# Generalization of pseudogroups

Pseudogroups are defined here: https://ncatlab.org/nlab/show/pseudogroup

One of the problems with defining manifolds in terms of pseudogroups is that it gives no notion of a morphism between manifolds, all you get is isomorphisms.

It seems like the definition of pseudogroup can be changed to get a concept of "pseudomonoid" which does encompass morphisms between manifolds: (source)

Let $$X$$ be a topological space and let $$N$$ be the monoid of partial continuous functions $$X\rightarrow X$$ (any element of $$N$$ has open domain by continuity), ordered by inclusion of graphs. Then define a pseudomonoid on $$X$$ to be a submonoid $$S$$ of $$N$$ such that

• If $$f, f_\alpha\in N$$ and $$f=\cup_\alpha f_\alpha$$, then $$f\in S$$ iff $$(\forall \alpha) (f_\alpha\in S)$$. (Equivalently, $$S$$ is a lower set of $$N$$ and the inclusion $$S\rightarrow N$$ lifts joins.)

(This condition implies that $$S$$ is closed under open restriction, and, being a submonoid of $$N$$, $$S$$ contains the identity map $$\text{id}_X$$, so $$S$$ contains all coreflexive elements of $$N$$.)

Let $$G$$ be the inverse semigroup $$G$$ of all elements of $$S$$ that are bijective and whose opposite (as a relation) is also in $$S$$.

Now let $$M$$ be a set. A chart on $$M$$ is a partial bijection $$M\rightarrow X$$. Two charts $$a,b$$ are $$S$$-compatible iff $$b^{\text{op}}a\in G$$ (or equivalently, $$b^{\text{op}}a, a^{\text{op}}b \in S$$). (A chart is compatible with itself iff its image is open.)

An $$S$$-atlas on $$M$$ is a collection of pairwise $$S$$-compatible maps whose domains cover $$M$$.

An $$S$$-manifold is a set $$M$$ equipped with an $$S$$-atlas. A map $$f$$ between $$S$$-manifolds $$M$$, $$N$$ with $$S$$-atlases $$A$$, $$B$$ is a morphism iff there exists a collection of pairs $$(\phi_\alpha, \psi_\alpha)\in A\times B$$ such that the collection $$\{\text{dom} \phi_\alpha \times \text{dom} \psi_\alpha\}$$ covers the graph of $$f$$ and for all $$\alpha$$, $$\psi_\alpha f \phi_\alpha^{\text{op}}\in S$$.

By using partial functions we don't need to worry about having all different domains involved and keeping track of restrictions and such.

Does this kind of thing exist anywhere in the literature?