Definition df-lan 49602

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 49622). 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 49622 (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 49629) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 49630) where 𝐿𝐹 is the composed functor. Intuitively, the first component 𝐿 can be regarded as the result of an "inverse" of pre-composition; the source category of 𝑋:𝐶⟶𝐸 is "extended" along 𝐹:𝐶⟶𝐷.

The left Kan extension is a generalization of many categorical concepts such as colimit. In § 7 of Chapter X of Categories 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 49603 for the dual concept.

(Contributed by Zhi Wang, 3-Nov-2025.)

Assertion

Ref	Expression
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)))

Distinct variable group:   𝑐,𝑑,𝑒,𝑓,𝑝,𝑥

Detailed syntax breakdown of Definition df-lan

Step	Hyp	Ref	Expression
1		clan 49600	. 2 class Lan
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3436	. . . 4 class V
5	4, 4	cxp 5617	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1539	. . . . 5 class 𝑝
8		c1st 7922	. . . . 5 class 1st
9	7, 8	cfv 6482	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7923	. . . . . 6 class 2nd
12	7, 11	cfv 6482	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1539	. . . . . . 7 class 𝑐
16	10	cv 1539	. . . . . . 7 class 𝑑
17		cfunc 17761	. . . . . . 7 class Func
18	15, 16, 17	co 7349	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1539	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7349	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4583	. . . . . . . 8 class ⟨𝑑, 𝑒⟩
22	13	cv 1539	. . . . . . . 8 class 𝑓
23		cprcof 49368	. . . . . . . 8 class −∘F
24	21, 22, 23	co 7349	. . . . . . 7 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25	14	cv 1539	. . . . . . 7 class 𝑥
26		cfuc 17852	. . . . . . . . 9 class FuncCat
27	16, 19, 26	co 7349	. . . . . . . 8 class (𝑑 FuncCat 𝑒)
28	15, 19, 26	co 7349	. . . . . . . 8 class (𝑐 FuncCat 𝑒)
29		cup 49168	. . . . . . . 8 class UP
30	27, 28, 29	co 7349	. . . . . . 7 class ((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))
31	24, 25, 30	co 7349	. . . . . 6 class ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)
32	13, 14, 18, 20, 31	cmpo 7351	. . . . 5 class (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥))
33	10, 12, 32	csb 3851	. . . 4 class ⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥))
34	6, 9, 33	csb 3851	. . 3 class ⦋(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	2, 3, 5, 4, 34	cmpo 7351	. 2 class (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)))
36	1, 35	wceq 1540	1 wff 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)))

This definition is referenced by:  lanfn  49604  reldmlan  49606  lanfval  49608  rellan  49618