Models of signatures

Content created by Fernando Chu, Fredrik Bakke and Louis Wasserman.

Created on 2023-03-20.
Last modified on 2025-05-14.

module universal-algebra.models-of-signatures where
Imports 
open import foundation.dependent-pair-types
open import foundation.sets
open import foundation.universe-levels

open import lists.tuples

open import universal-algebra.signatures

Idea

A model of a signature Sig in a type A is a dependent function that assigns to each function symbol f of arity n and n-tuple of elements of A an element of A.

Definitions

Models

module _
  {l1 : Level} (Sg : signature l1)
  where

  is-model : {l2 : Level} → UU l2 → UU (l1 ⊔ l2)
  is-model X =
    ( f : operation-signature Sg) →
    ( tuple X (arity-operation-signature Sg f) → X)

  is-model-signature : {l2 : Level} → (Set l2) → UU (l1 ⊔ l2)
  is-model-signature X = is-model (type-Set X)

  Model-Signature : (l2 : Level) → UU (l1 ⊔ lsuc l2)
  Model-Signature l2 = Σ (Set l2) (λ X → is-model-signature X)

  set-Model-Signature : {l2 : Level} → Model-Signature l2 → Set l2
  set-Model-Signature M = pr1 M

  is-model-set-Model-Signature :
    { l2 : Level} →
    ( M : Model-Signature l2) →
    is-model-signature (set-Model-Signature M)
  is-model-set-Model-Signature M = pr2 M

  type-Model-Signature : {l2 : Level} → Model-Signature l2 → UU l2
  type-Model-Signature M = pr1 (set-Model-Signature M)

  is-set-type-Model-Signature :
    { l2 : Level} →
    ( M : Model-Signature l2) →
    is-set (type-Model-Signature M)
  is-set-type-Model-Signature M = pr2 (set-Model-Signature M)

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