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

Image by Gidon Pico from Pixabay

THEORY CAT.

• bjects

morphisms

?

=

points in morphisms from

what is ?

any set

unique morphism

terminal

points in morphisms from

terminal

Abstraction

category theory type theory

Universal Properties

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

Categorical Aspects

• 1. connectives in type theory

Π⊤⊥ Path Σ

universal properties in some category

Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ

• 2. all objects

that look like a type theory

Contexua Categorie Comprehension categories Display map categories Categories ith aributes Categories with families

Π⊤ ⊥ Path Σ

Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ

interpretations as morphisms

i n t e r p r e t a t i

• n

Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ

A type theory is the most general

• bject that looks like

a type theory

Π⊤ ⊥ Path ΣΠ⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ

models of the type theory

(the most general model is the theory itself)

Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ Π⊤ ⊥ Path Σ

Normalization and

• ther meta-theorems

follow from the syntax being the most general

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

Further Readings