved objects inside 𝒯=PSh((kAlg finp) op)\mathcal{T} = PSh((k-Alg^{finp})^{op}): those that satisfy a sheaf condition with respect to a Grothendieck topology on (kAlg finp) op(k-Alg^{finp})^{op} and among those moreover those that are locally isomorphic to a representable: these are the schemes or algebraic spaces over kk.

Probably the category of sheaves localization of PSh((kAlg finp) op)PSh((k-Alg^{finp})^{op}) with respect to one of the standard topologies (Zariski or etale) is still a smooth topos, so that this conmdition can be enforced by passing to a more restrictive model.

But since schemes alone will never form a topos, and smooth topos axiomatization of algebraic geometry will always contain more general – also more “pathological” – objects than schemes. It’s an example of an old dictum by Grothendieck that it is useful to have a nicely behaved category that contains pathological objects than a badly-bahved category of only nice objects: the topos-general nonsense allows useful general constructions in 𝒯\mathcal{T} which may in each individual case be checked for whether they land in the sub-category of schemes or algebraic spaces or not.

A similar comment of course applies to the “well-adapted models” mentioned above: into these ordinary manifolds only embed, they contain “smooth space”s much more general than manifolds (such as diffeological spaces) but also possibly “pathological” ones, from some perspective or other.

See also

warning on preservation of (co)limits

Most of the examples above provided toposes into which a category NiceSpacesNiceSpaces of nicely behaved spaces, such as manifolds or schemes embeds full and faithfully.

NiceSpaces𝒯. NiceSpaces \hookrightarrow \mathcal{T} \,.

But the inclusion functor will in general not preserve all limits and colimits that exist in NiceSpacesNiceSpaces.

For instance for the well-adopted models the full and faithful inclusion of Diff typically respects only pullbacks of manifolds along transversal map. This is because this is the case for the inclusion Diff𝕃Diff \hookrightarrow \mathbb{L} of Diff into the category of smooth loci and the Grothendieck topology for these models is typically subcanonical. See the discussion at smooth locus for more on this.

This means that universal constructions in a smooth topos may yield different results than in a smaller category of more nicely behaved spaces. However, it is noteworthy that in the above example of manifolds, one may argue that the ordinary pullback of manifolds along non-transversal maps is the wrong pullback in any case: it doesn’t behave well with the cohomology of manifolds. One motivation for derived geometry, in the case of manifolds specifically the motivation for considering derived smooth manifolds, is to pass from objects in a topos instead more generally to objects in an (∞,1)-topos of stack ∞-stacks.

Zoran: In derived geometry we want to go to derived infinity-stacks, not just infinity stacks. The embedding into infinity stacks is commuting with pullbacks, but not the one into derived infinity stacks. At least in algebraic world, and if I understand what Spivak has for spectra it is the same story.

Revised on February 2, 2014 13:59:03 by John Dougherty? (108.72.3.0)