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 $R$ — is equipped with the structure of a commutative algebra, such that for infinitesimal objects $S \subset R^n$ all morphisms $S \to R$ are linear — i.e. such that the Kock-Lawvere axiom holds.
There is a standard definition and various straightforward variations.
For $(\mathcal{T},R)$ a lined topos there is the obvious notion of $R$-algebra objects $A$ in $T$.
For $A$ and $B$ any two $R$-algebra objects, there is the subobject $R Alg_T(A,B) \subset B^A$ of morphisms $A \to B$ that are algebra homomorphisms.
Write
for the algebra spectrum of $A$ in $\mathcal{T}$.
An $R$-Weil algebra $W$ is an $R$-algebra of the form $W = R \oplus J$, where $J$ is an $R$-finite-dimensional nilpotent ideal.
(smooth topos)
A lined topos $(\mathcal{T},R)$ is a smooth topos if
the algebra spectra $Spec(W)$ of all Weil algebras $W$ in $\mathcal{T}$ are infinitesimal objects in that the functor $(-)^{Spec W} : \mathcal{T} \to \mathcal{T}$ has a right adjoint (the “amazing right adjoint”);
it satisfies the Kock-Lawvere axiom in that for all $R$-Weil algebra objects $W$ the canonical morphism
is an isomorphism in $\mathcal{T}$.
A smooth topos $(\mathcal{T},R)$ is called a well adapated model if there is a full and faithful functor
from the category Diff of smooth manifolds into it, that takes the real line $\mathbb{R}$ to the line object $R$.
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.
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 $R$ 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.
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 $k$ be a field and let $(k-Alg^{finp})^{op}$ be the opposite category of the category of finitely presented $k$-algebras. Then the presheaf category $\mathcal{T} = PSh((k-Alg^{finp})^{op})$ equipped with the line object $R = k[T]$ (the algebra of polynomials over $k$ in one variable $T$)
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 $\mathcal{T} = PSh((k-Alg^{finp})^{op})$ produces something like formal schemes over $k$.
Indeed, much of algebraic geometry over $k$ 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 $\mathcal{T} = PSh((k-Alg^{finp})^{op})$: those that satisfy a sheaf condition with respect to a Grothendieck topology on $(k-Alg^{finp})^{op}$ and among those moreover those that are locally isomorphic to a representable: these are the schemes or algebraic spaces over $k$.
Probably the category of sheaves localization of $PSh((k-Alg^{finp})^{op})$ 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 $\mathcal{T}$ 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
Most of the examples above provided toposes into which a category $NiceSpaces$ of nicely behaved spaces, such as manifolds or schemes embeds full and faithfully.
But the inclusion functor will in general not preserve all limits and colimits that exist in $NiceSpaces$.
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 $Diff \hookrightarrow \mathbb{L}$ 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.