Definition df-lan 49862

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 49882). 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 49882 (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 49889) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 49890) 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 49863 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 49860	. 2 class Lan
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3440	. . . 4 class V
5	4, 4	cxp 5622	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1540	. . . . 5 class 𝑝
8		c1st 7931	. . . . 5 class 1st
9	7, 8	cfv 6492	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7932	. . . . . 6 class 2nd
12	7, 11	cfv 6492	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1540	. . . . . . 7 class 𝑐
16	10	cv 1540	. . . . . . 7 class 𝑑
17		cfunc 17778	. . . . . . 7 class Func
18	15, 16, 17	co 7358	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1540	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7358	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4586	. . . . . . . 8 class ⟨𝑑, 𝑒⟩
22	13	cv 1540	. . . . . . . 8 class 𝑓
23		cprcof 49628	. . . . . . . 8 class −∘F
24	21, 22, 23	co 7358	. . . . . . 7 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25	14	cv 1540	. . . . . . 7 class 𝑥
26		cfuc 17869	. . . . . . . . 9 class FuncCat
27	16, 19, 26	co 7358	. . . . . . . 8 class (𝑑 FuncCat 𝑒)
28	15, 19, 26	co 7358	. . . . . . . 8 class (𝑐 FuncCat 𝑒)
29		cup 49428	. . . . . . . 8 class UP
30	27, 28, 29	co 7358	. . . . . . 7 class ((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))
31	24, 25, 30	co 7358	. . . . . 6 class ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)
32	13, 14, 18, 20, 31	cmpo 7360	. . . . 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 3849	. . . 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 3849	. . 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 7360	. 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 1541	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  49864  reldmlan  49866  lanfval  49868  rellan  49878