> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/450-category-theory/topics/terms.md).

# Category Theory :: Terms

* CT pre-reqs
  * axioms
    * identity
    * closure
    * associativity
    * commutativity
    * distributivity
    * invertability
    * idempotence
  * sets
    * cardinality
    * powerset
    * subset
    * Cartesian product
    * ordered pair
    * types of sets
      * the empty set
      * singleton set
      * unordered pair
      * 𝔹, ℕ, ℤ, ℚ, ℝ
  * relations
    * types of relations
      * empty relation
      * total relation
      * functional
      * serial
      * euclidean
    * relational axioms
      * reflexivity
      * symmetry
      * transitivity
      * trichotomy
      * (right-)unique
      * (left-)serial
    * relational closures
      * reflexive closure
      * symmetric closure
      * transitive closure
  * functions
    * sets wrt functions
      * domain
      * codomain
      * range
      * image
      * pre-image
    * properties of functions
      * injectivity
      * surjectivity
      * bijectivity
    * types
      * total
      * partial
  * abstract algebra
    * algebraic structure, algebra
      * carrier set
      * operation over the set
      * algebra axioms
    * algebraic structures
      * magma
      * semigroup
      * monoid
      * group
      * abelian group
      * ring
      * field
* CT elementary
  * category theory
  * category
  * object
  * morphism, arrow
    * identity arrow
    * composition of arrows
    * associativity of composition
  * commutative diagram
  * functor
  * types of objects
    * initial object
    * terminal object
  * types of morphism
    * isomorphism
    * homomorphism
    * epimorphism
    * monomorphism
  * types of categories
    * empty category
    * monoid
    * discrete category
    * Named categories
      * 𝗦𝗲𝘁
      * 𝗥𝗲𝗹
      * 𝗚𝗿
      * 𝗚𝗿𝗽
      * 𝗩𝗲𝗰𝘁
      * 𝗖𝗮𝘁
      * 𝗛𝗮𝘀𝗸
  * types of functors
    * endofunctor
    * identity functor
* categorical constructions
  * product
  * coproduct
* natural transformation
  * naturality
  * naturality square
* monad
  * Kleisli triple
* algebra
  * initial algebra
  * f-algebra