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Restructured the manifold entry to avoid duplication with pseudogroup, and moved the section on the tangent bundle to tangent bundle
The definition via pseudogroups is supposed to have the following problem:
There is in general no notion of morphisms between manifolds. At best, we can only talk about isomorphisms of manifolds.
However, couldn’t this problem be solved by using a “pseudomonoid” instead of a pseudogroup? We can define a pseudomonoid $M$ on the topological space $X$ as a wide subcategory of $\text{Op}(X)$ such that for all $U, V, U_\alpha$ open in $X$, $f: U\rightarrow V$ such that $(U_\alpha)$ covers $U$, we have the condition $f\in M \iff (\forall \alpha) f|_{U_\alpha}:U\cap U_\alpha\rightarrow V\cap f(U_\alpha) \in M$.
I’m not sure right away. I’d have to think about it.
Here’s a new way of characterizing manifolds that I like better.
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
(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.
Joshua, is this in the literature someplace?
Not that I know of, but I haven’t looked very hard.
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