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Fix an identifier (UniMath#1830)
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nmvdw authored Jan 26, 2024
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`B` form a set.
As an intermediate construction, we first show that every bicategory `B` for which
the objects form a 1-type, gives rise to a double category ([bicat_to_double_bicat])
the objects form a 1-type, gives rise to a double category ([bicat_to_double_cat])
without any requirement on univalence or the objects/horizontal morphisms forming
a set. Afterwards we instantiate this to setbicategories ([setbicat_to_pseudo_double_bicat]).
a set. Afterwards we instantiate this to setbicategories ([setbicat_to_pseudo_double_cat]).
Such a construction would have limited usability for univalent bicategories. If the
objects in a univalent bicategory forms a 1-type, then there can be at most one
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Qed.

(** * 7. The double category corresponding to a bicategory *)
Definition bicat_to_double_bicat
Definition bicat_to_double_cat
: double_cat.
Proof.
use make_double_cat.
Expand All @@ -748,12 +748,12 @@ Section BicatToDoubleCat.
End BicatToDoubleCat.

(** * 8. The pseudo double setcategory corresponding to a bicategory *)
Definition bisetcat_to_pseudo_double_bicat
Definition bisetcat_to_pseudo_double_cat
(B : bisetcat)
: pseudo_double_setcat.
Proof.
use make_pseudo_double_setcat.
- refine (bicat_to_double_bicat B _).
- refine (bicat_to_double_cat B _).
use hlevelntosn.
exact (globally_strict_bisetcat B).
- split.
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