- Title: [[Three Realisms and The Idea of Sheaves]] - Type: #source/video - Author: David Jaz Myers - Reference: [David Jaz Myers: "Three Realisms and The Idea of Sheaves" - YouTube](https://www.youtube.com/watch?v=RPuWHN0BTio) - Published at: - Reviewed at: 2023-06-11 - Links: --- - Local realism - captured well by [[Sheaves]] Realism is a [[Philosophy|philosophical]] idea that there are real things in the world In math, we apply this to modeling. Our models are "about real things in the world". 3 Realisms (and their mathematical tools) 1) Fixed realism (sets) - one model - what's real is what's true in that model 2) Covariant Realism (group actions) - many equivalent models - what's real is how things change as you change your model (e.g. [[Tensors]] [[Vector space|Vector spaces]] and change under basis) - Group actions are rules for how to "change our point of view" 1) Local Realism ([[Sheaves]]) - Many inequivalent models - What's real is how you handle disagreement ([[Cohomology]]) "If we have different coordinates, we need to coordinate" - D. Spivak ![[_Media/Covariant Realism.png]] - We realize that not everyone is going to work from the same agreed upon constraints and ways of measuring - We need to codify what things change with (covariant^[I'm going to stick with co-variant because that's what he uses in the presentation but I really think he means variant both co and contra]) and what things stay the same (invariant) as we change our models - e.g. under a rigid transformation, the coordinates used to measure the points change (vary) but distance between them doesn't (invariant) - The equivalences form a [[Algebraic Group|Group]] where the equivalences (e.g. forward and backward [[Linear Transformation|linear transformations]]) are the objects and composition is the operation. - Klein: your group determines your geometry - Noether: your group determines your conserved quantities ## Local Realism - When you happen upon a contradiction, make a distinction (William James). -> distinct contexts. towards [[Sheaves]] ![[_Media/Realisms - Alice Bob disagree shared context.png]] Resolutions - "0-dimensional" Don't communicate - if the communication never happened there's no shared context to disagree about - "1-dimensional" "agree to disagree". Record each person's observation - "higher-dimensional" e.g. a 3-way conversation with gossip ![[_Media/Three Realisms - Sheaf.png]] - The stalk is a set of items (elements are called germs) - stalks are sets - Bundled together. - The site is the "context and how the elements of the context are related" - The site is a category - The sheaf is the arrangement of sets of situations living over the site - XXX these are actually pre-sheaves. ## Pre-Sheaves A pre-sheaf on a site $\mathcal{C}$ is a [[Contravariant Functor|contravariant functor]] $M:\mathcal{C^{op}} \to Set$ - Think of the site $\mathcal{C}$ as a category of contexts - The arrows in C can be though of as inclusions - $M(C)$ is a model of each context $C:\mathcal{C}$ - e.g. Alice has here model of how the conversation is going - for each inclusion of context $i:C \to C'$ we have a specialization of models $M(C') \to M(C)$ s.t. ```tikz \usepackage{tikz-cd} \begin{document} \begin{tikzcd} & {C'} &&& {M(C')} \\ C && {C''} & {M(C)} && {M(C'')} \arrow[from=2-1, to=1-2] \arrow[from=1-2, to=2-3] \arrow[from=2-1, to=2-3] \arrow[from=1-5, to=2-4] \arrow[from=2-6, to=1-5] \arrow[from=2-6, to=2-4] \end{tikzcd} \end{document} ``` Glueing: what separates a pre-sheaf from a sheaf. ## Sheaves on a simplicial complex - Think of a simplicial complex as a higher order generalization of a graph A simplicial complex with a set of points $P$ is a set of cells $C \subseteq 2^P$ where every subset of a cell is a cell. ![[_Media/Three Realisms - Sheaves on a simplicial complex.png]] Each cell is a context. Each inclusion of subsets is a way of including a context. ## Market Models A market model has: - (Site): **Agents** connected by **Channels** in a graph - (Sheaf): each agent has a set of **goods** or **baskets** it can trade and each channel has a set of **transactions** that can occur over it. ![[_Media/Three Realisms - Market Model Sheaf.png]] ## Cohomology: Consensus and Disagreement This is an impossible figure because it can't be built in 3d space. ![[_Media/Three Realisms - Cohomology Penrose Impossible figure.png]] Cover any corner and you could build it (local agreement). Cohomology lets us measure disagreement with linear algebra. For a sheaf $M: \mathcal{C} \to Set$ on a graph $\mathcal{C}$. ![[_Media/Three Realism - Cohomology.png]] Cochains - functions that assign values to each point and each edge Differential - the difference between cochains in 0 dimension and cochains in 1 dimension Cohomolgy let's us model the dynamics of disagreement Disagreement leads to change (**in** the sheaf) - Changes **of** sheaf (e.g. new goods, new transactions) - Changes **of** site (e.g. new markets) TODO(shawn) finish this?