Description: Definition of the (local) left Kan extension. Given a functor F : C --> D and a functor X : C --> E , the set ( F ( <. C , D >. Lan E ) X ) consists of left Kan extensions of X along F , which are universal pairs from X to the pre-composition functor given by F ( lanval2 ). See also § 3 of Chapter X in p. 240 of Mac Lane, Saunders, Categories for the Working Mathematician, 2nd Edition, Springer Science+Business Media, New York, (1998) [QA169.M33 1998]; available at https://math.mit.edu/~hrm/palestine/maclane-categories.pdf (retrieved 3 Nov 2025).
A left Kan extension is in the form of <. L , A >. where the first component is a functor L : D --> E ( lanrcl4 ) and the second component is a natural transformation A : X --> L F ( lanrcl5 ) where L F is the composed functor. Intuitively, the first component L can be regarded as the result of a "inverse" of pre-composition; the source category is "extended" along C --> D .
The left Kan extension is a generalization of many categorical concepts such as colimit. In § 7 of Chapter X ofCategories for the Working Mathematician, it is concluded that "the notion of Kan extensions subsumes all the other fundamental concepts of category theory".
This definition was chosen over the other version in the commented out section due to its better reverse closure property.
See df-ran for the dual concept.
(Contributed by Zhi Wang, 3-Nov-2025)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | df-lan | ⊢ Lan = ( 𝑝 ∈ ( V × V ) , 𝑒 ∈ V ↦ ⦋ ( 1st ‘ 𝑝 ) / 𝑐 ⦌ ⦋ ( 2nd ‘ 𝑝 ) / 𝑑 ⦌ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( 𝑐 Func 𝑒 ) ↦ ( ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) 𝑥 ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 0 | clan | ⊢ Lan | |
| 1 | vp | ⊢ 𝑝 | |
| 2 | cvv | ⊢ V | |
| 3 | 2 2 | cxp | ⊢ ( V × V ) |
| 4 | ve | ⊢ 𝑒 | |
| 5 | c1st | ⊢ 1st | |
| 6 | 1 | cv | ⊢ 𝑝 |
| 7 | 6 5 | cfv | ⊢ ( 1st ‘ 𝑝 ) |
| 8 | vc | ⊢ 𝑐 | |
| 9 | c2nd | ⊢ 2nd | |
| 10 | 6 9 | cfv | ⊢ ( 2nd ‘ 𝑝 ) |
| 11 | vd | ⊢ 𝑑 | |
| 12 | vf | ⊢ 𝑓 | |
| 13 | 8 | cv | ⊢ 𝑐 |
| 14 | cfunc | ⊢ Func | |
| 15 | 11 | cv | ⊢ 𝑑 |
| 16 | 13 15 14 | co | ⊢ ( 𝑐 Func 𝑑 ) |
| 17 | vx | ⊢ 𝑥 | |
| 18 | 4 | cv | ⊢ 𝑒 |
| 19 | 13 18 14 | co | ⊢ ( 𝑐 Func 𝑒 ) |
| 20 | 15 18 | cop | ⊢ ⟨ 𝑑 , 𝑒 ⟩ |
| 21 | cprcof | ⊢ −∘F | |
| 22 | 12 | cv | ⊢ 𝑓 |
| 23 | 20 22 21 | co | ⊢ ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) |
| 24 | cfuc | ⊢ FuncCat | |
| 25 | 15 18 24 | co | ⊢ ( 𝑑 FuncCat 𝑒 ) |
| 26 | cup | ⊢ UP | |
| 27 | 13 18 24 | co | ⊢ ( 𝑐 FuncCat 𝑒 ) |
| 28 | 25 27 26 | co | ⊢ ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) |
| 29 | 17 | cv | ⊢ 𝑥 |
| 30 | 23 29 28 | co | ⊢ ( ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) 𝑥 ) |
| 31 | 12 17 16 19 30 | cmpo | ⊢ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( 𝑐 Func 𝑒 ) ↦ ( ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) 𝑥 ) ) |
| 32 | 11 10 31 | csb | ⊢ ⦋ ( 2nd ‘ 𝑝 ) / 𝑑 ⦌ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( 𝑐 Func 𝑒 ) ↦ ( ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) 𝑥 ) ) |
| 33 | 8 7 32 | csb | ⊢ ⦋ ( 1st ‘ 𝑝 ) / 𝑐 ⦌ ⦋ ( 2nd ‘ 𝑝 ) / 𝑑 ⦌ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( 𝑐 Func 𝑒 ) ↦ ( ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) 𝑥 ) ) |
| 34 | 1 4 3 2 33 | cmpo | ⊢ ( 𝑝 ∈ ( V × V ) , 𝑒 ∈ V ↦ ⦋ ( 1st ‘ 𝑝 ) / 𝑐 ⦌ ⦋ ( 2nd ‘ 𝑝 ) / 𝑑 ⦌ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( 𝑐 Func 𝑒 ) ↦ ( ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) 𝑥 ) ) ) |
| 35 | 0 34 | wceq | ⊢ Lan = ( 𝑝 ∈ ( V × V ) , 𝑒 ∈ V ↦ ⦋ ( 1st ‘ 𝑝 ) / 𝑐 ⦌ ⦋ ( 2nd ‘ 𝑝 ) / 𝑑 ⦌ ( 𝑓 ∈ ( 𝑐 Func 𝑑 ) , 𝑥 ∈ ( 𝑐 Func 𝑒 ) ↦ ( ( ⟨ 𝑑 , 𝑒 ⟩ −∘F 𝑓 ) ( ( 𝑑 FuncCat 𝑒 ) UP ( 𝑐 FuncCat 𝑒 ) ) 𝑥 ) ) ) |