{-# OPTIONS --without-K --safe #-} -- 'Traditionally', meaning in nLab and in -- "Lectures on n-Categories and Cohomology" by Baez and Shulman -- https://arxiv.org/abs/math/0608420 -- (-2)-Categories are defined to be just a single value, with trivial Hom -- But that's hardly a definition of a class of things, it's a definition of -- a single structure! What we want is the definition of a class which turns -- out to be (essentially) unique. Rather like the reals are (essentially) the -- only ordered complete archimedean field. -- So we will take a -2-Category to be a full-fledge Category, but where -- 1. |Obj| is (Categorically) contractible -- 2. |Hom| is connected (all arrows are equal) -- Note that we don't need to say anything at all about ≈ module Categories.Minus2-Category where open import Level open import Categories.Category open import Data.Product using (Σ) import Categories.Morphism as M private variable o ℓ e : Level record -2-Category : Set (suc (o ⊔ ℓ ⊔ e)) where field cat : Category o ℓ e open Category cat public open M cat using (_≅_) field Obj-Contr : Σ Obj (λ x → (y : Obj) → x ≅ y) Hom-Conn : {x y : Obj} {f g : x ⇒ y} → f ≈ g