structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
An $(\infty,1)$-cohesive site is a site such that the (∞,1)-category of (∞,1)-sheaves over it is a cohesive (∞,1)-topos.
A site $C$ is $\infty$-cohesive over ∞Grpd if it is
and an ∞-local site.
In detail this means that $C$ is
a site – a small category $C$ equipped with a coverage;
with the property that
it has a terminal object $*$;
it is a cosifted category (for instance in that it has all finite products);
for every covering family $\{U_i \to U\}$ in $C$
the Cech nerve $C(U) \in [C^{op}, sSet]$ is degreewise a coproduct of representables;
the simplicial set obtained by replacing each copy of a representable by a point is contractible (weakly equivalent to the point in the standard model structure on simplicial sets)
the simplicial set of points in $C(U)$ is weakly equivalent to the set of points of $U$:
These conditions are stronger than for a cohesive site, which only guarantees cohesiveness of the 1-topos over it.
This definition is supposed to model the following ideas:
every object $U$ has an underlying set of points $Hom_C(*,U)$. We may think of each $U$ as specifying one way in which there can be cohesion on this underlying set of points;
in view of the nerve theorem the condition that $\lim_\to C(U)$ is contractible means that $U$ itself is contractible, as seen by the Grothendieck topology on $C$. This reflects the local aspect of cohesion: we only specify cohesive structure on contractible lumps of points;
in view of this, the remaining condition that $Hom_C(*,C(U))$ is contractible is the $\infty$-analog of the condition on a concrete site that $Hom_C(*,\coprod_i U_i) \to Hom_C(*, U)$ is surjective. This expresses that the notion of topology on $C$ and its concreteness over Set are consistent.
The site for a presheaf topos, hence with trivial topology, is $\infty$-cohesive if it has finite products.
All covers $\{U_i \to U\}$ consist of only the identity? morphism $\{U \stackrel{Id}{\to} U\}$. The Cech $C\{U\}$ is then the simplicial object constant on $U$ and hence satisfies its two conditions above trivially.
The following sites are $\infty$-cohesive:
the category CartSp with covering families given by open covers $\{U_i \hookrightarrow U\}$ by convex subsets $U_i$;
we can take the morphisms $\mathbb{R}^k \to \mathbb{R}^l$ in $CartSp$ to be
– in which case the sheaf topos over it models generalized topological spaces, the 2-sheaf 2-topos contains for instance topological stacks;
or smooth maps
– in which case the sheaf topos models generalized smooth spaces such as diffeological spaces, the (∞,1)-sheaf (∞,1)-topos is that of ∞-Lie groupoids;
the site ThCartSp $\subset \mathbb{L}$ of smooth loci consisting of smooth loci of the form $R^n \times D^l_{(k)}$ with the second factor infinitesimal, where covering families are those of the form $\{U_i \times D^l_{(k)} \to U \times D^l_{(k)}\}$ with $\{U_i \to U\}$ a covering family in $CartSp$ as above.
This is a site of definition for the Cahiers topos.
More discussion of these two examples is at ∞-Lie groupoid and ∞-Lie algebroid.
Since every star-shaped region in $\mathbb{R}^n$ is diffeomorphic to an open ball (see there for details) we have that the covers $\{U_i \to U\}$ on CartSp by convex subsets are good open covers in the strong sense that any finite non-empty intersection is diffeomorphic to an open ball and hence diffeomorphic to a Cartesian space. Therefore these are good open covers in the strong sense of the term and their Cech nerves $C(U)$ are degreewise coproducts of representables.
The fact that $\lim_\to C(U) \simeq *$ follows from the nerve theorem, using that a Cartesian space regarded as a topological space is contractible.
Let $C$ be an $\infty$-cohesive site. Then the (∞,1)-sheaf (∞,1)-topos $Sh_{(\infty,1)}(C)$ over $C$ is a cohesive (∞,1)-topos that satisfies the axiom “discrete objects are concrete” .
If moreover for all objects $U$ of $C$ we have that $C(*,U)$ is inhabited, then the axiom “pieces have points” also holds.
Since the (n,1)-topos over a site for any $n \in \mathbb{N}$ arises as the full sub-(∞,1)-category of the $(\infty,1)$-topos on the $n$-truncated objects and since the definition of cohesive $(n,1)$-topos is compatible with such truncation, it follows that
Let $C$ be an $\infty$-cohesive site. Then for all $n \in \mathbb{N}$ the (n,1)-topos $Sh_{(n,1)}(C)$ is cohesive.
To prove this, we need to show that
This follows with the discussion at ∞-connected site.
$Sh_{(\infty,1)}(C)$ is a local (∞,1)-topos.
This follows with the discussion at ∞-local site.
The fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos $\Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ preserves finite (∞,1)-products.
If $\Gamma(U)$ is not empty for all $U \in C$, then pieces have points in $Sh_{(\infty,1)}(C)$.
The last two conditions we demonstrate now.
The functor $Pi : Sh_{(\infty,1)}(C) \to \infty Grpd$ whose existence is guaranteed by the above proposition preserves products:
By the discussion at ∞-connected site we have that $\Pi$ is given by the (∞,1)-colimit $\lim_\to : PSh_{(\infty,1)}(C) \to \infty Grpd$. By the assumption that $C$ is a cosifted (∞,1)-category, it follows that this operation preserves finite products.
Finally we prove that pieces have points in $Sh_{(\infty,1)}(C)$ if all objects of $C$ have points.
By the above discussion both $\Gamma$ and $\Pi$ are presented by left Quillen functors on the projective model structure $[C^{op}, sSet]_{proj,loc}$. By Dugger’s cofibrant replacement theorem (see model structure on simplicial presheaves) we have for $X$ any simplicial presheaf that a cofibrant replacement is given by an object that in the lowest two degrees is
where the coproduct is over all morphisms out of representable presheaves $U_i$ as indicated.
The model for $\Gamma$ sends this to
whereas the model for $\Pi$ sends this to
The morphism from the first to the latter is the evident one that componentwise sends $C(*,U)$ to the point. Since by assumption each $C(*,U)$ is nonempty, this is componentwise an epi. Hence the whole morphism is an epi on $\pi_0$.
and
cohesive site, ∞-cohesive site