Definition df-lan 50405

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 50425). 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 50425 (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 50432) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 50433) 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 50406 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 50403	. 2 class Lan
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3455	. . . 4 class V
5	4, 4	cxp 5659	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1569	. . . . 5 class 𝑝
8		c1st 7980	. . . . 5 class 1st
9	7, 8	cfv 6536	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7981	. . . . . 6 class 2nd
12	7, 11	cfv 6536	. . . . 5 class (2nd ‘𝑝)
13		vf	. . . . . 6 setvar 𝑓
14		vx	. . . . . 6 setvar 𝑥
15	6	cv 1569	. . . . . . 7 class 𝑐
16	10	cv 1569	. . . . . . 7 class 𝑑
17		cfunc 17906	. . . . . . 7 class Func
18	15, 16, 17	co 7410	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1569	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7410	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4595	. . . . . . . 8 class ⟨𝑑, 𝑒⟩
22	13	cv 1569	. . . . . . . 8 class 𝑓
23		cprcof 50171	. . . . . . . 8 class −∘F
24	21, 22, 23	co 7410	. . . . . . 7 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25	14	cv 1569	. . . . . . 7 class 𝑥
26		cfuc 17997	. . . . . . . . 9 class FuncCat
27	16, 19, 26	co 7410	. . . . . . . 8 class (𝑑 FuncCat 𝑒)
28	15, 19, 26	co 7410	. . . . . . . 8 class (𝑐 FuncCat 𝑒)
29		cup 49971	. . . . . . . 8 class UP
30	27, 28, 29	co 7410	. . . . . . 7 class ((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))
31	24, 25, 30	co 7410	. . . . . 6 class ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)
32	13, 14, 18, 20, 31	cmpo 7412	. . . . 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 3853	. . . 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 3853	. . 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 7412	. 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 1570	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  50407  reldmlan  50409  lanfval  50411  rellan  50421