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数学笔记 All systems are the same: The Principle of Computational Equivalence [1, 2], due to S. Wolfram, is the heuristic statement that almost all processes (involving classical computations) that are not obviously simple are of equivalent sophistication. Binomial theorem. Derivative. There is no volume, just the faces, as with the crosspolytopes. Also, the derivative has the boundary conditions as in homology. Study TaiDanae Bradley's thesis to learn about the transition between classical and quantum probabilities, between sets (F1) and vector spaces (Fq), and look for a connection with N and N2 as per the Yoneda lemma. David Corfield: Spatial notions of cohesion as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere. Anyons are composite particles in twodimensions that have statistics in between fermion (object) and boson (arrow) statistics. How can they be understood in terms of category theory? Pushdown automata have a stack of priorities. In general, automata deal with concerns  rūpesčiai. Automata are important for the dynamics of the three languages. Study categories with a single initial state and a single final state. What does Yoneda Lemma mean for them? Yoneda Lemma. The set function {$\theta \rightarrow \alpha \theta \textrm{Hom}(f,\_)$} is a rule for a pushdown automaton. The {$\alpha$} comes from the finite automaton (the input) and the {$\textrm{Hom}(f,\_)$} should describe the stack of memory. This all, on the lefthand side, is compared to a finite automaton on the righthand side. Looking at the Standard Model: 12 fermions are the 12 topologies. 4 bosons are the 4 representations of the nullsome. Higgs boson is the nullsome. Mass is an indicator of subsystems. How do positionmomentum relate symplectic geometry and the entropy of phase space? Does entropy makes sense in a phase space without a notion of momentum? John Baez about toposes and Lawvere:
Is there a kind of mathematics that is behind every science, every house of knowledge, every person? Is every computation system of sufficient sophistication equivalent to a house of knowledge? And are lesser systems equivalent to a part of a house of knowledge? The Logical Foundations of Cognition John Mcnamara, Gonzalo E. Reyes, 1994. Functor is a contextualization. Is Z a group or an abelian group? "Form follows function". Relate content and context.
Functors preserve all paths With formula we have variables of certain types. And if we have another formula, then we may take all of the variables to be different, but we may allow some of the variables to be the same (refer to the same value) if they have the same type. But in the case of metaphor we see what happens if we make the identification without respecting the type. How much logic can carry over? We can test the boundaries and explore. Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. Relate these paths to a matrix and to symmetric functions on the eigenvalues of a matrix. Standard model: leptons mediate forces
Also:
Thus 34 or 37 particles. When you have an object, then you ignore the other objects, the "nonobjects", thus in the binomial theorem you don't have to deal with their relationships. Yoneda Lemma:
Entropy
Yoneda Lemma
Algebraic thinking is stepbystep, as with a PDA, statebystate. Yoneda Lemma: Covariant version and contravariant version are two PDAs that come together to form a Turing machine. What is the relationship between circumstances (12 topologies) and the meaning of context in automata theory. House of knowledge: Pushdown automata: Every question has an answer. Their two wings, entering the game and leaving the game, are linked by the threecycle. Could particles be extremal black holes? The twosome is an expression of the nature of entropy, the second law of thermodynamics. The mind shifts from a state of greater ambiguity (where opposites coexist) to a state of lesser ambiguity (where all is the same). Correspondingly, the mind shifts from a state of higher energy to a state of lower energy. The onesome defines order, whereas the twosome defines entropy, and so the two are related in that way. The reversal of entropy is the reversal of the twosome. Tobler's First Law of Geography: Everything is related to everything else, but near things are more related than distant things. Relate to the Yoneda lemma. In logic, the difference between a statement (such as "you owe me money") being true and its being true and not empty is the difference between the natural numbers and the whole numbers. It's likewise the difference between weakly increasing and strictly increasing, less than or equal and less than. Weak increase is fundamental for the Yoneda Lemma so that a relation is reflexively satisfied by an object. https://bartoszmilewski.com/2020/05/22/oncomposability/ JS Bell’s essay “Six possible worlds of quantum mechanics”; it’s in his anthology “Speakable and unspeakable in quantum mechanics” When are two types the same? When are two identity morphisms the same? How does the associativity of composition relate to factoring out an identity morphism? In type theory, implication is understood as a function (by the HowardCurry correspondence?) This is what I want to say as regards natural transformations  they set up a function that is a validation of a computation, an implication. Finite automata  truth tables. Push down automata  type theory. Truth tables don't scale, look at negation. 
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