This is to be an extension of the pseudogroup approach to defining general notions of manifold. The pseudogroup approach makes clear how to define the groupoid of manifolds, but as far as I know does not cover the general notion of morphism between manifolds. To rectify this, a new notion (cartology) is introduced, providing an abstract environment in which pseudogroups make sense and also one where morphisms may be naturally defined. Manifolds are seen as akin to torsors in this approach.
We recall some ideas (due to Carboni) on bicategories of partial maps, close in spirit to the Carboni-Walters notions of bicategories of relations and cartesian bicategories. (Note: compositions are given in traditional order, where the composite of $r: X \to Y$ and $s: Y \to Z$ is denoted $s r: X \to Z$.)
(Carboni) A structure of bicategory of partial maps is a symmetric monoidal locally posetal bicategory in which
Each object $X$ comes equipped with a (unique) cocommutative comonoid structure $(\delta_X: X \to X \otimes X, \varepsilon_X: X \to 1)$,
Every morphism $r \colon a \to b$ is compatible with the comultiplication in the strict sense and with the counit in the lax sense:
Given a 1-cell $r: X \to Y$ in such a structure, the domain $d(r)$ is the composite
where the last isomorphism in the left unit constraint. We say $r$ is total if $d(r) = 1_X$. The following lemma is easily proven using the method of string diagrams:
(Carboni) In a structure of bicategory of partial maps $\mathbf{B}$, we have that
For any map $r: X \to Y$, the arrow $d(r): X \to X$ is coreflexive: $d(r) \leq 1_X$.
$d(s r) = d(d(s)r)$. In particular, $d(s) = d(d(s))$.
$rd(r) = r$.
$d(s)d(r) = d(r)d(s) = d(d(r)d(s))$.
$d(r)d(r) = d(r)$.
$r: X \to Y$ is total if and only if $\varepsilon_Y r = \varepsilon_X$.
Total 1-cells (as composed in $\mathbf{B}$) are the morphisms of a category, denoted $Tot(\mathbf{B})$. Furthermore, if $s r$ is total, so is $r$.
Finally, the monoidal product $\otimes$ restricts to a cartesian monoidal product on $Tot(B)$. The comultiplications $\delta_X: X \to X \otimes X$ are the diagonal maps in $Tot(\mathbf{B})$, and $1$ is terminal via $\varepsilon_X: X \to 1$.
A domainal bicategory is a bicategory admitting a structure of bicategory of partial maps, such that
(This is a form of functional completeness.)
We can show
Left adjoints are total.
If $i \dashv i^\ast$, then $d(i^\ast) = i i^\ast$.
Assuming that all domains $d(r)$ split, every morphism $r$ is of the form $r = f i^\ast$, where $i \dashv i^\ast$ and $f \coloneqq r i$ is total.
Split $d(r)$ as $i i^\ast$, and put $f = r i$. We have
The following equations show $f$ is total:
(Call this $f i^\ast$ the canonical factorization of a partial map $r$.)
All pseudo-invertible maps $r$ can be written in the form $j i^\ast$.
For any pseudo-invertible $r = j i^\ast$, we have $d(r) = i j^\ast$ j i^\ast$.
If $i: V \to Y$ and $j: U \to Y$ are left adjoints, and we write $i^\ast j: U \to V$ as $k h^\ast$ where $h: W \to U$ is a left adjoint and $k$ is total, then we can show $i k = j h$ directly (I hope). Then we may calculate that $h^\ast j^\ast i$ is right adjoint to $k$. It follows (from a lemma below) that i^\ast j = k h^\ast$ is pseudo-invertible, and now we conclude that pseudo-invertible maps are closed under composition.
In a domainal bicategory of partial maps, the subbicategory whose 1-cells are total is locally discrete.
Referring to 3. above, we may conclude that $U \stackrel{h}{\leftarrow} W \stackrel{k}{\to} V$ is the pullback of $U \stackrel{j}{\to} Y \stackrel{i}{\leftarrow} V$.
The arrow $j^\ast i$ is pseudo-invertible, and thus can be written as $h k^\ast$ where $h \dashv h^\ast$ and $k \dashv k^\ast$. Taking pseudo-inverses of both sides of $j^\ast i = h k^\ast$, we have $i^\ast j = k h^\ast$. Now suppose $r: A \to U, s: A \to V$ are any arrows such that we have a commutative square $j r = i s$. From $j^\ast j = 1_U$ and $i^\ast i = 1_V$, we derive
From these equations, it is easy to deduce that $k^\ast s \leq h^\ast r$ and $h^\ast r \leq k^\ast s$, so $k^\ast g = h^\ast f: A \to W$. Call this $p: A \to W$. We then have $h p = h k^\ast g = f$ and $k p = k h^\ast r = s$. Thus $(h: W \to U, k: W \to V)$ is the pullback.
Observation: given a domainal bicategory of partial maps $\mathbf{B}$, the bicategory obtained by splitting all coreflexives is another domainal bicategory of partial maps.
Can we go through Carboni’s development and show that there is at most one such structure in the domainal case?
A regional bicategory is a domainal bicategory such that the join $\bigvee_\alpha f_\alpha$ exists for a family $\{f_\alpha: X \to Y\}_{\alpha \in F}$ whenever the sheaf condition
holds for all $\alpha, \beta \in F$.
In addition to the axioms above, two more axioms are useful to consider:
We say that a bicategory of partial maps is functionally complete if it satisfies these two axioms. In this case, it may be shown that the category of objects and maps (i.e., left adjoints) is a left exact category $C$, and the bicategory is in fact the bicategory of partial maps in $C$.
The bicategory of regions is the locally posetal bicategory where
Objects are topological spaces (or locales if you prefer);
Morphisms are partial functions with open domain, that is spans
where $f$ is continuous and $i$ is an open embedding. These are clearly closed under span composition.
Such spans of type $\hom(X, Y)$ are locally ordered by inclusion.
This is a functionally complete bicategory of partial maps, denoted $\mathbf{Reg}$.
These local posets of $\mathbf{Reg}$ are not cocomplete, but they admit certain obvious joins: given a family of regional maps
the join $\bigvee_\alpha (U_\alpha, f_\alpha)$ exists iff we have local compatibility:
for all $\alpha, \beta$. Composition on either side with a $1$-cell preserves any local joins which exist.
Every coreflexive morphism $r \leq 1_X$ in $\mathbf{Reg}$ splits: there is an object $\dom(r)$ and morphisms $i \colon \dom(r) \to X$, $j \colon X \to \dom(r)$ such that the equations
hold. The object $\dom(r)$ may be called the domain of $r$.
In any category, splittings of idempotents (of coreflexives in particular, for poset-enriched categories) are unique up to isomorphism. For splitting pairs $(i, j)$ of coreflexives, we have $i \vdash j$.
A cartology is a (locally full) subbicategory $i: C \hookrightarrow \mathbf{Reg}$ such that
(Closure under open subspaces) If $X \in Ob(C)$ and $r \leq 1_X$ in $\mathbf{Reg}$, then $r$ belongs to $C$ and splits in $C$.
(Closure under joins) If $f_j \colon X \to Y$ are maps of $C$ and the join of $i(f_j)$ exists in $\mathbf{Reg}(X, Y)$, then this join belongs to $C$ (i.e., is of the form $i(f)$ for some $f$ in $C(X, Y)$).
The notion of cartology makes sense for any bicategory of partial maps in place of $Reg$, and we will develop the theory with that generalization in mind.
Intended examples include the case where the objects of $C$ are Euclidean spaces $\mathbb{R}^n$, and morphisms are spans
where $f$ is smooth (or PL, etc.).
Let $C$ be a cartology. A morphism $r \colon X \to Y$ in $C$ is pseudo-invertible if there exists $s \colon Y \to X$ such that
$s r: X \to X$ and $r s: Y \to Y$ are coreflexive,
$r = r s r$ and $s = s r s$.
The morphism $s$ is called a pseudo-inverse of $r$.
A pseudo-inverse is unique when it exists.
Suppose $s$ and $t$ are pseudo-inverse to $r$. We have
and similarly $r s \leq r t$, so $r s = r t$. By a similar argument, $s r = t r$. In that case
as claimed.
Suppose $r \colon X \to Y$ and $s \colon Y \to X$ are such that $r s$ and $s r$ are coreflexive, and $(i \colon U \to X, i^\ast \colon X \to U)$ splits $s r$ and $(j \colon V \to Y, j^\ast \colon Y \to V)$ splits $r s$. Then $j^\ast r i \colon U \to V$ is inverse to $i^\ast s j \colon V \to U$.
We calculate
where the second equation may be inferred from idempotence of the coreflexive morphism $r s$. Similarly $i^\ast s j j^\ast r i = 1_U$.
An arrow $r: X \to Y$ is pseudo-invertible iff there are left adjoints $i: U \to X$, $j: U \to Y$ such that $r = j i^\ast$.
For the “if” clause, we have (in a bicategory of partial maps) that $i^\ast i = 1_U = j^\ast j$. In that case, $j i^\ast$ has a pseudo-inverse $i j^\ast$, since we have
and
For the “only if” clause, let $s$ be the pseudo-inverse of $r$. Let $(i, i^\ast)$ split $s r$, let $(j, j^\ast)$ split $r s$, and put $\phi = j^\ast r i$. By the previous lemma, $\phi$ is invertible, with $\phi^{-1} = \phi^\ast$, and so $(j\phi, (j\phi)^\ast)$ is a splitting for $r s$. Moreover, we have
Thus, if we rename $j\phi$ as $j$, we have that $r i = j$ and $(j, j^\ast)$ splits $r s$. Then
which completes the proof.
Pseudo-invertible morphisms are closed under composition.
Let $r \colon X \to Y$ and $s: Y \to Z$ be pseudo-invertible, and write $r = j_0 i_0^\ast$ and $s = j_1 i_1^\ast$ as in the preceding lemma. Also write $i_1^\ast j_0 = k h^\ast$ where $k$ is total. We may calculate that $h^\ast j_0^\ast i_1$ is right adjoint to $k$ (see above). It follows that
and since $j_1 k$ and $i_0 h$ are left adjoints, it follows from the preceding lemma that $s r$ is pseudo-invertible.
The following definition is well-known; see for example Thurston.
A pseudogroup on a topological space (or locale) $X$ is a groupoid $G$ each of whose objects is an open set of $X$, and whose morphisms are homeomorphisms between such open sets, satisfying the following conditions:
The objects cover $X$. (Equivalently, in light of the last axiom, every open set of $X$ is an object of $G$.)
If $g: V \to W$ belongs to $G$ and $U \subseteq V$ is an open set, then the restriction $g|_U: U \to g(U)$ belongs to $G$. (Equivalently, in light of the other axioms, every inclusion map $id|_U: U \to V$ belongs to $G$.)
(Sheaf property) If $g: U \to V$ is a homeomorphism and if there is a covering $U_\alpha$ of $U$ such that the restrictions $g|_{U_\alpha}: U_\alpha \to g(U_\alpha)$ are morphisms of $G$, then $g$ is also morphism of $G$.
In a cartology $C$, the core groupoid of the Cauchy completion of the monoid of pseudo-invertible morphisms from an object $X$ to itself is a pseudogroup.
In the first place, every coreflexive endomorphism is pseudo-inverse to itself, and their extensions range over all open subsets of $X$. These are the objects; obviously they cover $X$.
The notion of a $C$-manifold modeled on an object $X$ of $C$ is defined just as before, using the pseudogroup on $X$ implied by the previous lemma. In particular, we have $C$-charts of an atlas structure on $M$, which are morphisms in $\mathbf{Reg}$
satisfying the expected properties. We can thus speak of $C$-manifolds (or $(C, X)$-manifolds if we want to make explicit the modeling space $X$).
Now, given a cartology $C$, we define the category of $C$-manifolds. Let $M$ be a $(C, X)$-manifold and $N$ a $(C, Y)$-manifold. Then, a $C$-morphism from $M$ to $N$ is a continuous map $f: M \to N$ such that the $\mathbf{Reg}$-composite
belongs to $C$, for every pair of charts $(U, \phi): X \to M$ and $(V, \psi): Y \to N$.
These definitions need to be carefully checked against known examples (e.g., the categories $Top$, $PL$, and $Smooth$, among others).