1.2 KiB
1.2 KiB
Trinitarianism
By the end of this arc we will have 'a place to do maths'. The following features will have three interpretations:
- Proof theoretic, with types as propositions
- Type theoretic, with types as programs
- Category theoretic, 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?