lean2/hott/hit/hit.md

hit

Declaration and theorems of higher inductive types in Lean. We take two higher inductive types (hits) as primitive notions in Lean. We define all other hits in terms of these two hits. The hits which are primitive are n-truncation and quotients. These are defined in init.hit and they have definitional computation rules on the point constructors.

Here we find hits related to the basic structure theory of HoTT. The hits related to homotopy theory are defined in homotopy.

Files in this folder:

• quotient: quotients, primitive
• trunc: truncation, primitive
• colimit: Colimits of arbitrary diagrams and sequential colimits, defined using quotients
• pushout: Pushouts, defined using quotients
• coeq: Co-equalizers, defined using quotients
• set_quotient: Set-quotients, defined using quotients and set-truncation
• prop_trunc: The construction of the propositional truncation from quotients.

The following hits have also 2-constructors. They are defined only using quotients.

• two_quotient: Quotients where you can also specify 2-paths. These are used for all hits which have 2-constructors, and they are almost fully general for such hits, as long as they are nonrecursive
• refl_quotient: Quotients where you can also specify 2-paths
• groupoid_quotient: The realization or quotient of a groupoid.