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:
w[[Groupoid]] introduces idea of partially defined operations.
w[[Semigroup]] discards identity and inverses.
w[[Fiber bundle]] allows for a more general notion of function by projecting to some base space.
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July 6, 2022
Course outline # Lecture 1: Sets, bags, lists # Mathematical structures as data structures (5 min) # sets, bags, lists, tuples Multiplicity functions (10 min) # Representing data structures as oracles that answer questions about themselves
Example of a traditional duality
Union, intersection, chaining, in terms of multiplicity
Lifting functions (15min) # functions as uniform relations, relations as pixelated images
lifting functions: multi-images and multi-pre-images
multi-functions as images
multi-images as counting pixels in rows and columns
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July 6, 2022
Pathmap sheaf # Introduction # This chapter introduces the concept of the pathmap sheaf from a graph \( \graph{G} \)$ to a graph \( \graph{H} \)$. This will allow us to formalize the notion of an atlas on a graph \( \graph{G} \)$, and for special choices of \( \graph{H} \)$, define a kind of curvature on \( \graph{G} \)$ as a kind of sheaf cohomology.
The pathmap sheaf is a structure whose elements are partial pathmaps from \( \graph{G} \)$ to \( \graph{H} \)$ -- pathmaps that are not necessarily defined for all paths in \( \graph{G} \)$ -- those for which it is defined are called the support of the partial pathmap.
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