Sheaf Theory vs Human Memory. Community Participation!

Apr 2015
Hello Physics Forum,

The following is my research on sheaves and human memory. It is based on essay by Lovering. I have been extracting important points and generalizing his ideas.

Here is a snippet of my introduction.

Memory is Processing of Data
-memory what happens as a result of processing information
Depth of Processing of Information
Shallow Processing
-sensory memory: appearance, physical qualities, sound
Deep Processing
-encode meaning, relating similar environments and circumstances
-deeper level of processing, easier information recalled
source: Memory, Encoding Storage and Retrieval | Simply Psychology

Sheaf Theory - Lovering
General Learning Process
-certain field of interest
-collection of facts, ideas, examples
-maybe necessary to restrict area of study, specialized area
-sometimes apply interesting way, sometimes trivial
-similar ideas in different area of interest
-maybe two ideas contrast: deduce relationship
-arise general theory that envelops ideas
-glue different areas together depending on similarity

Sheaf Action A: Limited Data
-let topological space and attaching data to it
-restriction: only capable of perceiving some of the topology such that some of the data be hidden
-brain reconstructs data based on topology (incomplete information)
-end result: abstraction of data and reassembly of data
-unique, nicely-behaved sheaf arises from open data sets on a topology (after restriction) when gluing exists
-sheaves have structure: depending on restriction maps

Sheaf Action B: Local Properties
-whether that data on topological space is compatible (or gluable)
-study of smooth functions on manifold
-functions may not be explicitly defined
-we work with local properties
-sensory memory, preconceptions, judging factuality

Section Results
Data management: Sections, Data Restriction
Define a Presheaf pp2
Let presheaf F of a category
Topological space X
1) Every open set U subset X a group F(U),
2) Every inclusion U subset V a group homomorphism rho_{VU}: F(V) to F(U)

1)-F(U) is called Section
2)-Restriction map is a Morphism

A sheaf is a mathematical concept that holds local information on open sets. A google search will reveal that sheaf (math) ideas apply to memory models through computation. Please help generalize mathematical theorems starting from page 3 of Lovering. Work needs to be done regarding manifolds, schemes, and sheaf cohomology: modules and complexes.

This is not a school project. It is my hobby that can become something more. Any work will be accredited to your name. Making this a featured discussion would be great!

Thanks all for your time.
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