Definition df-ran 49603

Description: Definition of the (local) right Kan extension. Given a functor 𝐹:𝐶⟶𝐷 and a functor 𝑋:𝐶⟶𝐸, the set (𝐹(⟨𝐶, 𝐷⟩ Ran 𝐸)𝑋) consists of right Kan extensions of 𝑋 along 𝐹, which are universal pairs from the pre-composition functor given by 𝐹 to 𝑋 (ranval2 49625). The definition in § 3 of Chapter X in p. 236 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 49625 (retrieved 3 Nov 2025).

A right Kan extension is in the form of ⟨𝐿, 𝐴⟩ where the first component is a functor 𝐿:𝐷⟶𝐸 (ranrcl4 49634) and the second component is a natural transformation 𝐴:𝐿𝐹⟶𝑋 (ranrcl5 49635) 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 right Kan extension is a generalization of many categorical concepts such as limit. 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-lan 49602 for the dual concept.

(Contributed by Zhi Wang, 4-Nov-2025.)

Assertion

Ref	Expression
df-ran	⊢ Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))

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

Detailed syntax breakdown of Definition df-ran

Step	Hyp	Ref	Expression
1		cran 49601	. 2 class Ran
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	. . . . . . . . 9 class ⟨𝑑, 𝑒⟩
22	13	cv 1539	. . . . . . . . 9 class 𝑓
23		cprcof 49368	. . . . . . . . 9 class −∘F
24	21, 22, 23	co 7349	. . . . . . . 8 class (⟨𝑑, 𝑒⟩ −∘F 𝑓)
25		coppf 49117	. . . . . . . 8 class oppFunc
26	24, 25	cfv 6482	. . . . . . 7 class ( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))
27	14	cv 1539	. . . . . . 7 class 𝑥
28		cfuc 17852	. . . . . . . . . 10 class FuncCat
29	16, 19, 28	co 7349	. . . . . . . . 9 class (𝑑 FuncCat 𝑒)
30		coppc 17617	. . . . . . . . 9 class oppCat
31	29, 30	cfv 6482	. . . . . . . 8 class (oppCat‘(𝑑 FuncCat 𝑒))
32	15, 19, 28	co 7349	. . . . . . . . 9 class (𝑐 FuncCat 𝑒)
33	32, 30	cfv 6482	. . . . . . . 8 class (oppCat‘(𝑐 FuncCat 𝑒))
34		cup 49168	. . . . . . . 8 class UP
35	31, 33, 34	co 7349	. . . . . . 7 class ((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))
36	26, 27, 35	co 7349	. . . . . 6 class (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)
37	13, 14, 18, 20, 36	cmpo 7351	. . . . 5 class (𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥))
38	10, 12, 37	csb 3851	. . . 4 class ⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥))
39	6, 9, 38	csb 3851	. . 3 class ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥))
40	2, 3, 5, 4, 39	cmpo 7351	. 2 class (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))
41	1, 40	wceq 1540	1 wff Ran = (𝑝 ∈ (V × V), 𝑒 ∈ V ↦ ⦋(1st ‘𝑝) / 𝑐⦌⦋(2nd ‘𝑝) / 𝑑⦌(𝑓 ∈ (𝑐 Func 𝑑), 𝑥 ∈ (𝑐 Func 𝑒) ↦ (( oppFunc ‘(⟨𝑑, 𝑒⟩ −∘F 𝑓))((oppCat‘(𝑑 FuncCat 𝑒)) UP (oppCat‘(𝑐 FuncCat 𝑒)))𝑥)))

This definition is referenced by:  ranfn  49605  reldmran  49607  ranfval  49609  relran  49619