TheHoTTGame/Trinitarianism

Trinitarianism

By the end of this arc we will (almost) 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 (E.g. ℕ)
• recipes for making things inside objects (E.g. n + m for n and m in naturals.)
• propositions to reason with (E.g. n = 0 for n in naturals.)

In proof theory, types are propositions and terms of a type are their proofs. In type theory, types are programs / constructions and terms are algorithms / recipes. 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 / bundle
• substitution / substitution / pullback (of bundles)
• existence / Σ type / total space of bundles
• for all / Π type / space of sections of bundles

Something doesn't feel the Same

There will be one thing missing from this 'place to do maths' and that is a notion of equality. This is where HoTT deviates from its predecessors, and is the theme of the next arc.