Autoid
July 6, 2022
Autoid #
Motivation #
We want to define a flexible structure of "composable partial transformations", with a primary model being partial functions on some base set \( \setSymbol{X} \) to itself. It'll be a little like a semigroupoid.
First let's do a little catalogue of what's out there:
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w[[Groupoid]] introduces idea of partially defined operations.
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w[[Semigroup]] discards identity and inverses.
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w[[Fiber bundle]] allows for a more general notion of function by projecting to some base space.
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w[[Pseudogroup]] introduces sheaf-idea of of gluing and restriction, but has all this stuff about diffeomorphisms.
We have a base set \( \setSymbol{X} \) serving as a space, subsets of which are going to serve as domains and codomains of morphisms. Each morphism [EOF[\( \function{f} \), \( \setSymbol{F} \)]] will map a subset of \( \setSymbol{X} \) to another subset. Saying two morphisms [{\( \function{f} \),\( \function{g} \)}] cannot be composed is another way of saying the target of \( \function{f} \) has zero intersection with the source of \( \function{g} \). This just yields the null morphism [0].
Therefore there are two projections: [FSF[SrcFunction, \( \setSymbol{F} \), \( \subsets{X} \)]] and [FSF[TgtFunction, \( \setSymbol{F} \), \( \subsets{X} \)]].
\[ \functionSignature{\src}{\setSymbol{F}}{\subsets{X}} \] \[ \functionSignature{\tgt}{\setSymbol{F}}{\subsets{X}} \]Restriction is just composition with the identity map on some subset [\( \setSymbol{U} \) ⊆ \( \setSymbol{X} \)].
We can introduce a lattice structure on X, and then we can perform gluing etc.
But we could also think of the maps as multimaps. Not being defined is just zero cardinality on some \( \setSymbol{X} \). Being "overdefined" is having > 1 cardinality. Gluing is just pointwise sum -- actually not quit.