Category Theory

https://arxiv.org/pdf/1609.05382.pdf
Table 6.1: Analogies between variables in various physical settings
(Es como lo que vi en termodinámica del no equilibrio)
También habla de sistemas abiertos vs sistemas cerrados

• symmetric monoidal category in nLab
  
• rosetta.pdf
  
• 2301.11867.pdf
  
• rosetta_topos_web.pdf
  
• Categorical models for process planning - nihms-1629726.pdf
  
• A Graphical, Model-Based Representation for Classical Plans using Category Theory - 9_a_graphical_model_based_repres.pdf (PDDL)
  
• String diagram - Wikipedia
  

• prathyvsh/category-theory-resources: Resources for learning Category Theory for an enthusiast
  
• Resources for functional programming beginner - Software Engineering Stack Exchange
  
• Functional Programming and Category Theory [Part 1] - Categories and Functors
  
• [1912.10642] Notes on Category Theory
  

Lawvere Theories | Bartosz Milewski’s Programming Cafe

https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/

Applied Category Theory (@ MIT 2019) - YouTube
David Spivak - Category Theory - YouTube

Preorder

reflexive (\[a \le a \forall a \in P\]), transitive (\[\text{if } a \le b \text{ and } b \le c \text{ then } a \le \forall a,b,c \in P \])
an antisymmetric preorder is a partial order, and a symmetric preorder is an equivalence relation

Partial order

reflexive, transitive, and antisymmetric (\[\text{ if } a \le b \text{ and } b \le a \text { then } a = b\])
No two distinct elements precede each other. Some pairs are incomparable, with neither being greater/lesser
Lamport timestamps for example are partial orders

Total order

reflexive, transitive, antisymmetric and strongly connected (\[a < b \text{ if not } a \ge b\])

Strictness

All definitions become strict if equality is not allowed

• Async/Lazy functor
  
  • a → a | Promise/Future
    
  • function composition becomes callbacks
    
• A Group representation
  
• The transpose operator on vector spaces, the adjoint/hermitian transpose on Hilbert spaces
  
• The Laplace transform
  
• Every Homomorphism is a Functor
  The Functor is an extension of the Homomorphism to many other structures that either do not have an underlying set, or are not algebraic (for examples partial order, or relationships of inclusion in a set)
  
• Canonical (Legendre) transformations
  Between Lagrangian and Hamiltonian
  
• Functor, Applicative and Monad instances for Reader Full implementation
  

• Free monoid - Wikipedia
  The monoid whose elements are all the finite sequences (or strings) of zero or more elements from that set, with string concatenation as the monoid operation and with the unique sequence of zero elements
  Free objects are the direct generalization to categories of the notion of basis in a vector space → a concrete programming language implementation (?) where composition is simply executing one function after another (~“line by line”?)
  Docker for example considers each layer dependent on all the previous ones, instead of building a dependency tree
  Concatenative programming language - Wikipedia
  
• Trace monoid - Wikipedia
  
• History monoid - Wikipedia
  
• Dependency relation - Wikipedia
  I think it all comes from this dependency relations.
  Free monoid → your program is a string of characters, every character depends on anyone else, in principle. Dependency by default. Trace/history monoid → You have to explicitly define which operations are dependent on another ones. Independency by default
  
• If you can undo your operations, then you got a group (transactions)
  
• The union of convex hulls form a monoid
  
• A function or method with a return type that forms a monoid is itself a monoid
  

• https://www.cs.cmu.edu/~crary/819-f09/Moggi91.pdf
  
• Partiality: Computations that may not terminate
  
• Nondeterminism: Computations that may return many results
  
• Side effects: Computations that access/modify state
  
  • Read-only state, or the environment
    
  • Write-only state, or a log
    
  • Read/write state
    
• Exceptions: Partial functions that may fail
  
• Continuations: Ability to save state of the program and then restore it on demand
  
  • Continuation - Wikipedia (coroutines / generators)
    Continuations are useful for encoding other control mechanisms in programming languages such as exceptions, generators, coroutines, and so on.
    
  • On Recursion, Continuations and Trampolines - Eli Bendersky’s website
    

  • Continuation-passing style - Wikipedia
    In functional programming, continuation-passing style (CPS) is a style of programming in which control is passed explicitly in the form of a continuation. This is contrasted with direct style, which is the usual style of programming.
    

    A function written in continuation-passing style takes an extra argument: an explicit “continuation”; i.e., a function of one argument. When the CPS function has computed its result value, it “returns” it by calling the continuation function with this value as the argument. That means that when invoking a CPS function, the calling function is required to supply a procedure to be invoked with the subroutine’s “return” value
    «Like callbacks»
    

• Interactive Input
  
• Interactive Output
  
• https://ncatlab.org/nlab/show/monad+%28in+computer+science%29
  
• Task Monad
  
  • Build systems a la carte: theory and practice - Simon Peyton Jones
    
  • snowleopard/build: Build Systems à la Carte
    
  • build-systems-jfp.pdf
    
  • Build systems à la carte: Theory and practice | Journal of Functional Programming | Cambridge Core
    

Producer-consumer problem
Catamorphism → fold (consumers)
Anamorphism → unfold (generators, producers)
https://thealmarty.com/2022/09/06/anamorphisms-aka-unfolds-explained/
https://bartoszmilewski.com/2022/04/05/teaching-optics-through-conspiracy-theories/
Producers and consumers correspond to covariant and contravariant functors

https://news.ycombinator.com/item?id=5196708
Derivadas → permite añadir un subárbol a un árbol existente
Árboles
Negativos y fraccionales
Un logaritmo es una llamada a una función? Tiene que estar en la misma base en la que está para que coincidan sus tipos

https://en.wikipedia.org/wiki/Posetal_category

a posetal category, or thin category is a category whose homsets each contain at most one morphism. As such, a posetal category amounts to a preordered class (or a preordered set, if its objects form a set).

Functional Query Language

• Databases are categories
  
• Exploring Category-Theoretic Approaches to Databases
  
• CQL
  
• https://www.google.com/search?q=%22database%22+%22category+theory%22
  
• Graphical Regular Logic
  
• Algebraic Databases
  

• David Spivak - Sense-making: accounting for intelligibility - IPAM at UCLA - YouTube
  

Composition of Logs is the Writer Monad

• https://en.m.wikipedia.org/wiki/John_C._Baez
  
  • Network Theory
    
• John Baez in nLab
  
• John Baez’s Stuff
  
• John Baez - YouTube
  
• torsors relative quantities like energies, potential, gauges, phases…
  

• Categorical quantum mechanics - Wikipedia
  
• [0905.3010] Categories for the practising physicist
  

https://www.sci.unich.it/geodeep2022/slides/2022-07-25%20Intro%20to%20categories.pptx

• [2105.06332] Towards Foundations of Categorical Cybernetics
  
• General abstract nonsense.
  
  • Lax functors describe emergent effects – Jules Hedges