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Mathlib.CategoryTheory.Join.Final

(Co)Finality of the inclusions in joins of categories #

This file records the fact that inclLeft C D : C ⥤ C ⋆ D is initial if C is connected. Dually, inclRight : C ⥤ C ⋆ D is final if D is connected.

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def CategoryTheory.Join.costructuredArrowEquiv (C : Type u_1) (D : Type u_2) [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] (d : D) :
CostructuredArrow (inclLeft C D) (right d) ≌ C

The category of Join.inclLeft C D-costructured arrows with target right d is equivalent to C.

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Instances For
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    def CategoryTheory.Join.structuredArrowEquiv (C : Type u_1) (D : Type u_2) [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] (c : C) :
    StructuredArrow (left c) (inclRight C D) ≌ D

    The category of Join.inclRight C D-structured arrows with source left c is equivalent to D.

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    Instances For
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      instance CategoryTheory.Join.instInitialInclLeftOfIsConnected (C : Type u_1) (D : Type u_2) [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [IsConnected C] :
      (inclLeft C D).Initial
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      instance CategoryTheory.Join.instFinalInclRightOfIsConnected (C : Type u_1) (D : Type u_2) [Category.{v_1, u_1} C] [Category.{v_2, u_2} D] [IsConnected D] :
      (inclRight C D).Final