Connect type categories with comprehension categories #170
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…t_disp to disp_codomain
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| apply identity. | ||
| Defined. | ||
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| Definition typecat_idtoiso_dpr |
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This definition does not seem to be needed.
| Context {C : precategory}. | ||
| Context (TC : typecat_obj_ext_structure C). | ||
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| Definition typecat_comp_ext_compare |
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Maybe a different definition, that returned an isomorphism instead of a morphism, would be more useful.
| apply (isofhleveltotal2 2). | ||
| + apply homset_property. | ||
| + intro. apply isasetaprop. apply homset_property. | ||
| Defined. |
| : typecat_iso_triangle _ A B ≃ @iso_disp C (typecat_disp TC) _ _ (identity_iso Γ) A B | ||
| := (_,, typecat_is_triangle_idtoiso_fiber_disp_isweq A B). | ||
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| Definition typecat_disp_is_disp_univalent |
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This should work the other way round as well: if the displayed category is univalent, then type_cat_idtoiso_triangle should be an equivalence pointwise.
| Context (TC : typecat_structure C). | ||
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| (* NOTE: copied with slight modifications from https://github.com/UniMath/TypeTheory/blob/ad54ca1dad822e9c71acf35c27d0a39983269462/TypeTheory/Displayed_Cats/DisplayedCatFromCwDM.v#L114-L143 *) | ||
| Definition pullback_is_cartesian |
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Does this lemma need the full typecat_structure? It looks like it only uses the object-extension structure.
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