module Cat.Restriction.Total where

The Subcategory of Total MapsπŸ”—

Let C\mathcal{C} be a restriction category. We can construct a wide subcategory of C\mathcal{C} containing only the total maps of C\mathcal{C}, denoted Total(C)\mathbf{Total}(\mathcal{C}).

  Total-wide-subcat : Wide-subcat {C = C} β„“
  Total-wide-subcat .Wide-subcat.P = is-total
  Total-wide-subcat .Wide-subcat.P-prop = is-total-is-prop
  Total-wide-subcat .Wide-subcat.P-id = id-total
  Total-wide-subcat .Wide-subcat.P-∘ = total-∘

  Total-maps : Precategory o β„“
  Total-maps = Wide Total-wide-subcat

Furthermore, the injection from Total(C)\mathbf{Total}(\mathcal{C}) into C\mathcal{C} is pseudomonic, as all isomorphisms in C\mathcal{C} are total.

  Forget-total : Functor (Total-maps C↓) C
  Forget-total = Forget-wide-subcat

    : is-pseudomonic Forget-total
  Forget-total-pseudomonic =
    is-pseudomonic-Forget-wide-subcat invertible→total