Definition df-lan 49345

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⟩Lan𝐸)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 49363). 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 49363 (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 49369) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 49370) where 𝐿𝐹 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 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 49346 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 49343	. 2 class Lan
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3457	. . . 4 class V
5	4, 4	cxp 5650	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1538	. . . . 5 class 𝑝
8		c1st 7981	. . . . 5 class 1st
9	7, 8	cfv 6528	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7982	. . . . . 6 class 2nd
12	7, 11	cfv 6528	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1538	. . . . . . 7 class 𝑐
16	10	cv 1538	. . . . . . 7 class 𝑑
17		cfunc 17854	. . . . . . 7 class Func
18	15, 16, 17	co 7400	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1538	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7400	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4605	. . . . . . . 8 class ⟨𝑑, 𝑒⟩
22	13	cv 1538	. . . . . . . 8 class 𝑓
23		cprcof 49147	. . . . . . . 8 class −∘F
24	21, 22, 23	co 7400	. . . . . . 7 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25	14	cv 1538	. . . . . . 7 class 𝑥
26		cfuc 17945	. . . . . . . . 9 class FuncCat
27	16, 19, 26	co 7400	. . . . . . . 8 class (𝑑 FuncCat 𝑒)
28	15, 19, 26	co 7400	. . . . . . . 8 class (𝑐 FuncCat 𝑒)
29		cup 48974	. . . . . . . 8 class UP
30	27, 28, 29	co 7400	. . . . . . 7 class ((𝑑 FuncCat 𝑒)UP(𝑐 FuncCat 𝑒))
31	24, 25, 30	co 7400	. . . . . 6 class ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒)UP(𝑐 FuncCat 𝑒))𝑥)
32	13, 14, 18, 20, 31	cmpo 7402	. . . . 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 3872	. . . 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 3872	. . 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 7402	. 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 1539	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  49347  reldmlan  49349  lanfval  49351  rellan  49359