Definition df-lan 50444

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 50464). 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 50464 (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 50471) and the second component is a natural transformation 𝐴:𝑋⟶𝐿𝐹 (lanrcl5 50472) 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 50445 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 50442	. 2 class Lan
2		vp	. . 3 setvar 𝑝
3		ve	. . 3 setvar 𝑒
4		cvv 3457	. . . 4 class V
5	4, 4	cxp 5661	. . 3 class (V × V)
6		vc	. . . 4 setvar 𝑐
7	2	cv 1569	. . . . 5 class 𝑝
8		c1st 7990	. . . . 5 class 1st
9	7, 8	cfv 6540	. . . 4 class (1st ‘𝑝)
10		vd	. . . . 5 setvar 𝑑
11		c2nd 7991	. . . . . 6 class 2nd
12	7, 11	cfv 6540	. . . . 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 17935	. . . . . . 7 class Func
18	15, 16, 17	co 7419	. . . . . 6 class (𝑐 Func 𝑑)
19	3	cv 1569	. . . . . . 7 class 𝑒
20	15, 19, 17	co 7419	. . . . . 6 class (𝑐 Func 𝑒)
21	16, 19	cop 4597	. . . . . . . 8 class ⟨𝑑, 𝑒⟩
22	13	cv 1569	. . . . . . . 8 class 𝑓
23		cprcof 50210	. . . . . . . 8 class −∘F
24	21, 22, 23	co 7419	. . . . . . 7 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25	14	cv 1569	. . . . . . 7 class 𝑥
26		cfuc 18026	. . . . . . . . 9 class FuncCat
27	16, 19, 26	co 7419	. . . . . . . 8 class (𝑑 FuncCat 𝑒)
28	15, 19, 26	co 7419	. . . . . . . 8 class (𝑐 FuncCat 𝑒)
29		cup 50010	. . . . . . . 8 class UP
30	27, 28, 29	co 7419	. . . . . . 7 class ((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))
31	24, 25, 30	co 7419	. . . . . 6 class ((⟨𝑑, 𝑒⟩ −∘F 𝑓)((𝑑 FuncCat 𝑒) UP (𝑐 FuncCat 𝑒))𝑥)
32	13, 14, 18, 20, 31	cmpo 7421	. . . . 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 3854	. . . 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 3854	. . 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 7421	. 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 used by:  lanfn  50446  reldmlan  50448  lanfval  50450  rellan  50460