TheHoTTGame/Trinitarianism

Trinitarianism

By the end of this arc we will have 'a place to do maths'. The 'types' that will populated this 'place' will have three interpretations:

• Proof theoretically, with types as propositions
• Type theoretically, with types as programs
• Category theoretically, with types as objects in a category

Terms and Types

Here are some things that we could like to have in a 'place to do maths'

• objects to reason about (like ℕ)
• recipes for making things inside objects (like + 1)
• propositions to reason with (with the data of proofs) (like _ = 0)

In proof theory, types are propositions and terms of a type are their proofs. In type theory, types are programs and terms are algorithms. In category theory, types are objects and terms are generalised elements.

Non-dependent Types

• false / empty / initial object
• true / unit / terminal object
• or / sum / coproduct
• and / pairs / product
• implication / functions / internal hom

Dependent Types

• predicate / type family / over category
• substitution / substitution / pullback
• existence / Σ type / left adjoint to pullback
• for all / Π type / right adjoint to pullback

Question: how do we talk about equality?