CAT. THEORY Image by Gidon Pico from Pixabay = ? objects - PowerPoint PPT Presentation

CAT. THEORY Image by Gidon Pico from Pixabay

= ? objects morphisms

points in morphisms from

what is ? any set unique morphism terminal

points in morphisms from terminal

Abstraction category type theory theory

Universal Properties ℕ most general ℕ -algebra S1 most general S1-algebra Id(M; N) freely generated by refl n-truncation best n-type approximation

Categorical Aspects 1. connectives in type theory Π⊤⊥ Σ Path universal properties in some category

Π⊤ Σ ⊥ Path Π⊤ Σ ⊥ Path Π⊤ Σ ⊥ 2. all objects Path that look like a type theory

Comprehension categories Categories Contexua Π⊤ Σ ⊥ with families Categorie Path Categories Display map ith a  ributes categories

i n t e r p r e t a Π⊤ t i o Σ n ⊥ Path Π⊤ Σ ⊥ Path Π⊤ Σ ⊥ interpretations Path as morphisms

Π⊤ Σ ⊥ Path Π⊤ Σ ⊥ Path A type theory is the most general Π⊤ Σ ⊥ object that looks like Path a type theory

Π⊤ Σ ⊥ Path models of the ΣΠ⊤ Π⊤ Σ ⊥ ⊥ type theory Path Path Π⊤ Σ ⊥ Path (the most general model is the theory itself)

Π⊤ Σ ⊥ Path Π⊤ Σ ⊥ Path Normalization and other meta-theorems Π⊤ Σ ⊥ follow from the syntax Path being the most general

Further Readings Categorical Logic and Type Theory by Bart Jacobs Categorical Logic by Andrew Pi  s