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I would like to bring the discussion of pseudogroups at manifold into a more general abstract perspective.
I am not sure yet. This here is just to brainstorm a bit.
For instance, how if I said the following, not meant to be anywhere close to comprehensive, but just to get us started:
Let $\mathbf{H}$ be an $\infty$-topos.
Definition. Let $V \in \mathbf{H}$. We say that a $V$-atlas for an object $X \in \mathbf{H}$ is an effective epimorphism $a \colon \coprod_{i \in I} V \to X$ out of a coproduct of copies of $V$, such that the fiber product of $a$ with itself is again a coproduct of $V$s and so that the two projections out of it are both monomorphisms. An object $X$ that admits a $V$-atlas we call a $V$-manifold.
The two maps from $a \times a$ to $a$ (in $\mathbf{H}/X$) are monomorphisms? They're not. (They're epimorphisms, but presumably we already know this from the regularity properties of $\mathbf{H}$.)
I also had wanted a more general abstract perspective on manifolds, and had begun some notes on this in one of my webs, starting from a notion of bicategory of partial maps. I may release it soon.
Just a small comment for now: if you want to bring things like manifolds with corners into the picture, you might not want to constrain yourself to a single $V$. In my set-up, I was tending more along the lines of considering open subsets $U$ of a single $X$ as giving the local models; for example, $X$ could be a cube $I^n$ and various open subsets can contain various types of corners, or not.
They’re not.
True. I want to say that they are coproducts over monomorphisms.
$U_{i} \cap U_j \hookrightarrow U_i \,.$I may release it soon.
Okay, looking forward to it!
So here is maybe finally a point where the “remaining” Giraud axiom for $\infty$-toposes (that one which isn’t used in the theory of principal infinity-bundles) is relevant for internal cohesive geometry: the axiom that coproducts are disjoint coproducts:
given an object $U \in \mathbf{H}$ (or a family of such, with the evident generalizations in the following), we can first ask for an effective epimorphism $\coprod_i U \to X$ such that its Cech nerve is degreewise a coproduct of copies of $U$. Then by disjoint coproducts it follows that each face map is a union over component maps $U \to U$. Say that $X$ is a $U$-manifold if these component morphisms are all monomorphisms.
In particular, then, let $\mathbf{H}$ be a cohesive $\infty$-topos. Assume that there is an object $\mathbb{A}^1\in \mathbf{H}$ which “exhibits the cohesion” of $\mathbf{H}$ in that $\Pi : \mathbf{H} \to \infty Grpd$ exhibits the localization of $\mathbf{H}$ at $\{ X\times \mathbb{A}^1 \to X \}_{X \in \mathbf{H}}$.
Then we can say intrinsically that an $n$-dimensional cohesive manifold in $\mathbf{H}$ is an $\mathbb{A}^n$-manifold in $\mathbf{H}$.
In the smooth model $\mathbf{H} = Sh_\infty(CartSp)$, where $\mathbb{A}^1 \simeq \mathbb{R}$ is the standard smooth real line, this definition then reproduces the ordinary definition of smooth manifold.
The above axiomatizes possibly non-Hausdorff manifolds in cohesive homotopy type theory. To get Hausdorff manifolds we can pass further to differential homotopy type theory and demand that those compnent maps are moreover formally etale morphisms. (In the classical model of synthetic differential cohesion that makes them be local diffeomorphisms and hence open maps.)
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