Notions of Structure

Structures

Quoting the Homotopy Type Theory Book (with slight changes for notational consistency):

9.8 The structure identity principle

The structure identity principle is an informal principle that expresses that isomorphic structures are identical. We aim to prove a general abstract result which can be applied to a wide family of notions of structure, where structures may be many-sorted or even dependently-sorted, in-finitary, or even higher order. The simplest kind of single-sorted structure consists of a type with no additional structure. The univalence axiom expresses the structure identity principle for that notion of structure in a strong form: for types A, B, the canonical function (A = B) → (A ≃ B) is an equivalence. We start with a precategory X. In our application to single-sorted first order structures, X will be the category of U-small sets, where U is a univalent type universe.

Notion of Structure

Definition: A notion of structure (P,H) over X consists of the following. We use X₀ to denote the objects of X, and homₓ(x, y) to denote the morphisms morphism X x y of X. (i) A type family P : X₀ → Type. For each x : X₀ the elements of P x are called (P, H)-structures on x. (ii) For x y : X₀ and α : P x, β : P y, to each f : homₓ(x, y) a mere proposition H_{αβ}(f). If H_{αβ}(f) is true, we say that f is a (P, H)-homomorphism from α to β. (iii) For x : X₀ and α : P x, we have H_{αα}(1ₓ). (iv) For x y z : X₀ and α : P x, β : P y, γ : P z, if f : homₓ(x, y), we have H_{αβ}(f)→ H_{βγ}(g) → H_{αγ}(g ∘ f). Note: We rearrange some parameters to H to ease Coq's unification engine and typeclass machinery.

It would be nice to make this a class, but we can't:

    Existing Class is_structure_homomorphism.

gives

    Toplevel input, characters 0-41:
    Anomaly: Mismatched instance and context when building universe substitution.
    Please report.

When we move to polyproj, it won't anymore.

When (P, H) is a notion of structure, for α β : P x we define (α ≤ₓ β) := H_{αβ}(1ₓ).

By (iii) and (iv), this is a preorder with P x its type of objects.

Being a structure homomorphism is a preorder

Standard notion of structure

We say that (P, H) is a standard notion of structure if this preorder is in fact a partial order, for all x : X. A partial order is an antisymmetric preorder, i.e., we must have a ≤ b ≤ a → a = b.

Note that for a standard notion of structure, each type P x must actually be a set.

Precategory of structures

We now define, for any notion of structure (P, H), a precategory of (P, H)-structures, A = Str_{(P, H)}(X).

• The type of objects of A is the type A₀ := ∑ₓ P x. If a ≡ (x; α), we may write |a| := x.
• For (x; α) : A₀ and (y; β) : A₀, we define
  hom_A((x; α), (y; β)) := { f : x → y | H_{αβ}(f) }.

The composition and identities are inherited from X; conditions (iii) and (iv) ensure that these lift to A.

We use Records because they are faster than sigma types.