higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
A smooth manifold is a space that is locally isomorphic to a Cartesian space $\mathbb{R}^n$ equipped with its canonical smooth structure.
Traditionally, a smooth manifold is defined as follows.
A manifold is a smooth manifold if its transition functions are smooth functions $\mathbb{R}^n \to \mathbb{R}^n$, or in other words a $G$-manifold over the pseudogroup $G$ of $C^\infty$ diffeomorphisms between open sets of a Euclidean space.
So a smooth manifold is a $C^k$-differentiable manifold for all $k$.
A homomorphism of smooth manifolds is a smooth functions. Smooth manifolds and smooth functions form the category Diff.
A smooth manifold is equivalently a locally ringed space $(X,\mathcal{O}_X)$ which is locally isomorphic to the ringed space $(\mathbb{R}^n, C^\infty(-) )$.
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There is a more fundamental and general abstract way to think of smooth manifolds, which realizes their theory as a special case of general constructions in higher geometry.
In this context one specifies for instance $\mathcal{G}$ a geometry (for structured (∞,1)-toposes) and then plenty of geometric notions are defined canonically in terms of $\mathcal{G}$. The theory of smooth manifolds appears if one takes $\mathcal{G} =$ CartSp.
Alternatively one can specify differential cohesion and proceed as discussed at differential cohesion – structures - Cohesive manifolds (separated).
This is discussed in The geometry CartSp below.
Let CartSp be the category of Cartesian spaces and smooth functions between them. This has finite products and is in fact (the syntactic category) of a Lawvere theory: the theory of smooth algebras.
Moreover, $CartSp$ is naturally equipped with the good open cover coverage that makes it a site.
Both properties together make it a pregeometry (for structured (∞,1)-toposes) (if the notion of Grothendieck topology is relaxed to that of coverage in StrSp).
For $\mathcal{X}$ a topos, a product-preserving functor
is a $\mathcal{G}$-algebra in $\mathcal{X}$. This makes $\mathcal{X}$ is $\mathcal{G}$-ringed topos. For $\mathcal{G} =$ CartSp this algebra is a smooth algebra in $\mathcal{X}$. If $\mathcal{X}$ has a site of definition $X$, then this is a [sheaf] of smooth algebras on $X$.
If $\mathcal{O}$ sends covering families $\{U_i \to U\}$ in $\mathcal{G}$ to effective epimorphism $\coprod_i \mathcal{O}(U_i) \to \mathcal{O}(U)$ we say that it is a local $\mathcal{G}$-algebra in $\mathcal{X}$, making $\mathcal{X}$ a $\mathcal{G}$-locally ringed topos.
The big topos $Sh(\mathcal{G})$ itself is canonically equipped with such a local $\mathcal{G}$-algebra, given by the Yoneda embedding $j$ followed by sheafification $L$
It is important in the context of locally representable locally ringed toposes that we regard $Sh(\mathcal{G})$ as equipped with this local $\mathcal{G}$-algebra. This is what remembers the site and gives a notion of local representability in the first place.
The big topos $Sh(CartSp)$ is a cohesive topos of generalized smooth spaces. Its concrete sheaves are precisely the diffeological spaces. See there for more details. We now discuss how with $Sh(CartSp)$ regarded as a $CartSp$-structured topos, smooth manifolds are precisely its locally representable objects.
The representables themselves should evidently be locally representable and canonically have the structure of $CartSp$-structured toposes.
Indeed, every object $U \in \mathrm{CartSp}$ is canonically a CartSp-ringed space, meaning a topological space equipped with a local sheaf of smooth algebras. More generally: every object $U \in CartSp$ is canonically incarnated as the $CartSp$-structured (∞,1)-topos
given by the over-(∞,1)-topos of the big (∞,1)-sheaf (∞,1)-topos over $CartSp$ and the structure sheaf given by the composite of the (∞,1)-Yoneda embedding and the inverse image of the etale geometric morphism induced by $U$.
Say a concrete object $X$ in the sheaf topos $Sh(CartSp)$ – a diffeological space – is locally representable if there exists a family of open embeddings $\{U_i \hookrightarrow X\}_{i \in X}$ with $U_i \in CartSp \stackrel{j}{\hookrightarrow} Sh(CartSp)$ such that the canonical morphism out of the coproduct
is an effective epimorphism in $Sh(CartSp)$.
Let $LocRep(CartSp) \hookrightarrow Sh(CartSp)$ be the full subcategory on locally representable sheaves.
There is an equivalence of categories
of the category Diff of smooth manifolds with that of locally representable sheaves for the pre-geometry $CartSp$.
Define a functor $Diff \to LocRep(CartSp)$ by sending each smooth manifold to the sheaf over $CartSp$ that it naturally represents. By definition of manifold there is an open cover $\{U_i \hookrightarrow X\}$. We claim that $\coprod_i U_i \to X$ is an effective epimorphism, so that this functor indeed lands in $LocRep(CartSp)$. (This is a standard argument of sheaf theory in Diff, we really only need to observe that it goes through over CartSp, too.)
For that we need to show that
is a coequalizer diagram in $Sh(CartSp)$ (that the Cech groupoid of the cover is equivalent to $X$.). Notice that the fiber product here is just the intersection in $X$ $U_i \times_X U_j \simeq U_i \cap U_j$. By the fact that the sheaf topos $Sh(CartSp)$ is by definition a reflective subcategory of the presheaf topos $PSh(CartSp)$ we have that colimits in $Sh(CartSp)$ are computed as the sheafification of the corresponding colimit in $PSh(CartSp)$. The colimit in $PSh(CartSp)$ in turn is computed objectwise. Using this, we see that that we have a coequalizer diagram
in $PSh(CartSp)$, where $S(\{U_i\})$ is the sieve corresponding to the cover: the subfunctor $S(\{U_i\}) \hookrightarrow X$ of the functor $X : CartSp^{op} \to Set$ which assigns to $V \in CartSp$ the set of smooth functions $V \to X$ that have the property that they factor through any one of the $U_i$.
Essentially by the definition of the coverage on $CartSp$, it follows that sheafification takes this subfunctor inclusion to an isomorphism. This shows that $X$ is indeed the tip of the coequalizer in $Sh(CartSp)$ as above, and hence that it is a locally representable sheaf.
Conversely, suppose that for $X \in Conc(Sh(CartSp)) \hookrightarrow Sh(CartSp)$ there is a family of open embeddings $\{U_i \hookrightarrow X\}$ such that we have a coequalizer diagram
in $Sh(CartSp)$, which is the sheafification of the corresponding coequalizer in $PSh(CartSp)$. By evaluating this on the point, we find that the underlying set of $X$ is the coequalizer of the underlying set of the $U_i$ in $Set$. Since every plot of $X$ factors locally through one of the $U_i$ it follows that $X$ is a diffeological space.
It follows that in the pullback diagrams
the object $U_i \cap U_j$ is the diffeological space whose underlying topological space is the intersection of $U_i$ and $U_j$ in the topological space underlying $X$. In particular the inclusions $U_i \times_X U_j \hookrightarrow U_i$ are open embeddings.
We may switch from regarding smooth manifolds as objects in the big topos $X \in Sh(CartSp)$ to regrading them as toposes themselves, by passing to the over-topos $Sh(CartSp)/X$. This remembers the extra (smooth) structure on the topological space $X$ by being canonically a locally ringed topos with the structure sheaf of smooth functions on $X$: a CartSp-structured (∞,1)-toposes
For every choice of geometry (for structured (∞,1)-toposes) there is a notion of $\mathcal{G}$-locally representable structured (∞,1)-topos (StrSp).
Smooth manifolds are equivalently the 0-localic CartSp-generalized schemes of locally finite presentation.
The statement says that a smooth manifold $X$ may be identified with an ∞-stack on CartSp (an ∞-Lie groupoid) which is represented by a CartSp-structured (∞,1)-topos $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ such that
$\mathcal{X}$ is a 0-localic (∞,1)-topos;
There exists a family of objects $\{U_i \in \mathcal{X}\}$ such that the canonical morphism $\coprod_i U_i \to *_{\mathcal{X}}$ to the terminal object in $\mathcal{X}$ is a regular epimorphism;
For every $i \in I$ there is an equivalence
The second and third condition say in words that $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ is locally equivalent to the ordinary cannonically CartSp-locally ringed space $\mathbb{R}^n$ (for $n \in \mathbb{N}$ the dimension. The first condition then says that these local identifications cover $\mathcal{X}$.
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A textbook reference is
Discussion of smooth manifolds as colimits of the Cech nerves of their good open covers is also at
The general abstract framework of higher geometry referred to above is discussed in