Commit 2024-12-13 19:15 5aaaf8b3
View on Github →feat(CategoryTheory/Enriched): Add eHomCongr and related lemmas (#19932)
Recall that when C
is both a category and a V
-enriched category, we say it is an EnrichedOrdinaryCategory
if it comes equipped with an equivalence between morphisms X ⟶ Y
in C
and morphisms 𝟙_ V ⟶ (X ⟶[V] Y)
in V
that preserves identities and composition in C
.
In such a V
-enriched ordinary category C
, isomorphisms in C
induce isomorphisms between hom-objects in V
. We define this isomorphism, CategoryTheory.Iso.eHomCongr
, and prove that it behaves nicely with respect to composition in C
.
The lemmas here parallel those for unenriched categories in Mathlib/CategoryTheory/HomCongr.lean
and those for sorts in Mathlib/Logic/Equiv/Defs.lean
(cf. Equiv.arrowCongr
). Note, however, that they construct equivalences between Type
s and Sort
s, respectively, while in this PR we construct isomorphisms between objects in V
.