Category Theory
 Axioms
 Examples
 Categories and Polymorphism
 Combinators
 Encodings
 Diagram Chasing
 Constructions
 Functors
 Natural Transformations
 Monoidal Categories
 String Diagrams
 Higher Category
 Topos
 Processes
 Categorical Databases
 Sheaves
 Profunctors
 Optics
 Logic
 Computational Category Theory
 Internal Language
 Resources
EPR?
Doctrines
Abstract Finite categories Kind of like a graph data structure. Kind of like a finite group. A list of objects, a list of morphisms with domain and codomain, and a multiplication table. Any particular question can be answered via enumeration
Finitely Generated Categories
Axioms
Category is objects and morphisms. Morphisms have a partial operation called composition and there is an identity morphism for every object
Examples
Groups and Monoids
The algebraic laws of monoid basically work for a category with one object, where multiplication is composition
A group can be modeled as a category with one object, where ever morphism has an inverse. Then the axioms of cateogory corresponds to the axioms of a group
PoSet
partially ordered sets. There is at most one morphism between each object
Lattices are an interesting further subcase.
https://en.wikipedia.org/wiki/Partially_ordered_set
FinSet
Finite sets as objects, finite maps (dicitonaries) as morphisms
Can also consider partial finite maps.
FinVect
vector spaces are objects (dimensionality or labelled). Linear maps are morphisms (~ matrices)
FinRel
LinRel
Categories and Polymorphism
Part of the appeal of category theory comes from it’s relationship to polymorphism. Polymorphism is counterintuitive really in regards to the simple model of types being sets.
https://bartoszmilewski.com/2014/10/28/categorytheoryforprogrammersthepreface/
https://arxiv.org/pdf/2209.01259.pdf Category Theory for Programming
Combinators
https://en.wikipedia.org/wiki/Combinatory_logic
Combinators are little functional pieces you can combine like lego blocks. When you have a language of combinators, variable binding issues are gone. They are hard to program with
Combinators are not synonymous with categories, but categories can inform a particular style of combinators
Encodings
Encoding category theory to first order, higher order logic, and dependent type theory. Generalized algebraic theories.
Relational Composition comp(F,G,H)
Comp as Partial function comp(F,G) = Some(H)
Objects as Sort
Blog posts
 Egglog 2: Automatically Proving the Pullback of a Monic is Monic https://www.philipzucker.com/egglog2monic/
 Egglog Examples: Pullbacks, SKI, Lists, and Arithmetic https://www.philipzucker.com/egglog3/
 Rewriting Monoidal Categories in the Browser with Egg
Diagram Chasing
Constructions
Products
CoProducts
Initial Objects
Final
Equalizers
Pullbacks
PushOuts
Functors
Functors are mappings between categories. This means they are a tuple of map from objects to objects and morphism to morphism such that composition plays nice (commutes sort of) Functors are mappings between categories (a object to object map and morphism to morphism map) that plays nice with composition and identity. F(id) = id. F(f.g) = F(f) . F(g)
F = (Fo, Fm) Fm(f . g) = Fm(f) . Fm(g)
Representations of groups. Functor from group as a category to
Adjunctions
Free/Forgetful
Galois Connections. Abstraction and concretization
Closure/Rounding and lift
Abstract interpretation Intervals <> sets polytopes convexsets <> sets
Natural Transformations
Monoidal Categories
Vect
String Diagrams
Higher Category
https://en.wikipedia.org/wiki/Higher_category_theory
Morphisms go between objects 2morphisms go between morphisms
An example I like is Rel. Morphisms are binary relations. binary relations can be compared via inclusion (they form a partial order themselves).
Topos
https://twitter.com/johncarlosbaez/status/1461346819963686920?s=20 John Baez
Use functions instead of a notion of element being member of set. functions can essentially select elements. functions into “truth value” set (indicator functions) are analogs of subsets.
https://jonsterling.notion.site/Topoiinsideandout7b0b86e39eeb43aeaee3c3af1dd91f2a jon sterling blog post
To say something is a topos is to say it is
 a category
 a special category with some special constructs
Weird notions of truth value are fun.
Processes
Time
Categorical Databases
schema is a finite category Data lives over it Mappings between schema describe
Sheaves
Profunctors
Optics
Logic
Rising Sea
A proof of A  B
is the basic “morphism”. A proof is a tree. We can perhaps annotated this sequent with free variables and unification variables in play (signature). This makes this morphism floating over some kind of variable set, something sheafy? Or a set floats of variables floats over the proof
 Objects are propositions.
 Cut is compose.
 Axiom is id.
 Existentials and universals as adjunctions Different proofs are different morphisms between the same objects.
Computational Category Theory
homotopy.io globular dicopy cartographer
From a programming perspective I think there are a couple contributions:

Category theory has a number of very intuitive looking graphical notations which nevertheless translate to very exact algebraic expressions. This is pretty dang nice. Category theory is a road to a very principled formulation of things that are already done in dataflow languages, tensor diagrams, and UML and things like that. These graphical languages are a reasonable user facing interface.

Empirically in mathematics, category theory has been found to be a flexible and “simple” language to describe and interconnect very disparate things. In programming terms that means that it is a plausible source of inspiration for consistent interfaces that can do the same.

Point free DSLs are really nice to work with as library/compiler writer. It makes optimizations way easier to recognize and implement correctly. Named references are phantom links attaching different points in a syntax tree and make it hard to manipulate them correctly. In slightly different (and vague) terms, category theory gives a methodology to turn abstract syntax graphs into trees by making all links very explicit.
I can’t comment much on what category theory contributes to mathematics, but I think there are similar points. I think category theory can make precise analogies between extremely disparate fields like logic, set theory, and linear algebra. I think categorical intuitions lead to clean definitions when specialized to some particular field. Combinatory categorical logic does make some aspects of logic less spooky.
Internal Language
What is this? Cody talks about this a lot.
Resources
 Bartosz Milewski
 Computational Category Theory

Maclane
 A Categorial Theory of Objects as Observed Processes

You know more about presheaves than you think sheaves are databases The underlying category is a schema. It maps up to FinSet. Processes are a timey database.

Sheaves in Geometry and logic

Seven Sketches in Compositionality: An Invitation to Applied Category Theory
 pattern matching: a sheaf theortic approach thesis
topology a category theory approach https://news.ycombinator.com/item?id=33601318